Quantum Coding with Low-Depth Random Circuits

Quantum Coding with Low-Depth Random Circuits

Michael J. Gullans ,1,2 Stefan Krastanov,3,4 David A. Huse ,2 Liang Jiang ,5,6 and Steven T. Flammia61Joint Center for Quantum Information and Computer Science, NIST/University of Maryland,

College Park, Maryland 20742, USA2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

3John A. Paulson School of Engineering and Applied Sciences, Harvard University,Cambridge, Massachusetts 02138, USA

4Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139, USA

5Pritzker School of Molecular Engineering, The University of Chicago, Illinois 60637, USA6AWS Center for Quantum Computing, Pasadena, California 91125, USA

(Received 13 November 2020; revised 11 June 2021; accepted 12 July 2021; published 24 September 2021)

Random quantum circuits have played a central role in establishing the computational advantages ofnear-term quantum computers over their conventional counterparts. Here, we use ensembles of low-depthrandom circuits with local connectivity in D ≥ 1 spatial dimensions to generate quantum error-correctingcodes. For random stabilizer codes and the erasure channel, we find strong evidence that a depth OðlogNÞrandom circuit is necessary and sufficient to converge (with high probability) to zero failure probability forany finite amount below the optimal erasure threshold, set by the channel capacity, for any D. Previousresults on random circuits have only shown that OðN1=DÞ depth suffices or that Oðlog3 NÞ depth sufficesfor all-to-all connectivity (D → ∞). We then study the critical behavior of the erasure threshold in the so-called moderate deviation limit, where both the failure probability and the distance to the optimal thresholdconverge to zero with N. We find that the requisite depth scales like OðlogNÞ only for dimensions D ≥ 2

and that random circuits require Oð ffiffiffiffiN

p Þ depth for D ¼ 1. Finally, we introduce an “expurgation”algorithm that uses quantum measurements to remove logical operators that cause the code to fail byturning them into either additional stabilizers or into gauge operators in a subsystem code. With suchtargeted measurements, we can achieve sublogarithmic depth in D ≥ 2 spatial dimensions below capacitywithout increasing the maximum weight of the check operators. We find that for any rate beneath thecapacity, high-performing codes with thousands of logical qubits are achievable with depth 4–8 expurgatedrandom circuits in D ¼ 2 dimensions. These results indicate that finite-rate quantum codes are practicallyrelevant for near-term devices and may significantly reduce the resource requirements to achieve faulttolerance for near-term applications.

DOI: 10.1103/PhysRevX.11.031066 Subject Areas: Quantum Physics,Quantum Information,Statistical Physics

I. INTRODUCTION

Achieving reliable simulations of many-body quantumdynamics remains a central challenge across different areasof science. Quantum computers offer a natural computa-tional advantage for such problems in near-term devices, asexemplified by recent experiments on random circuitsampling [1]. However, despite remarkable advances inquantum control and measurement [1–10], many platforms

face daunting resource requirements when accounting forscalable quantum error correction [11–14]. On the otherhand, it is now understood that one can significantly lowerthe resource requirements for fault tolerance through, e.g.,hardware efficient encodings [15–17], accurate noise esti-mation [18–20], noise-bias-preserving gates [21–23], long-range interactions [24–26], and better choices of codes withassociated decoding algorithms [27–30]. These develop-ments suggest that fault tolerance with much lower over-head is possible in near-term devices [26].A common technique in classical error correction is to

study random codes, which often nearly saturate thebounds for the optimal codes [31–33]. Moreover, practical,near-optimal codes with efficient encoders and decodersare possible through random constructions of low-density

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

PHYSICAL REVIEW X 11, 031066 (2021)

2160-3308=21=11(3)=031066(23) 031066-1 Published by the American Physical Society

https://orcid.org/0000-0003-3974-2987

https://orcid.org/0000-0003-1008-5178

https://orcid.org/0000-0002-0000-9342

https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.11.031066&domain=pdf&date_stamp=2021-09-24

https://doi.org/10.1103/PhysRevX.11.031066




https://creativecommons.org/licenses/by/4.0/

https://creativecommons.org/licenses/by/4.0/

parity check (LDPC) codes [32,33]. In the quantum case,the decoding problem tends to be more difficult to solve(including for LDPC codes), but analogous random codingresults have been obtained for stabilizer codes, where thedecoding problem is similar to the classical case. Two-localrandom Clifford circuits with all-to-all connectivity havebeen shown to achieve an extensive code distance on Nqubits at a depth upper bounded byOðlog3NÞ [34,35]. Thisscaling is comparable to provably optimal constructions fortwo-designs from Clifford gates at depth OðlogNÞ withaccess to OðNÞ additional ancillae [36]. Spatial locality isoften an important constraint in quantum computingarchitectures. In D spatial dimensions, the depth for localcircuits to achieve an approximate two-design is upperbounded by OðN1=DÞ [37,38]. Such constructions are onlyrequired if the code needs to correct all errors up untilthreshold. Achieving optimal performance for local noisemodels requires fewer resources because the code onlyneeds to correct typical errors in the thermodynamic limit.In this paper, we develop the general theory of optimal

decoding with low-depth random encodings that includeboth unitaries and targeted measurements for one of thesimplest error models given by the erasure channel. Manyof the results apply for more general error channels, butoptimal recovery probabilities are easy to compute forerasure errors, making it a useful error model for bench-marking quantum codes [39,40]. We show that, in anyspatial dimension, random Clifford encodings of finite-ratecodes converge to zero failure probability below theoptimal erasure threshold, set by the channel capacity,for depths OðlogNÞ, thus improving on the random circuitbounds described above. We then introduce an “expurga-tion” algorithm to surpass this logarithmic barrier andachieve convergence at a sublogarithmic depth in D > 1dimensions. This method works by using quantum mea-surements to remove (expurgate) logical operators from thecode that have a high probability of failure until either asteady-state code is reached or target coding parameters areobtained. These low-quality logicals are either turned intoadditional stabilizers or gauge operators to form a sub-system code. This expurgation process monotonicallyincreases the code distance and recovery probability ofany stabilizer subsystem code. At a practical level, one canuse random coding and expurgation to generate high-performance, finite-rate codes for thousands of logicalqubits with depth 4–8 circuits in two dimensions.Our results also establish several connections between

quantum error-correction thresholds, random matrix theory(RMT), and statistical physics. Using a RMT ansatz, wedevelop a complete critical theory for optimal decoding oferasure errors for random stabilizer codes. We numericallybenchmark this ansatz to a high degree of precision in thecritical region of the erasure threshold. These scalingresults guide our numerical analysis of optimal decodingfor finite-depth encoders in finite-size systems. Focusing on

the critical scaling theory of random codes at low depths,we find that random Clifford circuits can achieve thecapacity of the erasure channel only at parametricallylarger depthOð ffiffiffiffi

Np Þ in 1D. InD > 1 dimensions, however,

random Clifford circuits retain the depth ≤ OðlogNÞscaling at capacity. The marginal dimension being 2D isconsistent with Imry-Ma-type arguments regarding therelevance of randomness in the error patterns at the optimalthreshold [41].We also analyze the case of Haar random circuit

encoders at a high depth greater OðNÞ, where optimaldecoding is likely exponentially hard. We find similarresults as for the high-depth Clifford encoders but withsmall quantitative differences that indicate Haar randomcodes are slightly more optimal than random stabilizercodes. Through an approximate mapping to an Ising model,we argue that the erasure threshold with local randomcircuits can be generally understood as a type of first-orderdomain-wall pinning phase transition.

A. Relation to previous work

In this section, we discuss the relation of our results tosome of the prior work on quantum error-correcting codesand random quantum circuits.

1. Quantum error-correcting codes

Starting in the early days of quantum error correction,a common strategy for proving fault tolerance was tostudy concatenated codes [42–44]. These codes reducedecoherence by successively encoding quantum informa-tion in nested chains of small codes. Unfortunately, thisapproach typically suffers from large space-time resourcecosts and low error thresholds [11,13,26]. A paradigmaticexample of a code that, in balance, requires minimalresources is the 2D surface code [45,46]. This topologicalcode saturates the capacity for the erasure channel at a zerocode rate on a square lattice [46], is provably fault tolerantunder more general noise models [46,47], and has highlyefficient decoding algorithms [27–30,39,40,46–51] as wellas a large variety of fault-tolerant strategies for implement-ing gates [12,14,52–58]. However, despite its remarkableproperties, the surface code requires a prohibitively largeoverhead in the number of physical qubits for applicationson near-term devices [14]. With issues of this nature inmind, it remains a central goal to develop more space-efficient, ideally finite-rate, codes that achieve similarlevels of performance to the surface code [26].Extending topological codes, or, more generally, low-

density parity check (LDPC) codes, to finite rate facesvarious theoretical obstructions in spatially local models[59]. Two routes to overcome this obstacle are to usesubsystem codes [60] or remove the constraint of geometriclocality while keeping the LDPC condition. In all-to-allcoupled systems, a large variety of finite-rate LDPC codeshave been developed by extending the surface code to

GULLANS, KRASTANOV, HUSE, JIANG, and FLAMMIA PHYS. REV. X 11, 031066 (2021)

031066-2

nonlocal geometries or adapting classical codes based onexpander graphs [24,25,61,62]. Furthermore, severalthreshold theorems have been proven for a large familyof these codes [26,63,64]. Maintaining all-to-all connec-tivity in the thermodynamic limit eventually runs intoprohibitive resource constraints, but these codes are appli-cable to near-term ion-trap quantum computers [65,66] andquantum networks [67]. Another interesting class of finite-rate codes that retain some locality structure, but are not ofthe LDPC type, are provided by holographic codes thatoriginally arose in the study of quantum gravity and theAdS/CFT correspondence [68,69]. The quasilocal codesconsidered here differ from these various classes of codesbecause their properties emerge from generic, local scram-bling dynamics instead of concatenation, topology,expander graphs, or hyperbolic geometry.

2. Random quantum circuits

The theoretical methods in this work draw from a varietyof recent results in random quantum circuits, which haveserved as a powerful tool to examine quantum many-bodydynamics in nonperturbative limits [70–79]. Notable exam-ples are their extensive applications to quantum gravity[70–72] and quantum chaos and equilibration [73–80].Despite the intricate structure of a particular circuit, whichforms the basis for average-case hardness of random circuitsampling [1,6,81–84], there is a notion of universality in thedynamics of extensive quantities such as the entanglement[74] or the distance of the code generated by the circuit[35]. Such notions of universality build on the randommatrix theory or eigenstate thermalization approach todescribing late-time equilibration in closed quantum sys-tems [85–88].More specifically, our results have direct relevance to a

recently discovered phase transition that arises in moni-tored random circuits, where unitary gates are interspersedwith random projective measurements [89,90]. Thesemodels have attracted interest in condensed matter theorydue to the potential connections to chaos, thermalization,conformal field theory, and the many-body localizationphase transition [91–116]. In the context of quantuminformation, their study has led to novel insights intoemergent quantum error correction [93,94,117–119], aswell as the sampling complexity of constant depth circuitsin 2D [120]. Because of the repeated rounds of measure-ments acting on a code-space density matrix, the dynamicsduring our expurgation algorithm display a similar phe-nomenology to the “purification” dynamics of a mixed statein the unitary-measurement models [93,103,107,119];however, there are several important differences in thepresent case due to the nonrandom, targeted choice ofmeasurements. Furthermore, since we show that logarith-mic depth random circuits are sufficient to reliably encodequantum information, our results may provide guidance forrigorous existence proofs of the volume-law phase in some

models. They may also help guide efforts in developingfault-tolerant strategies for monitored random circuits thatincorporate feedback.

B. Structure of the paper

The paper is organized as follows: In Sec. II, we outlineour theoretical approach for studying random quantumcodes. We then summarize our main results and theoreticalmethods. In Sec. III, we provide some background on thebasic concepts and terminology used to describe quantumerror-correction thresholds. In Sec. IV, we present the RMTsolution to the erasure threshold for random stabilizercodes. In Sec. V, we present our results on the behaviorof low-depth random circuit encoders for the erasurechannel. In Sec. VI, we present our expurgation algorithmto surpass the depthOðlogNÞ barrier inD > 1 dimensions.In Sec. VII, we present an analysis of the erasure thresholdfor general Haar random codes. In Sec. VIII, we describe anapproximate mapping of the erasure threshold to a first-order domain-wall pinning transition that occurs in theordered phase of the Ising model. We provide furtherdiscussions and present our conclusions in Sec. IX.We remark that the arguments in the paper use a

combination of rigorous proofs, large-scale numerics,conjectures, and some occasional heuristics. To test ourideas as strongly as possible with this approach, we analyzethe problem from multiple perspectives and systematicallycompare our results across different spatial dimensions.What emerges from this analysis is a consistent frameworkto describe quantum coding with local random circuits.

II. SUMMARY

In this section, we outline the theoretical approach takenin this work and summarize our main results.

A. Theoretical approach

This paper is focused on developing a theory of optimaldecoding for finite-rate codes generated by random circuits.To approach this problem, we directly investigate theprobability of successful recovery PðRÞ of the encodingand decoding scheme for the specific error model oferasures. This type of observable is complementary toother performance metrics that are agnostic to the errormodel, for example, the code distance. One advantage ofstudying recovery or failure probabilities is that it allows usto obtain a more detailed understanding of the codeperformance near the optimal threshold. For coding belowthe optimal threshold, we find that focusing on thisobservable often suggests methods to tailor the codes tothe detailed properties of the noise, as we explore with ourexpurgation algorithm in Sec. VI.The qualitative behavior of the optimal (minimal) failure

probability PðFÞ ¼ 1 − PðRÞ is shown in Fig. 1(a). Here, eis a parameter that characterizes the strength of noise in a

QUANTUM CODING WITH LOW-DEPTH RANDOM CIRCUITS PHYS. REV. X 11, 031066 (2021)

031066-3

given error model (or class of error models), and we assumethat the implemented code is optimal for this error model.Below threshold, the failure probability converges to zeroin the limit of large N. Past a critical error rate ec (set by thechannel capacity limit for the optimal code), for e > ec, thefailure probability instead converges to 1 in the large-Nlimit. This discontinuous behavior in the large-N limit ischaracteristic of a phase transition. Motivated by resultsfrom classical error correction [32,33], we assume that thefailure probability for an optimal code for large N is wellapproximated by the average behavior of a high-depthrandom stabilizer code under optimal decoding [121]. Oneprimary question that we address is what minimal depth ofa random encoding Clifford circuit is needed for large butfinite N to achieve near-optimal failure probability for thespecific case of erasure errors [see Fig. 1(b)].More specifically, in finite-size systems, the failure

probability for the optimal code will generically be afunction of both the error rate e and the number of (qu)bits N per code block. However, in the thermodynamiclimit of large N, the failure probability will approach ascaling form in the vicinity of the critical error rate ec [seeshaded region in Fig. 1(a)]

− logPðFÞ ¼ Nafopt½ðe − ecÞN1=b�; ð1Þ

where a and b are critical exponents and the correctionsare assumed to be subleading in powers of 1=N. Wetake the logarithm of the failure probability as it generallyscales like free energy, e.g., in the surface code [47]. Incoding theory, properties of the scaling functions for theoptimal codes fopt have been extensively studied underoptimal decoding of Markovian error channels (e.g., seeRef. [122]). The finite-size scaling behavior is important

because it determines the rate of convergence to the idealbehavior below threshold. The underlying idea of this workis to use the scaling properties of the optimal codes for agiven error channel as an ideal performance benchmark.We effectively define a code as optimal if it achievescapacity at threshold and its threshold behavior lies in thesame universality class as the truly optimal codes for thiserror channel.Of course, finding explicit and efficiently implementable

representations for encoding and decoding maps of optimalcodes is generally a difficult problem [123]. To approxi-mate this paradigm in a setting that allows for moretheoretical progress and potential practical implicationsfor quantum computing, we relax the benchmark criticalbehavior from that of optimal codes to the average behaviorof high-depth random stabilizer codes. As mentionedabove, random codes typically achieve similar levels ofperformance as optimal codes. In quantum error correction,even random stabilizer codes are often sufficient. Wepresent numerical evidence on small systems that theHaar random code transition is in the same universalityclass as the random stabilizer code transition. However, wealso see small quantitative differences between the scalingfunctions for the two cases, with slightly more optimalperformance for the Haar codes. Random stabilizer codesare thus better classified as “near-optimal” codes for theerasure channel, which is similar to a well-known result forthe depolarizing channel [124,125].

B. Main results

As discussed in the Introduction, our main results centeraround the resource requirements (in terms of encodingcircuit depth) to achieve zero failure probability or approachcapacity for finite-rate codes generated by random circuits.In particular, we study stabilizer codes generated by two-local random Clifford circuits on hypercubic lattices in Dspatial dimensions or on all-to-all coupled networks. Thebasic setup is illustrated in Fig. 1(b). In this example, everyother qubit is mapped to an encoded or “logical” qubit at acode rate ofR ¼ 1=2, and the random circuit is implementedin 1D. We provide a summary of the scalings found in thiswork in Table I.The error model is taken to be an erasure model where

eN sites of an N-qubit system are randomly selected andtraced out of the system, with those sites heralded to thedecoder but unknown to the encoder. The failure proba-bility for the more physically relevant case of independent,identically distributed (IID) erasures at each site withprobability e can be determined from the failure probabilityfor the fixed-fraction erasure model, which is why wemostly focus on the latter. [126] For the random stabilizercodes, we show that the transition is, in a certain sense, firstorder since for e < ec, the logarithm of the failure prob-ability is proportional to −ðec − eÞN in the limit of large N.If we interpret this as free energy, it is extensive, and its

(a)Above

threshold

Below threshold

Cri

tical

(b)

ErrorEncoding Decoding

Random unitary

Encoded qubit

FIG. 1. (a) Probability of decoding failure PðFÞ vs error rate ethrough an error-correction threshold for an optimal code. In thiswork, we probe the optimality of a given code ensemble bycomparing the location and universality class of the critical pointto random stabilizer codes. (b) Illustration of the models westudy. The encoding circuit is a low-depth random unitary circuit,and the error is an erasure of a fixed fraction eN of N qubits. Thedecoding proceeds via generalized measurements with outcomess and recovery operators Rs. We mostly focus on stabilizer codes,where optimal decoding of Pauli error channels like erasure errorsis possible with stabilizer syndrome measurements, followed bythe conditional application of single-site Clifford gates.


031066-4

density has a discontinuity in the first derivative withrespect to e at ec, as is the case for first-order phasetransitions. This first-order transition is rounded out forfinite N. This finite-size rounding is minimal if we take anerror model with erasures on a fixed fraction eN of sites.The finite-size rounding of the transition in the IID model ismuch stronger (by a factor of

ffiffiffiffiN

p).

We find that the best-known analytical bounds for theconvergence rate to a two-design strongly overestimate thecircuit depth d required for convergence of the failureprobability towards the high-depth ½d ¼ OðNÞ� limit. Mostnotably, in any D ≥ 2, at the critical point, the depthrequired scales as d ≤ OðlogNÞ, which is comparable tothe optimal depth for generating a two-design in systemswithout spatial locality constraints OðlogNÞ [36]. Even inD ¼ 1, we find that removing the randomness in erasurelocations by taking regularly spaced erasures leads to arequired depth to approach zero failure probability belowthe optimal threshold of OðlogNÞ. Spatial randomness inthe erasure locations seems to only be a relevant perturba-tion for the finite-size scaling behavior in D ¼ 1 and notfor D ≥ 2.To simplify the analysis, we fix the initial code rate at

precisely R ¼ 1=2 in most of our discussion and drop thisargument from the scaling functions. Also fixing the initialspatial arrangement of the logical qubits to be every othersite in the lattice, the failure probability has a four-parameter dependence

− logPðFÞ ¼ Fðe;D; d; NÞ: ð2Þ

We first consider the high-depth limit d ¼ OðNÞ of thefailure probability, which does not depend onD. Through aRMT ansatz, we obtain an asymptotic form for the failureprobability in the fixed fraction model that depends only onthe total number of erasures eN (an integer) relative to thethreshold number ecN (which does not have to be an

integer): limN→∞ limd→∞ Fðe; d;D;NÞ ¼ f∞½ðe − ecÞN�for e near ec. Here, ec ¼ ð1 − RÞ=2 coincides with thechannel capacity limit for the erasure threshold [127]. Forthis fixed-fraction erasure error model, the scaling functionf∞ðxnÞ is only well defined on a countably infinite set ofvalues in the thermodynamic limit. The RMT solutionpredicts a value for the critical failure probability PðFÞ ¼0.38968… that is independent of R for 0 < R < 1; forR ¼ 1=2, we verify this value numerically to a precisionof 10−4.To understand the scaling with depth, we first consider a

simple mean-field model for the below-threshold behaviorin which we break the system up into individual blocks ofsizeOðlogNÞ. A simple analysis of this model based on theresults of Ref. [37] shows that the convergence to the high-depth behavior of the failure probability inD dimensions istypically OðlogNÞ for random Clifford encodings, but itcan be made as low as depth O½ðlogNÞ1=D� through theoptimized encodings of Ref. [36]. In the latter case, there isa reduction in the effective rate of the code due to the use ofOðlogNÞ ancilla qubits per block in the encoding scheme.Using an Imry-Ma-type argument [41], we then argue

that the positional randomness of the erasures is irrelevantfor the finite-size scaling in D ≥ 2. As a result, weconjecture that the critical points for all D ≥ 2 have thesame leading-order scaling with depth as the below-thresh-old behavior predicted from the mean-field model,

− limN→∞

logPðFÞje¼ec;D≥2 ¼ fDcðd − A logNÞ:

Using this ansatz, we find a consistent scaling collapse inour numerics.We also study the scaling behavior with depth d in the

1D case (D ¼ 1), which has to be treated separately. Bystudying the convergence of the critical failure probabilityPðFÞje¼ec to the RMT prediction, we find numericalevidence for a leading-order scaling behavior of the form

− limN→∞

logPðFÞje¼ec;D¼1 ¼ f1cðd=ffiffiffiffiN

pÞ: ð3Þ

In contrast, below the critical error rate, we find that thefailure probability for d > OðlogNÞ exhibits exponentialdecay with the depth PðFÞje<ec;D¼1 ∼ e−d=AðeÞ for somefunction AðeÞ that diverges as ðec − eÞ−1 upon approachingec. This behavior leads to an overall OðlogNÞ depth forconvergence to zero failure probability below threshold butwith a rate that goes to zero at the optimal erasure threshold.We argue that the

ffiffiffiffiN

pscaling at ec has an intuitive

explanation as arising from the Poisson fluctuations inthe number of excess erasures in a given extensive region.After establishing these scaling results, we introduce our

expurgation algorithm based on measuring logical oper-ators in the system that are likely to lead to failures. Weprove that the code distance and recovery probability for

TABLE I. Random Clifford circuit encoding depths required toreach zero failure probability for finite-rate codes under erasureerrors in D dimensions. Here, ec −Oð1=NbÞ denotes codingarbitrarily close to the critical region of the optimal erasurethreshold in the thermodynamic limit. We find b ¼ 1 for thefixed-fraction erasure model and b ¼ 1=2 for the IID model.Here,D ¼ 2 is the marginal dimension for the relevance of spatialrandomness in the errors to the threshold behavior, which makesthe scaling at the optimal threshold difficult to reliably determinefrom numerics or Imry-Ma arguments. The last column shows theresults upon expurgation of bad logical operators using quantummeasurements (see Sec. VI).

D e < ec e¼ec−Oð1=NbÞ Expurgation e<ec

1 ð1=ec−eÞ logN N1=2 logN2 logN logN (conj.) ðlogNÞc, c < 1> 2 logN logN ðlogNÞc, c < 1


031066-5

Pauli error channels will monotonically increase with thisexpurgation strategy. We then numerically study the per-formance of the algorithm in 2D and all-to-all coupledsystems. In both cases, we see strong evidence that a giventarget failure probability can be achieved with a sub-log-Ndepth circuit.Finally, to test the generality of these results obtained for

stabilizer codes, we study the erasure threshold for Haarrandom circuits. We first study the high-depth limit usingsmall-scale numerics. We find consistent critical behaviorwith the random stabilizer code threshold but small quanti-tative differences in the scaling functions (as noted above).Using well-studied mappings of two-local Haar randomcircuits to statistical mechanics models [71,74,128], wedescribe an approximatemapping of the erasure threshold toa first-order pinning transition for domain walls that occursin the ordered phase of Dþ 1-dimensional Ising models.Such transitions display similar phenomenology to ournumerically observed results for random Clifford circuits.

III. PRELIMINARIES

In this section, we introduce the basic terminology andconcepts underlying quantum error-correcting codes, opti-mal decoding, and quantum error-correction thresholds. Wederive a formula used throughout the paper for the recoveryprobability of stabilizer subsystem codes under erasureerrors.

A. Optimal decoding

The general setup we consider follows the illustration inFig. 1(b). Information is first mapped into a nonlocal codespace; it is then subjected to local errors and decoded. In thetheory of fault tolerance, one needs to consider errors inboth the encoding and decoding steps; however, we do notaddress such issues here, and we assume both the encodingand decoding operations are implemented perfectly.In the quantum case, these three operations are typically

described using the language of quantum channels, whichare linear maps that are completely positive and tracepreserving [129]. Denoting the encoding, error, and decod-ing channels by E, N , and D, respectively, the centralobject of interest is the composite channel

D∘N ∘E: ð4Þ

Error correction can be done perfectly when this compositechannel acts as the identity on the allowed input statesD∘N ∘EðρÞ ¼ ρ or is unitarily equivalent to the identityD∘N ∘EðρÞ ¼ UρU† for a known unitary U.When this map is not exactly unitarily equivalent to the

identity, then it is convenient to use a fidelity metric toquantify its proximity to the identity. One natural fidelitymetric that we study in this work is the max-average statefidelity [129]

Favg ¼ maxD

Zdψhψ j½D∘N ∘Eðjψihψ jÞ�jψi; ð5Þ

where dψ is taken as a uniform measure over pure inputstates for E and the maximum is taken over all possibledecoding mapsD. This fidelity metric quantifies the degreeto which a randomly drawn codeword can be recoveredback to its initial state under optimal decoding.A fidelity metric closely related to this average state

fidelity is the entanglement fidelity, which measures thedegree to which the map preserves entanglement with areference system [130]. Given an initial density matrix ρSon the system S, we purify it to the state ρSR ¼ jΨSRihΨSRjby introducing entanglement with a reference system Rsuch that ρS ¼ TrR½ρSR�. Then, the entanglement fidelityunder optimal decoding is

FeðρSÞ ¼ maxD

hΨSRj½D∘N ∘EðjΨSRihΨSRjÞ�jΨSRi; ð6Þ

where the maps act as the identity on the reference systemand FeðρSÞ is independent of the choice of purification.Conveniently, the max-average fidelity is equivalent to themax-entanglement fidelity of the completely mixed statethrough the formula Favg ¼ ½qFentðI=qÞ þ 1�=ðqþ 1Þ,where q is the dimension of the input space [131,132].In cases where the optimization over decoders is difficult

to compute, we can still gain insight into the quantum error-correction threshold by studying the coherent quantuminformation [133,134]

IcðρS;N ∘EÞ ¼ SðρS0 Þ − SðρRS0 Þ; ð7Þ

where SðρÞ ¼ −Tr½ρ log2 ρ� is the von Neumann entropy,ρS0 ¼ N ∘EðρSÞ, and ρRS0 ¼ N ∘EðρSRÞ. The coherent quan-tum information is closely related to the entanglementfidelity because when jIcðρ;N ∘EÞ − SðρÞj < ϵ, it impliesthat FeðρÞ ≥ 1–2

ffiffiffiϵ

p[134]. In our analysis of random

stabilizer codes, we directly compute Favg, while forHaar random codes, we use the coherent quantum infor-mation to bound FeðI=qÞ.The coherent quantum information is fundamentally

related to the quantum channel capacity through thelimiting formula [135,136]

QðN Þ ¼ limN→∞

1

Nmaxρ

Icðρ;N⊗NÞ: ð8Þ

In this work, we study erasure errors, which, for a singlequbit, are defined by the channel

N ðρÞ ¼ ð1 − eÞρ ⊗ j0ih0j þ e=2I ⊗ jeihej: ð9Þ

The states j0i=jei are orthogonal states that herald theabsence/occurrence of the erasure on this site. Note that, inmany physically relevant scenarios, the state of the system


031066-6

itself is mapped to an orthogonal state under an erasureerror, which is an equivalent description of this channel, forour purposes. We choose the representation in Eq. (9) tosimplify the notation in later discussions.The heralded nature of the erasure locations dramatically

simplifies the decoding problem, as we discuss below forstabilizer codes. Furthermore, because of this classicalregister, the capacity of the erasure channel is additive[i.e.,QðN Þ ¼ maxρ Icðρ;N Þ] and is derivable from the no-cloning theorem [127]. It is also easy to compute thechannel capacity from the maximization of the coherentquantum information QðN Þ ¼ ð1 − 2eÞ, where e is thelocal erasure probability on each site. Equivalently, for acode rate R ¼ k=N ¼ Q, the optimal erasure threshold inthe thermodynamic limit is ec ¼ ð1 − RÞ=2.

B. Optimal decoding: Stabilizer codes

These concepts of optimal decoding are illustrated moreconcretely by considering the example of qubit stabilizercodes. An ½N; k� qubit stabilizer code encodes k logicalqubits in N physical qubits. The codewords are spanned bythe set of 2k stabilizer states that are the simultaneouseigenstates of a stabilizer group S ⊂ PN , which is anAbelian subgroup of the Pauli group on N qubits PN suchthat −I ∉ S. Given a generating set fZ1;…; ZN−kg for S,optimal decoding is possible through projective measure-ments of these generators (called syndrome measurements)for Pauli error channels. These are channels that have aKraus representation of the form

N ðρÞ ¼XE;k

pðE; kÞEρE† ⊗ jkihkj; ð10Þ

where E is an element of PN , jki are orthogonal quantumstates that are used to store classical data (e.g., the erasurelocations), and pðE; kÞ ≥ 0 is a joint probability distribu-tion over the allowed error operators E and register indicesk. Such quantum-classical channels are sometimes called a“quantum instrument” [137]. Erasure errors can be repre-sented in this form because of the following identity for thepartial trace operation on site n:

Trn½ρ� ⊗ In=2 ¼ 1

4ðρþ XnρXn þ YnρYn þ ZnρZnÞ; ð11Þ

where In, Xn, Yn, and Zn are the four Pauli operators.The Pauli group operation can be represented by

standard matrix multiplication of the N-qubit Pauli oper-ators. For two Pauli group elements P1;2, we use thenotation ½½P1; P2�� ¼ Tr½P1P2P−1

1 P−12 �=2N to denote their

scalar commutator: If P1 and P2 commute, then½½P1; P2�� ¼ 1: otherwise, ½½P1; P2�� ¼ −1. We can extendthe generating set for S to a complete generating set for PN

by appending destabilizer operators fX1;…; XN−kg thatsatisfy ½Xi; Xj� ¼ 0 and ½½Zi; Xj�� ¼ ð−1Þδij , a generating

set for the logical operators Li (these are Pauli groupelements that commute with S but are not contained in S[129]), and the Pauli group element iI. Since each E is anelement of the Pauli group, we can decompose them basedon the outcomes they produce in the syndrome measure-ments Esk ¼ gskLEsk

, where s ¼ ðs1;…; sN−kÞ is a vector ofsyndrome bits ðsi ¼ 0=1Þ, gsk is Pauli group elementsatisfying ½½gsk; Zi�� ¼ ð−1Þsi , and LEsk

is a logical operator.The gsk is nonunique and is allowed to be linearlydependent on elements of S, the destabilizers, logicaloperators, and iI.After applying the error channel N and performing a

projective measurement of the syndrome bits s and registerstate k, the state is mapped to

Msk∘N ∘EðρÞ ¼ XEsk

pðEsk; kÞgskLEskρL†

Eskg†sk; ð12Þ

where we traced over the classical register for notationalconvenience. Applying the correction operator g†sk, which isa product of single-site Clifford gates, the state becomes amixture of states in the code space

g†sk½Msk∘N ∘EðρÞ�gsk ¼XEsk

pðEsk; kÞLEskρL†

Esk: ð13Þ

Below threshold in the thermodynamic limit, all the LEsk

must converge in probability to the same logical operatorLsk up to multiplication by elements of the stabilizer groupS, i.e., LEsk

¼ gEskLsk for some gEsk

in S. Since the operatorsgEsk

act trivially in the code space, the initial state can thenbe perfectly recovered by applying the additional unitarycorrection operator L†

sk.In finite-size systems, where perfect decoding is not

generally possible, an optimal decoding strategy is anymaximum-likelihood decoder based on the observed s and k[47]. In this approach, we further break up the set of all Esk

into logical operator classes Eisk ¼ gskgEi

skLisk, such that the

gEiskare in S and the Li

sk cannot be related (modulo a phase)through multiplication by elements of S. Conditioned on sand k, the decoder applies a unitary correction operatorR†sk ¼ gskL

imsk to the state Rsk½Msk∘N ∘EðρÞ�R†

sk, with Limsk

corresponding to the most likely logical error equivalenceclass. This operator can be computed by finding the value ofi that maximizes the probability

Ziðs; kÞ ¼XEisk

pðEisk; kÞ; ð14Þ

i.e., Zmaxðs; kÞ ¼ maxi Ziðs; kÞ. The probability of a perfectrecovery for all input states under optimal decoding is thengiven by


031066-7

PðRÞ ¼Xs;k

Zmaxðs; kÞ: ð15Þ

For general codes and Pauli error channels, finding Limsk is

likely to be exponentially hard, but, in the case of erasureerrors, an efficient optimal decoding strategy has beenderived by Delfosse and Zemor [40]. Briefly reviewingtheir argument, one can use the fact that erasure errorlocations are heralded, which implies that the decoder onlyneeds to find the most likely error operator that acts on theerased sites. Once the sites are known, according to Eq. (11),all error operators are equally likely, which implies that theprobabilities in Eq. (14) are all equal for a fixed k; therefore,amaximum-likelihood strategy is to simply chooseRsk to beany Pauli operator that lives on the erased sites and producesthe observed syndrome measurement. Using the standardrepresentation for stabilizer states from the Gottesman-Knilltheorem [138,139], such a Pauli operator can be found giventhe syndrome check operators, erasure locations, and syn-drome measurement outcomes using Gaussian eliminationin a time of at most OðN3Þ.Here, we derive an explicit formula for the recovery

probability under erasure errors that is convenient for ourpurposes. We make use of a generating matrix for the erroroperators that act in the erased region e,

MðS; L; eÞ ¼

0BBBBBBBBBBBBB@

szi1 lzi1

sxi1 lxi1

..

. ...

szine lzine

sxine lxine

1CCCCCCCCCCCCCA

: ð16Þ

The first ns ¼ N − k columns sE are vectors of syndromebits for a local basis of error operators defined by therelation ð−1ÞsE;i ¼ ½½Zi; E��. The last 2k columns lE sim-ilarly encode the scalar commutator of the local errors witha generating set for the logical operators. Crucially,stabilizer codes are additive codes, which implies that ifE ¼ E1 þ E2, then sE ¼ sE1

þ sE2and lE ¼ lE1

þ lE2.

As a result, the row vectors ðsμi jlμiÞ act as a generating setfor all possible syndromes and their associated logicalerrors in the erased region.We now show how to compute the recovery probability

from the matrix M. Performing row reduction on Midentifies all errors that map to the all-zero syndromebut have a nontrivial logical operator content. Errors of thistype can be used to enumerate all uncorrectable errors forthat set of erasure locations. For each matrixMðS; L; eÞ, wedefine

rMðS; L; eÞ ¼ rankðMÞ − rankðMSÞ; ð17Þ

where MS is the submatrix of M consisting of the first nscolumns ofM. Here, rM counts the number of basis vectorsfor errors that have a zero syndrome but act nontrivially onthe logical subspace.For each syndrome, the decoder can only apply a

single recovery operator; however, this recovery strategywill always fail with some probability if the error islinearly dependent on one of the rM basis vectors withtrivial syndrome and nontrivial logical operator content.Since all the errors occur with equal probability forerasures, the optimal recovery probability can then bedirectly computed as

PðRjS; L; eÞ ¼ 1

2rMðS;L;eÞ : ð18Þ

Incidentally, k − rM is also equal to the coherent quantuminformation of this encoding scheme under erasure errors.As we take advantage of in Sec. VI, these formulasdirectly generalize to stabilizer subsystem codes byremoving columns of M associated with generators forthe gauge group.

IV. RANDOM STABILIZER CODE THRESHOLD

In this section, we present a solution to the critical theoryof the erasure threshold for random stabilizer codes basedon a RMTansatz. Establishing the basic phenomenology ofthe random stabilizer erasure threshold transition is stan-dard material in quantum information theory [140]. Ourmain contribution is to derive analytic predictions for thecode-averaged probability of perfect recovery PðRjneÞaccording to Eq. (15), where ne is the number of erasedsites in the fixed-fraction model and k is the number oflogical encoded qubits. In the Appendix A, we further showthat PðRjneÞ is equal to the code-averaged max-averagefidelity Favg in the thermodynamic limit. We use this resultto argue that the Haar random erasure threshold, where weonly approximate Favg, is in the same universality class asthe random stabilizer erasure threshold.The encoding circuit U for a random stabilizer code

is a random Clifford unitary on N qubits. Since spatiallocality is irrelevant in this discussion, we take the initialunencoded logical qubits to be given by the last blockof k qubits, which implies that the stabilizer group hasgenerators

Zi ¼ UZiU†; i ¼ 1;…; ns; ð19Þ

where ns ¼ N − k is the number of stabilizer generators.We use the optimal decoding strategy for erasure errorsdescribed in Sec. III B: Given a set of erased sites e andsyndromes s, the decoder applies any Pauli operator Rse

that lives on e and flips each stabilizer generator Zi to have


031066-8

the same sign as si [40]. The circuit-averaged probabilityfor perfect recovery under optimal decoding satisfies

PðRjneÞ ¼ EU

Xe

PðRjU; eÞpðeÞ ¼ EUPðRjU; eÞ ð20Þbecause EUPðRjU; eÞ depends on e only through ne ¼ jejfor a fully random Clifford circuit.Since random stabilizer codes are nondegenerate in the

large-N limit, every correctable error needs to map to aunique syndrome. We can find the number of uniquesyndromes for a given U and e by determining the F2-rank nse of the syndrome matrix MSðS; L; eÞ formed fromthe first ns columns of M from Eq. (16). Intuitively, 2nse issimply the total number of unique syndromes available tothe decoder for this erasure pattern. Thus, the averagerecovery probability is

PðRjneÞ ¼ EU½2nse−2ne � ¼ 2nse−2ne ; ð21Þ

where nse ≡ log2 EU½2nseðU;eÞ� is just a function of ne,ns ¼ N − k, and N.We can approximate the behavior of nse for a random

stabilizer code in the two limits ns ≪ 2ne or ns ≫ 2ne[140]. In the former case, the number of possible errors isexponentially larger than the number of available syn-dromes. For a randomU, each syndrome occurs with nearlyequal probability; thus, each syndrome will be occupiedwith high probability and result in the scaling nse ¼ ns. Inthe opposite limit, the number of available syndromes isexponentially larger than the number of errors. As a result,there is a high probability that each error gets mapped to adistinct syndrome, resulting in the scaling nse ¼ 2ne. Theseestimates show that PðFÞ ¼ 1 − PðRÞ has a discontinuityat the channel capacity bound ns=N ¼ 1 − R ¼ 2ne=N ¼2e in the large-N limit.In the RMT approach described below, we can explicitly

calculate PðFÞ for all values of ne and ns, includingarbitrarily close to threshold,

PðFÞ ≈

8><>:

2−jδj−1 2ne ≪ ns1 − rc 2ne ¼ ns1 − 2−jδj 2ne ≫ ns;

ð22Þ

where δ ¼ 2ne − ns ¼ 2ðe − ecÞN is the distance fromthe critical point and rc is the recovery rate at the criticalpoint. As we show in the section below, rc ¼ 0.610322…in the RMT model. From this RMT solution, we also findthat the higher-order corrections to this formula areexponentially suppressed in the distance from the criticalpoint Oð2−2jδjÞ.

A. RMT solution

The exact formula for the code-averaged recoveryprobability is given by

PðRjneÞ ¼ EU

X2nem¼0

P½MðS; L; eÞ has rM ¼ m� 12m

: ð23Þ

There is no need to average over e for a fixed erasure numberbecause the circuit average removes the dependence on thespatial locations of errors. In the RMTapproach, we assumethat the syndrome matrices MðS; L; eÞ and MSðS; L; eÞ aregiven by random 2ne × ðns þ 2kÞ and 2ne × ns matrices,respectively. We do not expect this result to be true, exactly,because it ignores the constraint that the time evolutionpreserves commutation relations; however, we conjecturethat it is accurate up to exponentially small corrections inN.The reason it is a probable hypothesis is that M and MScan be constructed by taking submatrices of a much larger2N × 2N tableau representation forU [138,139]. The RMTansatz is based on the assumption that these submatricesare insensitive to the “global” constraint on the otherwiserandom U that it preserves commutation relations of Pauligroup elements.In the RMT anstaz and for e < 1=2, the matrix M will

have full rank 2ne with a probability that converges to 1exponentially in N because the number of columns is muchgreater than the number of rows. The average recoveryprobability then reduces to a combinatorial formula regard-ing the rank distribution of the MS matrix,

PðRjneÞ ¼Xm

#2ne × ns matrices of rankm#2ne × nsmatrices

2m

22ne:

The denominator is the number of matrices over F2 of size2ne × ns, which is equal to 22nens since each entry can takeone of two independent values. Finding the number of2ne × ns matrices of rank m is a less trivial, but familiar,result in combinatorics that also has applications inclassical error correction [141]. For completeness, weprovide a derivation in Appendix B.Using this formula, the probability of successful recov-

ery has the analytic expression

PðRjneÞ ¼Xnmsem¼0

Qm−1l¼0 ð22ne − 2lÞð2ns − 2lÞ22nens

Qm−1l¼0 ð2m − 2lÞ

2m

22ne;

where nmse ¼ minðns; 2neÞ. Note that PðFÞ ¼ 1 − PðRÞ hasthe asymptotic behavior given by Eq. (22) with the criticalparameter

rc ¼X∞m¼0

Q∞l¼1ð1 − 1

2mþlÞ22mðmþ1Þ Q∞

l¼1ð1 − 12lÞ ; ð24Þ

which is approximately rc ≈ 0.610322…. By numericallysampling MS matrices generated by depth N 1D localcircuits with periodic boundary conditions, we have veri-fied that these RMT predictions accurately approximate thetrue failure probability on these sizes. The results are shown


031066-9

in Fig. 2 up to size N ¼ 128 for R ¼ k=N ¼ 1=2. We findexcellent agreement between the exact numerical resultsand the RMT prediction throughout the critical region, evenfor sizes down to N ¼ 16. To obtain a more precisecomparison, we estimate the success probability withhigher precision at the critical point. Randomly generating108 MS matrices from a depth 2N circuit (N ¼ 40) in 1Dwith periodic boundary conditions provides our currentbest estimate

rc ≈ 0.61029� 3.7 × 10−5 ¼ 0.61029ð4Þ; ð25Þ

which agrees with the RMT value at a precision of 10−4. InAppendix D, we further show that the recovery probabilityis self-averaging at ec in the sense that a typical randomcode has a recovery probability that converges to rc in thelarge-N limit.

V. QUASILOCAL RANDOM STABILIZERCODE THRESHOLD

In this section, we investigate the erasure threshold forrandom stabilizer codes generated by finite-depth quantumcircuits in finite-size systems.

A. Block model: Mean-field limit

We can gain a surprising amount of insight into this localrandom coding problem by first considering a toy modelwith the simplified block encoding scheme illustrated inFig. 3. Furthermore, the basic arguments in this section arenot specific to the erasure channel. In this model, weremove gates that couple different blocks of qubits suchthat each block undergoes completely independent randomunitary dynamics. Intuitively, this model can be interpreted

as a type of mean-field model for the random codetransition. At large depth, the average failure probabilityfor this model becomes an upper bound on the averagefailure probability of the random code transition.Specifically, we break up a system of N qubits into cubic

blocks of size Nb ¼ LD, whereD is the space dimension ofthe encoding Clifford circuit in each block and L is thelinear size of the block. Each block has approximatelyð1 − RÞNb stabilizers and RNb logical qubits. Running ahigh-depth ½d ¼ OðNbÞ� random Clifford circuit on eachblock results in a rate R random stabilizer code on thisblock of qubits. If we apply an erasure error below therandom code threshold, then the average recovery proba-bility is just the product of the average recovery probabilityfor each block (since the codes between blocks areuncorrelated)

PðRjneÞ ¼YN=Nb

i¼1

ð1 − h2−δi−1ieÞ þOðN2−Nb=NbÞ; ð26Þ

where δi ¼ ð1 − RÞNb − 2nei is the distance from thecritical point in block i with nei erased sites. In orderfor our approximations to be valid, we require that Nbgrows as OðlogNÞ or faster. We make use of the fact thatthe fluctuations in the number of erasures in each region aredetermined by the central limit theorem

nbe¼eNbþΔb; Δb∼N ð0;σ2bÞ; σb¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieð1−eÞNb

p:

ð27Þ

As a result, the average failure probability is given by

PRMTðFÞ ≤ PðFÞ ≈ N2Nb

h2−δiie; ð28Þ

h2−δiie ≈ 22ðe−ecÞNb

Z0

−∞

dxffiffiffiffiffiffi2π

p 22σbx expð−x2=2Þ

¼ 22ðe−ecÞNbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πeð1 − eÞNb

pln 4

; ð29Þ

FIG. 2. Failure probability for 1D circuit of depth N withperiodic boundary conditions obtained via numerically sampling103 random codes. We take a fixed fraction erasure error at a coderate R ¼ 1=2, for which the critical erasure rate is ec ¼ N=4. Theblack line shows the RMT prediction computed for N ¼ 128,which agrees with the numerical results to within the statisticalerror bars. The inset shows an excellent collapse for this full rangeof sizes according to the predicted scaling.

FIG. 3. Toy model for the below-threshold behavior of finite-depth, random, unitary-encoding circuits in which we removegates that couple different blocks of qubits.


031066-10

where x ¼ ðe − ecÞffiffiffiffiN

p=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieð1 − eÞp

. Thus, for e < ec, thefailure probability converges to zero when

limN→∞

log2ðN=N3=2b Þ

Nb< 2ðec − eÞ; ð30Þ

which is satisfied for Nb ¼ OðlogNÞ, i.e., when the blocklength scales as L ¼ O½ðlogNÞ1=D�.To achieve a random code on each block, we naively

need to apply a depth d ¼ OðLÞ circuit [37,38]; however,this neglects the fact that there are rare Clifford circuitswhere Pauli operators remain localized to a given site. Inparticular, to preserve commutation relations, every two-qubit Clifford gate has to map at least one single-site Paulioperator for each site to another single-site Pauli operator.The probability of such localized logicals appearing in asystem of size N scales as N=Ad for a constant A thatdepends on the ensemble of gates used in the randomcircuit. Note that even if all two-qubit gates are entangling,A will still be finite. This constraint implies that one needsto apply a depth OðlogNÞ random Clifford circuit regard-less of dimensionality to avoid these rare localized oper-ators. As a result, the block model only converges to zerofailure probability for depth d ¼ OðlogNÞ for all spatialdimensions. Interestingly, a similar type of argument wasrecently used in Corollary 4 of Ref. [142] to prove a lowerbound of OðlogNÞ on the depth required to achieve a formof anticoncentration in random circuits. The ensemble wasformed from two-local circuits with the gates drawnrandomly from a two-design. Developing a more completeunderstanding of the relation between encoding propertiesof low-depth random circuits and other observables, e.g.,anticoncentration or sampling complexity, is an interestingsubject for future work.In the case of the channel coding problem considered

here, there are two routes to overcome the OðlogNÞ lowerbound on the depth required to achieve zero failureprobability below capacity. One simple approach withinthe block model picture is to apply an optimized imple-mentation of a two-design following Ref. [36], but includ-ing SWAP gates to map the all-to-all circuit to a localgeometry. This approach also requires the use of OðlogNÞancilla qubits per block, which, by our conventions, wouldeffectively reduce the overall rate of the code. With suchoptimized circuits, one can deterministically encode eachblock into a high-performance code in depth OðN1=D

b Þ,thereby allowing convergence of the full system to zerofailure probability at depth O½ðlogNÞ1=D�. This argumentshows that, in principle, one can surpass the OðlogNÞscaling by introducing long-range correlations into theencoding and allowing for additional ancilla qubits. Inpractice, however, the block model will always have arelatively weak convergence with depth because it does nottake advantage of correlations that can build up between

blocks. To achieve the sublogarithmic scaling in practice,we therefore use the expurgation strategy described inSec. VI below. In this approach, these rare localized logicaloperators are directly removed from the code by theexpurgation process.

B. Critical scaling

In the vicinity of the critical point for the randomstabilizer code, it is clear that the block encoding schemefails because each individual block fails with a largeprobability. As mentioned in the previous section, weexpect the original model to achieve better performancebecause the “blocks” formed by the finite depth circuit areeffectively correlated with each other. This implies that theerror correction in regions with excessive numbers oferasures can be assisted by nearby regions. As shown inFig. 4(a), we numerically observe that the convergence tothe critical properties of the random code behavior occurs atdepth Oð ffiffiffiffi

Np Þ in 1D. On the other hand, for D ≥ 2, the

convergence, even at the critical point, occurs at depthOðlogNÞ. As we show below, this distinction betweenD ¼ 1 and D ≥ 2 can be traced to the familiar fact that theboundary of a contiguous region in 1D is effectively zerodimensional. In the discussion below, we assume we areworking at depth greater than OðlogNÞ so that largeinhomogeneities in the quality of the random code aresmoothed out, while what is left over is the randomness inthe error pattern.We first give an argument for the

ffiffiffiffiN

pscaling in 1D

based on a mapping to a random walk for the IID model. Ifwe sum up the number of erasures relative to the criticalnumber along the length of the system, this is a biasedrandom walk that travels a certain distance on summingaround the full system. The random walker’s time is the

FIG. 4. (a) Recovery probability vs scaled depth d=ffiffiffiffiN

p(d ¼ number of two-qubit gates per site) for a 1D random circuitin a brickwork arrangement at the channel capacity limitðR; eÞ ¼ ð1=2; ecÞ. (b) Recovery probability vs scaled depth(A ¼ 6.5) for a nonrandom erasure error in which every fourthsite is erased from the system. In this case, the d ¼ 0 failureprobability is 1=2. Each two-qubit gate in these circuits is arandom Clifford gate.


031066-11

system’s space, while the random walker’s space is anexcess number of erasures in that segment of the system’sspace. The failures occur where this random walk “back-tracks” a distance d. Thus, the characteristic d ¼ d�, wherethe failure probability converges towards its high-depth½d ¼ OðNÞ] value, is the d where these backtracks becomerare in the system of lengthN. From the statistics of randomwalks, this has a probability of occurring that falls off asexpð−AN=d2Þ for some constant A, but the region that isdense can be of order d2=N distinct locations [143]. Byconsidering only regions where these local fluctuationsare above threshold, we arrive at the scaling formd� ¼ N1=2g½ðec − eÞN1=2�, with gðxÞ ∼ ð1=xÞ logðxÞ atlarge x and gð0Þ of order one. Thus, well below thresholdin 1D (x ≫ 1), the scaling for the critical depth isd� ∼ ðec − eÞ−1 logN. The depth required to converge tozero failure probability is always OðlogNÞ in 1D, but theprefactor diverges as one approaches the optimal threshold.A related argument that connects more directly to the

syndrome matrix MðS; L; eÞ proceeds as follows: Imaginewe apply a fixed number of erasures at the critical pointne ¼ ns=2 but distributed randomly throughout the system.If we cut the system into two halves, then one half of thesystem will effectively be above threshold with about

ffiffiffiffiN

pextra erasures, while the other half will be below threshold.In order to correct the above-threshold region, we need to“borrow” a sufficiently large number of error syndromebasis elements in MðS; L; eÞ from the region that is belowthreshold, which requires that the minimum support of ourerror syndrome basis elements is about

ffiffiffiffiN

pto satisfy this

condition. Thus, we need to run a depth on the order offfiffiffiffiN

pcircuit to generate sufficiently long-range error syndromesin the syndrome matrix M.To test our argument that it is only the local fluctuations

in the erasure number that determine the required depth, wecompare the convergence to the RMT prediction forrandom erasures in Fig. 4(a) against regularly arrangederasures in Fig. 4(b). In the spatially nonrandom case, theerror is chosen randomly from one of the four regularlyspaced erasure patterns with ne ¼ ecN ¼ N=4. In contrastto the random error model, we see convergence to the largedepth limit with an OðlogNÞ scaling.We remark that the recovery probability for a depth zero

circuit with this nonrandom error and our layout of logicalqubits is equal to 1=2. Thus, the recovery probability isnonmonotonic with depth: It is 1=2 at d ¼ 0, drops close tozero for0 < d ≪ A logN, and then is approximately equal to0.6 for d > A logN; the coefficient is found to be A ≅ 6.5.The situation changes dramatically in higher integer

dimensions where the prefactor of the logN scaling of d�does not need to diverge as one approaches the optimalerasure threshold. In this case, the random fluctuations inerasure number within a given region can be overcome bythe overlapping syndromes near the boundary whenever

LD−1d ∼ffiffiffiffiN

p∼ LD=2 → d ∼ L1−D=2: ð31Þ

This tension between random fluctuations and orderingtendencies is familiar from Imry-Ma arguments. Thisscaling indicates that D ¼ 2 is the marginal dimensionfor the relevance of random erasure locations. ForD > 2, atthe depth d ≥ A logN needed to produce a near-optimalcode, the effect of this erasure-location randomness issubdominant. This result appears to remain true in themarginal dimension D ¼ 2, where the subdominance isonly by factors of logN. In Fig. 5, we show the numericalresults for the recovery probability through the erasurethreshold at different values of the depth in two dimensions.We clearly see the exponential convergence to the RMTprediction throughout the critical region.In the case of intermediate dimensions 1 < D < 2, such

as can be realized in fractal lattices and critical percolationclusters, the perimeter of a region with excess erasures mayhave a nontrivial scaling with N that is also not spatiallyuniform. As a result, it would be an interesting subject forfuture work to precisely determine the fate of the criticalscaling on particular real space lattices with these inter-mediate dimensions.

C. Spatial correlations of uncorrectable errors

When used as a toy model for the low-depth regimelogN ≪ d ≪ N1=D, the block model suggests that errorswill generally be bunched in space. In particular, this modelleads to the intuition that regions with excess erasures willfail first with an uncorrectable error of weight close to dD.

FIG. 5. Recovery probability vs erasure fraction for a two-dimensional random circuit in a brickwork arrangement of gateswith periodic boundary conditions for different depths d for N ¼256 and R ¼ 1=2. Different sizes collapse to the same curve forthis way of scaling except within a region of width je − ecj ∼1=N near the critical point. We sequentially cycle through fourlayers so that each site interacts with its north, east, south, andwest neighbors for each four units of depth. The scaling behaviorconverges to the RMT prediction exponentially with depththroughout the critical region. The inset shows the same dataon a logarithmic scale, illustrating the scaling PðFÞ ∼ e−d=A fore < ec. Each two-qubit gate in the circuit consists of an iSWAPgate followed by a random single-site Clifford on each site.


031066-12

To test this argument, we consider a setup inspired by theentanglement fidelity: Two of the logical qubit sites areinitially entangled with external reference qubits, and theother logical qubits are in a random pure product state.Using these reference qubits as local probes, we define

an error as occurring in the vicinity of location i, ifreference qubit i loses its entanglement with the systemfollowing the full encoding, error, and decoding procedure.Specifically, we study the change in mutual informationbetween each probe and the system

ΔIðRi∶SÞ ¼ IðRi∶SÞ − IðRi∶S0Þ; ð32Þ

where IðA∶BÞ ¼ SðρAÞ þ SðρBÞ − SðρABÞ, ρRiS0 ¼D∘N ∘EðρRiSÞ is the density matrix of the system andreference probe i following the decoded error channel,ρRi

¼ TrS0 ½ρRiS0 �, and ρS0 ¼ TrRi½ρRiS0 �. Initially, the mutual

information IðS∶RiÞ ¼ 2. In these stabilizer code modelswith Pauli error channels, the mutual information changesin discrete integer steps. For two reference qubits entangledwith the system at sites x1 and x2, we then define code-averaged local error profiles

Piðx12; dÞ ¼ P½ΔIðRi∶SÞ > 0�; ð33Þ

P12ðx12; dÞ ¼ P½ΔIðR1∶SÞ þ ΔIðR2∶SÞ > 0�; ð34Þ

where Pð·Þ ¼ EUPð·Þ and x12 ¼ jx1 − x2j is the distancebetween the probes.Numerical results for these error profile functions are

shown in Fig. 6(a) for D ¼ 1 with length L ¼ N ¼ 128.We take R ¼ 1=2 and ne ¼ ecN ¼ N=4 so that uncorrect-able errors occur relatively frequently. We see convincingevidence that spatial locality plays an important rolefor these low-depth codes, despite the potential fornonlocal effects induced by the syndrome measurements.In particular, when x12 ¼ L=2, then the joint failuredistributionP12ðL=2; dÞ factorizes into a product distribution½P1ðL=2; dÞ�2 at low depths d. On the other hand, whenx12 < d, there is a clear bunching effect wherebyP12ðx12; dÞ > ½PiðL=2; dÞ�2. We study this more quantita-tively in Fig. 6(b) in terms of the conditional failureprobability of reference probe 2 given that reference probe 1failed: P2j1ðx; dÞ ¼ P12ðx; dÞ=P1ðx; dÞ. Rather intuitively,we see a collapse of the curves for different depths when thisconditional profile is plotted as a function of x=d.These spatial correlations in the uncorrectable errors are

an indication that these low-depth codes retain featuresassociated with spatial locality despite achieving the criticalbehavior of fully random or high-depth codes. Thus, inmany respects, they are a truly distinct class of codes fromfully random stabilizer codes.

VI. EXPURGATION ALGORITHM

As discussed in Sec. VA, there are strategies in higherdimensions to overcome the logN depth scaling found forrandom Clifford circuits. In this section, we introduce anatural method to improve the performance of these low-depth codes based on the fact that the dominant failuremodes at depths ½logN�1=D ≤ d ≤ logN are rare regionswith bad logical qubits.The basic ingredient in our algorithm is the efficient

implementation of quantum measurements of stabilizercode-space density matrices [139]. We assume that weare given a single logical operator g. We can update agenerating set for the stabilizer code and its logicaloperators by making a projective measurement of thecode-space density matrix following the tableau rulesoutlined by Aaronson and Gottesman [139],

ρS ¼1

2N

YN−k

i¼1

ðI þ ZiÞ → ð1� gÞρSð1� gÞ=2

¼ 1

2N

YN−k

i¼1

ðI þ ZiÞðI � gÞ; ð35Þ

where the sign of the measurement outcome is randomlychosen. This projective measurement operation will notaffect the original generating set for S except to add g to thelist of generators; however, it will modify the logicaloperators to ensure that all of the remaining logicaloperators commute with g, which implies that the “desta-bilizer” operator g associated with g is no longer a logicaloperator. As a result, we can form an ½N; k − 1� stabilizer

FIG. 6. (a) Local uncorrectable error probability of oneP1ðx12; dÞ or both P12ðx12; dÞ reference probe qubits entangledwith the system vs d. Here, we take D ¼ 1, R ¼ 1=2,ne=N ¼ ec ¼ 1=4, and N ¼ 128. Each two-qubit gate in thecircuit is a random Clifford gate. The local error probability isdefined as the probability that the mutual information of areference qubit is less than maximal after the optimal decoding.Note that the red curve almost perfectly coincides with the yellowcurve, indicating an absence of connected correlations for thesefar-separated uncorrectable errors. (b) Conditional error proba-bility of probe 2 when an error affects probe 1 vs scaled distancefor different d. When a probe fails in a given region, it implies thatthe second probe at a distance on the order of d also fails withhigh probability.


031066-13

code or stabilizer subsystem code by converting g or g intoa stabilizer or gauge operator, respectively. This procedurecan be iterated to successively convert logical operatorsinto additional stabilizers or gauge operators while leavingthe original syndrome stabilizers unaffected.Specifically, in our expurgation algorithm, we begin with

an ½N; k� stabilizer code with stabilizer group S and logicaloperators L. We then randomly generate an erasurepattern e and compute the matrix MðS; L; eÞ. Performingrow reduction allows us to form a basis fgig of linearlyindependent errors that map to the zero syndrome but havenontrivial logical operator content. We then perform asequence of projective measurements of these operators asdescribed above to form a new stabilizer or subsystemcode. This procedure is iterated many times until eitherthe rate of the code approaches a specified target value, thefailure probability reaches a certain threshold, or thenumber of logical operators goes to zero (i.e., expurgationfails).To put this algorithm on firmer mathematical footing, we

prove the following two simple propositions:Proposition 1: Let S be the stabilizer group for a

stabilizer subsystem code with logical operators L andgauge group G. For every g ∈ L ⊗ G that acts nontriviallyon L, the distance of S after expurgating g into S or Gmonotonically increases.Let us order all 4N − 1 Pauli group operators by their

Hamming weight (number of nontrivial sites) and computethe anticommutator of every Pauli group element with agenerating set for S and L,

0BBBBB@

sE1lE1

..

. ...

sE4N−1

lE4N−1

1CCCCCA: ð36Þ

We assume that g and an anticommuting logical g are twoof the generators and that they commute with all othergenerators for L. The distance of the subsystem code can befound by finding the first Pauli group element ED in this listthat has sED

¼ 0 and a nontrivial anticommutator vectorlED

. If we expurgate g, then this removes g and g from thelist of generators, which amounts to removing two of thecolumns from lED

and adding one column to sEDor not

(depending on the expurgation strategy). If lEDbecomes

trivial, then the distance might increase, depending on whathappens to the next Pauli group element in the list orderedby the Hamming weight. If sED

becomes nontrivial, then thedistance might also increase. If sED

remains trivial and lED

remains nontrivial, then the distance stays the same.Therefore, the distance is monotonic. ▪Essentially, we use the following two properties of

expurgation: (i) The stabilizer group never shrinks in size

(it can even grow, depending on the strategy), and (ii) thenumber of logical operators only decreases. Hence, therelevant set of operators that commute with stabilizersand anticommute with some logical operator never grows.Therefore, the code distance—defined as the minimumHamming weight of elements in the relevant set ofoperators—never decreases.A related proposition that follows a similar line of

reasoning is as follows:Proposition 2: Let S be the stabilizer group for a

stabilizer subsystem code with logical operators L andgauge group G. For every g ∈ L ⊗ G that acts nontriviallyon L, the optimal decoding recovery probability of S afterexpurgating g into S or G monotonically increases for allPauli error channels.For each Pauli group element E in the list from

Proposition 1, we let their probability of appearing inthe error channel be pðEÞ. We then group this list ofanticommutator vectors into subsets with the same syn-drome vector si, which each occur with total probabilityPðsiÞ. We further break up these groups into error classesEij of errors with identical values of lEij

. The conditionalrecovery probability is the probability of the most likelyerror class

pi ¼ maxj

XE∈Eij

pðEÞ ð37Þ

divided by PðsiÞ, such that the total recovery probability isPðRÞ ¼ P

i pi. Expurgation of g will never decrease thetotal value of this sum. In the case where g is turned into agauge operator, the syndrome classes and their totalprobabilities are unchanged, while the logical equivalenceclasses for that syndrome can only combine with each otheror stay the same. As a result, pi is monotonically increasingfor each i, which makes PðRÞ monotonically increaseunder expurgation. A similar argument holds when g isturned into a check operator. ▪The dynamics during this expurgation process bears close

resemblance to the purification dynamics of ρS for randomcircuit models with measurements studied by two of theauthors [93] and developed further in Refs. [103,107,119].In that case, though, the measurements are not selectivelychosen to project out certain logical operators, but rather,they are chosen as random, few-site projective measure-ments. In both dynamics, however, we observe a similartrend that the entropy of the code-space density matrixprogressively decreaseswithmeasurements until it reaches aplateau value. The plateau can either be at a subextensivevalue (a “pure” phase) or at a finite entropy density(a “mixed” phase). What is common between both typesof dynamics is that, whenever there is residual entropy in thecode-space density matrix, the expurgated code is able tobetter protect the remaining logical qubits against future


031066-14

errors in the system that are statistically independent fromthe errors that helped form the code.In Fig. 7, we provide an illustrative example of the

performance improvements that are possible with thisexpurgation strategy for 2D and all-to-all random circuitencodings. In both cases, all expurgated logicals are turnedinto gauge qubits; this process has the advantage that thesyndrome check operators are unchanged. In this case, thesupport of each check operator is determined by the initialencoding circuit. Maintaining low-weight check operatorshas advantages for fault tolerance by limiting the effects ofmeasurement errors. For both geometries, we see nearlylinear scaling of d� with logN before expurgation. Afterexpurgation, d� has a strongly sublinear scaling with logN.We also study the performance of these expurgated codes

in 1D, but we do not find improvement of the logN depthscaling upon expurgation. It is an interesting subject forfuture work to better characterize the full range of pos-sibilities that result from this type of targeted expurgationprocess for quantum codes that begin with many logicalqubits.

VII. HAAR RANDOM CODE THRESHOLD

In this section, we study the Haar random erasurethreshold. We find a similar threshold erasure rate andcritical scaling behaviors as the random stabilizer erasurethresholds; however, we observe small quantitativedifferences in the scaling functions near the critical pointfor the two codes. These results indicate that Haar randomcodes are more optimal than random stabilizer codes forerasure errors.In contrast to our analysis of the stabilizer codes, we do

not perform an optimal decoding analysis and only test for

the existence of an erasure threshold. We consider thecoherent quantum information of an initial state in the codespace after application of the erasure channel in Eq. (9) on arandom set e of the sites

Ic¼SðρQ0 Þ−SðρeÞ; ρQ0 ¼Tre½ρQ�; ρe¼Tre½ρQ�; ð38Þ

where ρQ is an initial encoded density matrix and ρe is thereduced density matrix on e. We study the purified channelwhere a reference system R is used to purify ρQ and anenvironment E purifies the error operation [see inset toFig. 8(a)]. In the case of the erasure error, the interaction ofthe system with the environment is through a SWAPoperation of each erased qubit in e with a qubit in E.The mutual information between the reference and thefictitious environment is equal to

IðR0∶E0Þ ¼ SðρR0 Þ þ SðρE 0 Þ − SðρR0E0 Þ¼ SðρQÞ − Ic ¼ RN − Ic ≥ 0; ð39Þ

where R is the rate of the code. As we discussed inSec. III A, when IðR0∶E0Þ ¼ jSðρQÞ − Icj < ϵ, then themax-entanglement fidelity for that input state satisfiesFeðρQÞ ≥ 1–2

ffiffiffiϵ

p; i.e., for ϵ sufficiently small, the error

channel can be approximately decoded.In Fig. 8, we show the results of numerical simulations

for IðR0∶E0Þ ¼ EUIðR0∶E0Þ for the Haar random code witha ρQ that acts trivially on the code space. We show theresults for both fixed fraction and IID erasure errors. We seeconsistent scaling results with the random stabilizer code:The fixed-fraction error model leads to a finite-size round-ing of the transition over a region scaling as je − ecj ∼ 1=N.The random fluctuations in the total number of erasures inthe IID model then round out the threshold even more,producing a “critical” region of width je − ecj ∼ 1=

ffiffiffiffiN

pand

amplitude I ∼ffiffiffiffiN

pat ec.

FIG. 7. (a) Interpolated depth d� to reach 50% failure proba-bility for a 2D random Clifford circuit with periodic boundaryconditions vs log-system size. All logical operators are turnedinto gauge degrees of freedom during expurgation. We removethe small amount of N=32 to aid in extracting the scaling vslog2 N to large sizes and small depths. We take an erasure fractionne=N ¼ 1=8 for both the expurgation algorithm and the calcu-lation of the failure probability. (b) Same as panel (a), but for anall-to-all circuit in which N=2 pairs of sites are randomly selectedto apply a two-qubit gate for each unit of depth. Each two-qubitgate in both geometries is a random Clifford gate.

FIG. 8. (a) IðR0∶E0Þ for the Haar random encoding following afixed-fraction erasure error at R ¼ 1=2. For e < ec ¼ ð1 − RÞ=2,IðR0∶E0Þ rapidly decays to zero, whereas it grows to an extensivevalue for e > ec. (b) Finite-size scaling in the IID erasure model.Inset: scaled IðR0∶E0Þ with an unscaled erasure rate. The curvesfor different system sizes cross near the channel capacity boundec ¼ 1=4.


031066-15

To obtain a more direct comparison between the Haarrandom and random stabilizer codes, we show the averagecoherent quantum information of each code ensemble inFig. 9. We see remarkably close quantitative agreementbetween the code performance; however, there are signifi-cant differences that appear at the critical point. Inparticular, in Fig. 9(b), we see that the two codes appearto be converging to substantially different values ofIðR0∶E0Þ in the large-N limit of 0.720(5) (Haar) and0.848(5) (Clifford). Thus, a Haar random code is slightlymore optimal than a random stabilizer code in this regionwhere the code fails. These quantitative differences in thescaling function indicate that the random stabilizer codedoes not necessarily saturate the performance of an optimalcode, even at leading order in the large-N limit. In the caseof the depolarizing channel, the optimal decoding thresholdfor a random stabilizer code is expected to be smaller thanthe channel capacity limit [124,125].

VIII. STATISTICAL MECHANICS MAPPING:DOMAIN-WALL PINNING

In this section, we present an approximate mapping ofthe erasure threshold to a first-order domain-wall pinningtransition in a related statistical mechanics description. Thisdiscussion applies to both Clifford and Haar models.We consider the quenched average of the purity of a

subregion A,

−log2PA ≡ −log2½EUTr½ρ2A��≤ −EUlog2Tr½ρ2A� ≤ EUSðρAÞ: ð40Þ

A natural approximation to the coherent quantum infor-mation is the difference in log-average purity of eachsubregion

Ip ¼ − log2 PQ0 þ log2 Pe: ð41ÞAlthough this quantity does not have a clear significancefor error correction in general systems, we expect that, for

deep Haar random circuits, the fluctuations in Ic overcircuits are small enough that it is well approximated by Ip[73,74,80]. For any U constructed of local two-qubitgates distributed according to a two-design, we cancompute Ip after circuit averaging using a well-studiedmapping between the average purity of subregions ofa D-dimensional random circuit to a Dþ 1-dimensionalpartition function of an Ising model with certain boundaryconditions at late times [74]. The condition that the initialstate is mixed on the logical qubit degrees of freedomcorresponds to a spin-polarized bottom boundary conditionon the logical qubit sites [97]. In this mapping, Ip becomesthe free energy cost of flipping the polarization of the toperased boundary condition in the presence of the polarizedboundary condition due to the logical qubits (see Fig. 10).The temperature of the effective Ising model is well

below the transition temperature, which implies that thefree energy is minimized primarily through energy mini-mization. Using a minimal energy surface approximation,we obtain a direct estimate for the analog of the mutualinformation between the reference and the environment forthe log-average purity,

IpðR0∶E0Þ ¼ −log2PQ − Ip ¼ RN − Ip

≈

8><>:

0 2ne ≪ ns2ne − ns ns ≪ 2ne ≪ ð1þ RÞN2RN ð1þ RÞN ≪ 2ne;

ð42Þ

where ns ¼ ð1 − RÞN. This quantity undergoes a phasetransition at the same point as the optimal erasure threshold;

Logical qubits

FIG. 10. Approximate description of below-threshold behaviorin the Ising model. The coherent quantum information is the freeenergy cost of flipping the top boundary condition on e and e inthe ordered phase of the Ising model. Below threshold, the topboundary condition on e polarizes the system into the orderedphase aligned along the same direction. The error-correctionthreshold occurs when the bulk of the system on the left flips to bepolarized in theþ direction as the size of e is increased, which is afirst-order domain-wall pinning phase transition.

FIG. 9. (a) IðR0∶E0Þ for the fixed fraction erasure errors in thevicinity of the critical point for the Haar random (H) and randomstabilizer (C: Clifford) codes. (b) Comparison of the codeperformance at the critical point. The Haar codes have slightlybetter performance than the random stabilizer codes withIðR0∶E0Þ ¼ 0.720ð5Þ and 0.848(5), respectively.


031066-16

thus, we suspect it captures some essential features ofthe threshold for the optimal code. In particular, the point2ne ¼ ns corresponds to a transition in the left half of Fig. 10where the top boundary condition is no longer sufficientlystrong to polarize the bulk of the system. In this case, themiddle domain flips to align with the logical qubits.Although it is clear that our codes will not be robust

against erasure errors that occur during the encodingcircuit, we can gain some additional insight into thebreakdown of the threshold using this statistical mechanicsmodel. In the Ising model mapping, erasures in the bulkcorrespond to fixing a finite density of spins in the bulk topoint along the þ direction, which will overcome thesurface pinning effect and prevent the formation of theordered− phase in the left of Fig. 10. As a result, in order tohave a fault-tolerant encoding, some form of error correc-tion should be applied during the evolution itself.

IX. CONCLUSIONS

In this paper, we revisited the study of quantum error-correcting codes generated by low-depth random circuits.In any spatial dimension, we found that a depth OðlogNÞrandom circuit is necessary and sufficient to achieve high-performance coding against erasure errors below theoptimal erasure threshold, set by the channel capacity.However, in 1D, coding arbitrarily close to the optimalthreshold requires a depth Oð ffiffiffiffi

Np Þ circuit because of the

relevance of spatial randomness in errors near codecapacity. The marginal dimension for high-performance,low-depth coding at capacity is 2D, where spatial random-ness becomes an irrelevant perturbation.Although spatial randomness in the errors becomes irrel-

evant above 1D, there are still large inhomogeneities in thequality of the randomcode due to randomcircuit fluctuations.Using a simple block model, we showed that the effects ofcode randomness in D > 1 can be mitigated through corre-lated coding and the use of additional ancilla qubits thateffectively reduce the rate of the code. An alternative strategy,which works better in practice, is to expurgate low-weightlogical operators from the code using quantum measure-ments. With these methods, we found that good codingbecomes possible at sub-log-N depths. Codes with rates near1=2 generated by our random coding algorithms can achievehigh performance at depth 4–8 in 2D for large erasure ratesand block sizes of thousands of qubits.The results in this work open up many directions for

future research. To develop these codes for use on near-term devices, a more general theory of optimal decoding forPauli error channels should be developed. Efficient optimaldecoding can likely be implemented for these low-depthcodes by taking advantage of their strongly local nature.For example, a brute force method is sufficient in the blockencoding model with logarithmic block sizes. It will also beinteresting to consider the performance of these codes in

conventional threshold theorems, including strategies forachieving full fault tolerance, e.g., as can always beachieved with concatenation.Another promising avenue of research is to further

develop the expurgation algorithm, which we used tosignificantly reduce the required depth to achieve success-ful decoding of erasure errors. It has now been wellestablished that fault-tolerant thresholds can be signifi-cantly improved by tailoring codes to the detailed proper-ties of the noise [27–30]. The expurgation algorithmprovides a wide variety of additional techniques to tailorcodes to specific noise models. In addition, it may bepossible to further improve the expurgation by usingquantum measurements that explicitly implement entan-glement swapping, similar to techniques used for themeasurement-based preparation of the surface code states[144,145].As mentioned in the Introduction, developing more

concrete connections between the results here and meas-urement-induced phase transitions is also promising toexplore. Unitary-measurement models that include botherrors and active error correction may realize a differentuniversality class of these transitions that might be moreresilient in near-term quantum computing devices.

ACKNOWLEDGMENTS

We thank Steve Girvin, Pradeep Niroula, and SarangGopalakrishnan for helpful discussions. M. J. G. andD. A. H. were supported in part by the DefenseAdvanced Research Projects Agency (DARPA) Drivenand Nonequilibrium Quantum Systems (DRINQS) pro-gram. L. J. acknowledges support from the Army ResearchOffice (ARO) (Grants No. W911NF-18-1-0020 andNo. W911NF-18-1-0212), ARO MURI (GrantNo. W911NF-16-1-0349), Air Force Office of ScientificResearch (AFOSR) MURI (Grant No. FA9550-19-1-0399),Department of Energy (DOE) (Grant No. DE-SC0019406),National Science Foundation (NSF) (Grants No. EFMA-1640959, No. OMA-1936118, and No. EEC-1941583),and the Packard Foundation (Grant No. 2013-39273).

APPENDIX A: MAX-AVERAGE FIDELITY FORRANDOM STABILIZER ERASURE THRESHOLD

In this Appendix, we prove that the max-average fidelityconverges to the perfect recovery probability for the randomstabilizer erasure threshold in the thermodynamic limit.For an initial random pure state j0ijψi on the unencoded

logical qubits at the k sites i ¼ ns þ 1;…; N forns ¼ N − k, the probability of successful error correctionfollowing the encoding by the Clifford unitary U, erasure atsites e, syndrome measurements with outcome s, andmaximum-likelihood recovery is given by


031066-17

PðRjU;ψ ; s; eÞPðsjU;ψ ; eÞ

¼ h0jhψ jU†Rse

Ynsi¼1

Psii Tre½Uj0ijψih0jhψ jU†� ⊗ Ie

2ne

×YnSi¼1

Psii R

†seUj0ijψi

Xse

δsse ; ðA1Þ

where Psii ¼ ðI − ð−1ÞsiUZiU†Þ=2 is a syndrome projector

for sites i ¼ 1;…; ns and Rse is the conditional recoveryoperator. We include a sum over Kroenecker delta functionsδsse , where fseg are the set of possible syndrome outcomesfor e. This term is nonzero only when the observedsyndrome is allowed for a given e; thus, it serves as aprojector onto the space of allowed syndromes. Themaximum possible size of fseg is 22ne as this is the numberof Pauli group elements with support only on e (moduloa phase).The precise form of Rse depends on the encoding circuit

U in addition to s and e; therefore, it cannot be calculated,in general, without completely specifying U. On the otherhand, we can use the fact that it can be moved past thesyndrome projectors by turning this into a projector ontothe perfect syndrome outcome. Since it has its supportentirely on e, we can then cancel the product of the tworecovery operators to arrive at the much simpler formula

PðRjU;ψ ;s;eÞPðsjU;ψ ;eÞ

¼Xse

δsseh0jhψ jU†Tre½Uj0ijψih0jhψ jU†�⊗ Ie2ne

Uj0ijψi:

ðA2Þ

Summing over syndrome measurements gives the recoveryprobability

PðRjU;ψ ; eÞ ¼ 1

22ne−nseþ 1

22nE−nse

×XPe≠I

jh0jhψ jU†PeUj0ijψij2; ðA3Þ

nse ¼ log2jfsegj ≤ 2ne: ðA4Þ

Here, we use the identity

Tre½ρ� ⊗Ie2ne

¼ 1

22ne

XPe

PeρP†e ; ðA5Þ

where Pe runs over a basis of Pauli group elements that acton sites in the subset e. Since the Clifford group forms a two-design, we have the identity from random matrix theory

EUUa0aU�b0bUc0cU�

d0d

¼EμUa0aU�b0bUc0cU�

d0d1

4N−1

�δa0b0δc0d0δabδcd

þδa0d0δb0c0δadδbc−1

2Nðδabδcdδa0d0δb0c0 þδa0b0δc0d0δadδbcÞ

�;

ðA6Þwhere Eμ is an average over the Haar measure on the unitarygroup on N qubits. This formula can be used to bound theaverage of the second term in Eq. (A3) as

EU;ψ

�1

22ne−nse

XPe≠I

jh0jhψ jU†PeUj0ijψij2�

ðA7Þ

≤1

22ne−nmse

XPe≠I

Eμ;ψ jh0jhψ jU†PeUj0ijψij2 ðA8Þ

¼ 1

22ne−nmse

XPe≠I

Eμjh0jh0jU†PeUj0ij0ij2 ðA9Þ

¼ ð22ne − 1Þð2N − 1Þ22ne−n

mseð4N − 1Þ ¼ Oð2−NþnmseÞ; ðA10Þ

where nmse ¼ minðns; 2neÞ, and in Eq. (A8), we used thefact that Ujψi ¼ UUψ j0i for Uψ distributed accordingto the Haar measure. As a result, up to corrections thatdecay exponentially withN for any erasure rate e < 1=2, wefind the formula for the code-averaged max-average statefidelity

Favg ¼ EU;ψ ½PðRjU;ψ ; eÞ� ¼ 2nse−2ne ; ðA11Þ

2nse ¼ EU½2nse �: ðA12Þ

This expression for Favg is equal to PðRÞ fromEq. (21).

APPENDIX B: COUNTING RANK-mMATRICES OVER F2

In this Appendix, we reproduce the standard formula forthe number of rank-m 2ne × ns matrices over F2. To findthe formula, we first use the fact that the number of m × nsmatrices of rank m is given by

Ym−1

k¼0

ð2ns −2kÞ¼ ð2ns −1Þð2ns −2Þ� � �ð2ns −2m−1Þ ðB1Þ

because we have 2ns − 1 choices for the first row and2ns − 2i−1 choices for row i to ensure that they are linearlyindependent from the first i − 1 rows. When 2ne > m, wehave to account for linear dependence between rows of the


031066-18

matrix, which leads to a degeneracy that is equal to thenumber of m-dimensional subspaces of a 2ne-dimensionalvector space over F2,

Qm−1k¼0 ð22ne − 2kÞQm−1k¼0 ð2m − 2kÞ : ðB2Þ

Here, the numerator counts the total number ofbases of an m-dimensional subspace of a vector space ofdimension 2ne, and the denominator counts the numberof bases for each subspace of dimension m. The number of2ne × nS matrices of rank m is given by the product ofEqs. (B1) and (B2),

Qm−1k¼0 ð22ne − 2kÞð2ns − 2kÞQ

m−1k¼0 ð2m − 2kÞ : ðB3Þ

APPENDIX C: INDEPENDENT IDENTICALLYDISTRIBUTED ERASURE ERRORS

In this Appendix, we derive the leading-order RMTsolution for the recovery probability for IID erasure errorswith error rate e. As noted in Sec. II B, the failureprobability for IID errors can be obtained from the failureprobability for the fixed-fraction model with ne ¼ eN.After averaging over all possible ne, there is additionalrounding of the transition due to Poisson fluctuations in thetotal number of erasures. To evaluate the associated finite-size scaling, we make use of the fact that the total numberof erasures is an extensive variable whose fluctuations aregoverned by the central limit theorem

ne ¼ eNþΔ; Δ∼N ð0;σ2Þ; σ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieð1−eÞN

p: ðC1Þ

In averaging over the erasure errors, we can ignore thecritical region for fixed ne because it has a width of about 1that is much less than the typical fluctuations in ne ∼

ffiffiffiffiN

p,

hlog2 PðFÞie ≈Z

∞

Δ0

dΔexpð−Δ2=2σ2Þffiffiffiffiffiffi

2πp

σ2ðΔ0 − ΔÞ; ðC2Þ

where ec ¼ ð1 − RÞ=2 and Δ0 ¼ ðe − ecÞN. After intro-ducing the scaling variable x ¼ ðe − ecÞ

ffiffiffiffiN

p=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieð1 − eÞp

,we find

−hlog2 PðFÞie ¼ffiffiffiffiN

pfðx; eÞ; ðC3Þ

fðx; eÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieð1 − eÞ

p �expð−x2=2Þffiffiffiffiffiffiffiffi

π=2p − xerfc

�xffiffiffi2

p��

; ðC4Þ

where erfcð·Þ is the complementary error function andfðx; eÞ is the scaling function for this random code tran-sition. At the critical point, fð0;ecÞ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ecð1−ecÞ=π

p. This

analysis implies that the critical region after averaging overne has a width scaling as je − ecj ∼ 1=

ffiffiffiffiN

pthat arises from

the width of the probability distribution of ne. Similarly, theaverage log-failure probability at the critical erasure rate ecscales as

ffiffiffiffiN

p.

APPENDIX D: SELF-AVERAGING OF RANDOMSTABILIZER CODE TRANSITION

One of the central assumptions in this work is that thefinite-size scaling behavior of random stabilizer codes nearthreshold well approximates the behavior of the optimalcodes. A necessary condition for this to be true isthat the random codes are self-averaging in the sense thata single realization of a random code has the sameproperties as the average over codes in the large-N limit.To test this self-averaging condition, we investigate theconvergence with N towards the RMT prediction for thecritical recovery probability rc for single realizations.Numerical Monte Carlo results for the standard deviationare shown in Fig. 11. We fix a random Clifford unitary Ugenerated by a high-depth circuit (depth 2N in 1D). For thatcircuit, we then estimate PðRÞ at the critical point of theoptimal codes for the fixed-fraction erasure model. Bygenerating many codes, we can then estimate the varianceEU½PðRÞ − rc�2 through sampling. Over the range of sizesshown in the figure, we see clear exponential decay of thestandard deviation with N, indicating that PðRÞ self-averages to the RMT prediction rc in the large-N limit.

[1] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin,R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao,D. A. Buell et al., Quantum Supremacy Using a Program-

FIG. 11. Fluctuations in the recovery probability over randomcodes vs N for the fixed-fraction erasure model with R ¼ 1=2 ate ¼ ec. The recovery probability for a random code appears to beself-averaging towards the RMT prediction rc at largeN. We takea depth 2N encoding circuit in 1D with a brickwork arrangementof gates. Each two-site gate in the circuit is a random Cliffordgate.


031066-19

mable Superconducting Processor, Nature (London) 574,505 (2019).

[2] J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C.Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried et al.,High-Fidelity Universal Gate Set for 9Beþ Ion Qubits,Phys. Rev. Lett. 117, 060505 (2016).

[3] S. Debnath, N. M. Linke, C. Figgatt, K. A. Landsman, K.Wright, and C. Monroe, Demonstration of a SmallProgrammable Quantum Computer with Atomic Qubits,Nature (London) 536, 63 (2016).

[4] A. D. Córcoles, E. Magesan, S. J. Srinivasan, A. W. Cross,M. Steffen, J. M. Gambetta, and J. M. Chow, Demonstra-tion of a Quantum Error Detection Code Using a SquareLattice of Four Superconducting Qubits, Nat. Commun. 6,6979 (2015).

[5] N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas,B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jianget al., Extending the Lifetime of a Quantum Bit with ErrorCorrection in Superconducting Circuits, Nature (London)536, 441 (2016).

[6] C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, S. V. Isakov,V. Smelyanskiy, A. Megrant, B. Chiaro, A. Dunsworth, K.Arya et al., A Blueprint for Demonstrating QuantumSupremacy with Superconducting Qubits, Science 360,195 (2018).

[7] X. Mi, M. Benito, S. Putz, D. M. Zajac, J. M. Taylor, G.Burkard, and J. R. Petta, A Coherent Spin–Photon Inter-face in Silicon, Nature (London) 555, 599 (2018).

[8] W. Huang, C. H. Yang, K. W. Chan, T. Tanttu, and B.Hensen, Fidelity Benchmarks for Two-Qubit Gates inSilicon, Nature (London) 569, 532 (2019).

[9] H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T.Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuletić, H.Pichler et al., Parallel Implementation of High-FidelityMultiqubit Gates with Neutral Atoms, Phys. Rev. Lett. 123,170503 (2019).

[10] L. Egan, D. M. Debroy, C. Noel, A. Risinger, D. Zhu, D.Biswas, M. Newman, M. Li, K. R. Brown, M. Cetina et al.,Fault-Tolerant Operation of a Quantum Error-CorrectionCode, arXiv:2009.11482.

[11] E. Knill, Quantum Computing with Realistically NoisyDevices, Nature (London) 434, 39 (2005).

[12] S. Bravyi and A. Kitaev, Universal Quantum Computationwith Ideal Clifford Gates and Noisy Ancillas, Phys. Rev. A71, 022316 (2005).

[13] K. M. Svore, D. P. DiVincenzo, and B. M. Terhal, NoiseThreshold for a Fault-Tolerant Two-Dimensional LatticeArchitecture, Quantum Inf. Comput. 7, 297 (2007).

[14] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N.Cleland, Surface Codes: Towards Practical Large-ScaleQuantum Computation, Phys. Rev. A 86, 032324(2012).

[15] Z. Leghtas, G. Kirchmair, B. Vlastakis, R. J. Schoelkopf,M. H. Devoret, and M. Mirrahimi, Hardware-EfficientAutonomous Quantum Memory Protection, Phys. Rev.Lett. 111, 120501 (2013).

[16] M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J.Schoelkopf, L. Jiang, and M. H. Devoret, DynamicallyProtected Cat-Qubits: A New Paradigm for UniversalQuantum Computation, New J. Phys. 16, 045014 (2014).

[17] B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L.Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, andR. J. Schoelkopf, Deterministically Encoding QuantumInformation Using 100-Photon Schrödinger Cat States,Science 342, 607 (2013).

[18] J. M. Martinis, Qubit Metrology for Building a Fault-Tolerant Quantum Computer, npj Quantum Inf. 1, 15005(2015).

[19] R. Harper, S. T. Flammia, and J. J. Wallman, EfficientLearning of Quantum Noise, Nat. Phys. 16, 1184 (2020).

[20] S. T. Flammia and J. J. Wallman, Efficient Estimation ofPauli Channels, ACM Trans. Quantum Comput. 1, 1(2020).

[21] P. Aliferis and J. Preskill, Fault-Tolerant Quantum Com-putation Against Biased Noise, Phys. Rev. A 78, 052331(2008).

[22] S. Puri, L. St-Jean, J. A. Gross, A. Grimm, N. E. Frattini,P. S. Iyer, A. Krishna, S. Touzard, L. Jiang, A. Blais et al.,Bias-Preserving Gates with Stabilized Cat Qubits, Sci.Adv. 6, eaay5901 (2020).

[23] J. Guillaud and M. Mirrahimi, Repetition Cat Qubits forFault-Tolerant Quantum Computation, Phys. Rev. X 9(2019).

[24] J.-P. Tillich and G. Zemor, Quantum LDPC Codes withPositive Rate and Minimum Distance Proportional to n1=2,IEEE Trans. Inf. Theory 60, 1193 (2014).

[25] M. B. Hastings, Decoding in Hyperbolic Spaces: LDPCCodes With Linear Rate and Efficient Error Correction(2013), arXiv:1312.2546; QIC 14, 1187 (2014).

[26] D. Gottesman, Fault-Tolerant Quantum Computation withConstant Overhead, arXiv:1310.2984.

[27] D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, UltrahighError Threshold for Surface Codes with Biased Noise,Phys. Rev. Lett. 120, 050505 (2018).

[28] D. K. Tuckett, A. S. Darmawan, C. T. Chubb, S. Bravyi,S. D. Bartlett, and S. T. Flammia, Tailoring Surface Codesfor Highly Biased Noise, Phys. Rev. X 9, 041031(2019).

[29] D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J.Brown, Fault-Tolerant Thresholds for the Surface Codein Excess of 5% under Biased Noise, Phys. Rev. Lett. 124,130501 (2020).

[30] J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T.Flammia, and B. J. Brown, The XZZX Surface Code, Nat.Commun. 12, 2172 (2021).

[31] C. E. Shannon, A Mathematical Theory of Communica-tion, Bell Syst. Tech. J. 27, 379 (1948).

[32] R. G. Gallager, Low-Density Parity-Check Codes, IRETrans. Inf. Theory 8, 21 (1962).

[33] R. Gallager, The Random Coding Bound Is Tight for theAverage Code, IEEE Trans. Inf. Theory 19, 244(1973).

[34] W. Brown and O. Fawzi, Decoupling with RandomQuantum Circuits, Commun. Math. Phys. 340, 867 (2015).

[35] W. Brown and O. Fawzi, Short Random Circuits DefineGood Quantum Error Correcting Codes, Proc. ISIT 346(2013).

[36] R. Cleve, D. Leung, L. Liu, and C. Wang, Near-LinearConstructions of Exact Unitary 2-Designs, Quantum Inf.Comput. 16, 0721 (2016).


031066-20

https://doi.org/10.1038/s41586-019-1666-5

https://doi.org/10.1038/s41586-019-1666-5

https://doi.org/10.1103/PhysRevLett.117.060505

https://doi.org/10.1038/nature18648

https://doi.org/10.1038/ncomms7979




https://doi.org/10.1126/science.aao4309

https://doi.org/10.1126/science.aao4309


https://doi.org/10.1038/s41586-019-1197-0



https://arXiv.org/abs/2009.11482


https://doi.org/10.1103/PhysRevA.71.022316


https://doi.org/10.26421/QIC7.4-2





https://doi.org/10.1088/1367-2630/16/4/045014

https://doi.org/10.1126/science.1243289

https://doi.org/10.1038/npjqi.2015.5

https://doi.org/10.1038/npjqi.2015.5

https://doi.org/10.1038/s41567-020-0992-8

https://doi.org/10.1145/3408039

https://doi.org/10.1145/3408039



https://doi.org/10.1126/sciadv.aay5901




https://doi.org/10.1109/TIT.2013.2292061








https://doi.org/10.1038/s41467-021-22274-1

https://doi.org/10.1038/s41467-021-22274-1

https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

https://doi.org/10.1109/TIT.1962.1057683

https://doi.org/10.1109/TIT.1962.1057683

https://doi.org/10.1109/TIT.1973.1054971

https://doi.org/10.1109/TIT.1973.1054971

https://doi.org/10.1007/s00220-015-2470-1

https://doi.org/10.1109/ISIT.2013.6620245

https://doi.org/10.1109/ISIT.2013.6620245

https://doi.org/10.26421/QIC16.9-10-1

https://doi.org/10.26421/QIC16.9-10-1

[37] F. G. S. L. Brandao, A. W. Harrow, and M. Horodecki,Local Random Quantum Circuits are ApproximatePolynomial-Designs, Commun. Math. Phys. 346, 397(2016).

[38] A. Harrow and S. Mehraban, Approximate Unitary t-Designs by Short Random Quantum Circuits Using Near-est-Neighbor and Long-Range Gates, arXiv:1809.06957.

[39] N. Delfosse, P. Iyer, and D. Poulin, A Linear-TimeBenchmarking Tool for Generalized Surface Codes,arXiv:1611.04256.

[40] N. Delfosse and G. Zemor, Linear-Time Maximum Like-lihood Decoding of Surface Codes over the QuantumErasure Channel, Phys. Rev. Research 2, 033042 (2020).

[41] Y. Imry and S.-k. Ma, Random-Field Instability of theOrdered State of Continuous Symmetry, Phys. Rev. Lett.35, 1399 (1975).

[42] P. W. Shor, Fault-Tolerant Quantum Computation, arXiv:quant-ph/9605011.

[43] D. Aharonov and M. Ben-Or, Fault-Tolerant QuantumComputation With Constant Error Rate, arXiv:quant-ph/9906129.

[44] D. Gottesman, Fault-Tolerant Quantum Computation withLocal Gates, J. Mod. Opt. 47, 333 (2000).

[45] A. Y. Kitaev, Quantum Computations: Algorithms andError Correction, Russ. Math. Surv. 52, 1191 (1997).

[46] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topo-logical Quantum Memory, J. Math. Phys. (N.Y.) 43, 4452(2002).

[47] C. T. Chubb and S. T. Flammia, Statistical MechanicalModels for Quantum Codes with Correlated Noise, Ann.Henri Poincare D 8, 269 (2021).

[48] G. Duclos-Cianci and D. Poulin, Fast Decoders forTopological Quantum Codes, Phys. Rev. Lett. 104,050504 (2010).

[49] J. R. Wootton and D. Loss, High Threshold ErrorCorrection for the Surface Code, Phys. Rev. Lett. 109,160503 (2012).

[50] A. Hutter, J. R. Wootton, and D. Loss, Efficient MarkovChain Monte Carlo Algorithm for the Surface Code, Phys.Rev. A 89, 022326 (2014).

[51] N. Delfosse and N. H. Nickerson, Almost-Linear TimeDecoding Algorithm for Topological Codes, arXiv:1709.06218.

[52] H. Bombin and M. A. Martin-Delgado, Topological Quan-tum Distillation, Phys. Rev. Lett. 97, 180501 (2006).

[53] H. Bombin and M. A. Martin-Delgado, Topological Com-putation without Braiding, Phys. Rev. Lett. 98, 160502(2007).

[54] H. Bombin, Clifford Gates by Code Deformation, New J.Phys. 13, 043005 (2011).

[55] C. Horsman, A. G. Fowler, S. Devitt, and R. Van Meter,Surface Code Quantum Computing by Lattice Surgery,New J. Phys. 14, 123011 (2012).

[56] B. J. Brown, K. Laubscher, M. S. Kesselring, and J. R.Wootton, Poking Holes and Cutting Corners to AchieveClifford Gates with the Surface Code, Phys. Rev. X 7,021029 (2017).

[57] P. Webster and S. D. Bartlett, Fault-Tolerant QuantumGates with Defects in Topological Stabilizer Codes, Phys.Rev. A 102, 022403 (2020).

[58] B. J. Brown, A Fault-Tolerant Non-Clifford Gate for theSurface Code in Two Dimensions, Sci. Adv. 6, eaay4929(2020).

[59] S. Bravyi, D. Poulin, and B. Terhal, Tradeoffs for ReliableQuantum Information Storage in 2D Systems, Phys. Rev.Lett. 104, 050503 (2010).

[60] D. Bacon, Operator Quantum Error-Correcting Subsys-tems for Self-Correcting Quantum Memories, Phys. Rev. A73, 012340 (2006).

[61] A. Leverrier, J.-P. Tillich, and G. Zemor, QuantumExpander Codes, IEEE 56th Annual Symposium onFoundations of Computer Science (2015), pp. 810–824.

[62] A. Bolt, D. Poulin, and T. M. Stace, Decoding Schemes forFoliated Sparse Quantum Error-Correcting Codes, Phys.Rev. A 98, 062302 (2018).

[63] A. A. Kovalev and L. P. Pryadko, Fault Tolerance ofQuantum Low-Density Parity Check Codes withSublinear Distance Scaling, Phys. Rev. A 87, 020304(R) (2013).

[64] O. Fawzi, A. Grospellier, and A. Leverrier, ConstantOverhead Quantum Fault-Tolerance with QuantumExpander Codes, arXiv:1808.03821.

[65] C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P.Maunz, L.-M. Duan, and J. Kim, Large-Scale ModularQuantum-Computer Architecture with Atomic Memoryand Photonic Interconnects, Phys. Rev. A 89, 022317(2014).

[66] K. A. Landsman, Y. Wu, P. H. Leung, D. Zhu, N. M. Linke,K. R. Brown, L. Duan, and C. Monroe, Two-QubitEntangling Gates within Arbitrarily Long Chains ofTrapped Ions, Phys. Rev. A 100, 022332 (2019).

[67] H. J. Kimble, The Quantum Internet, Nature (London) 453,1023 (2008).

[68] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill,Holographic Quantum Error-Correcting Codes: Toy Mod-els for the Bulk/Boundary Correspondence, J. High EnergyPhys. 06 (2015) 149.

[69] R. J. Harris, N. A. McMahon, G. K. Brennen, and T. M.Stace, Calderbank-Shor-Steane Holographic QuantumError-Correcting Codes, Phys. Rev. A 98, 052301(2018).

[70] P. Hayden and J. Preskill, Black Holes as Mirrors:Quantum Information in Random Subsystems, J. HighEnergy Phys. 09 (2007) 120.

[71] N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P.Hayden, Towards the Fast Scrambling Conjecture, J. HighEnergy Phys. 04 (2013) 022.

[72] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaosin Quantum Channels, J. High Energy Phys. 02 (2016)004.

[73] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, QuantumEntanglement Growth under Random Unitary Dynamics,Phys. Rev. X 7, 031016 (2017).

[74] A. Nahum, S. Vijay, and J. Haah, Operator Spreadingin Random Unitary Circuits, Phys. Rev. X 8, 021014(2018).

[75] C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, andS. L. Sondhi, Operator Hydrodynamics, OTOCs, andEntanglement Growth in Systems without ConservationLaws, Phys. Rev. X 8, 021013 (2018).


031066-21

https://doi.org/10.1007/s00220-016-2706-8

https://doi.org/10.1007/s00220-016-2706-8



https://doi.org/10.1103/PhysRevResearch.2.033042



https://arXiv.org/abs/quant-ph/9605011




https://doi.org/10.1080/09500340008244046

https://doi.org/10.1070/RM1997v052n06ABEH002155

https://doi.org/10.1063/1.1499754

https://doi.org/10.1063/1.1499754

https://doi.org/10.4171/AIHPD/105














https://doi.org/10.1088/1367-2630/13/4/043005

https://doi.org/10.1088/1367-2630/13/4/043005

https://doi.org/10.1088/1367-2630/14/12/123011





















https://doi.org/10.1007/JHEP06(2015)149




https://doi.org/10.1088/1126-6708/2007/09/120

https://doi.org/10.1088/1126-6708/2007/09/120









[76] A. Nahum, J. Ruhman, and D. A. Huse, Dynamics ofEntanglement and Transport in One-Dimensional Systemswith Quenched Randomness, Phys. Rev. B 98, 035118(2018).

[77] V. Khemani, A. Vishwanath, and D. A. Huse, OperatorSpreading and the Emergence of Dissipative Hydro-dynamics under Unitary Evolution with ConservationLaws, Phys. Rev. X 8, 031057 (2018).

[78] T. Rakovszky, F. Pollmann, and C.W. von Keyserlingk,Diffusive Hydrodynamics of Out-of-Time-Ordered Corre-lators with Charge Conservation, Phys. Rev. X 8, 031058(2018).

[79] A. Chan, A. De Luca, and J. T. Chalker, Solution of aMinimal Model for Many-Body Quantum Chaos, Phys.Rev. X 8, 041019 (2018).

[80] T. Zhou and A. Nahum, Emergent Statistical Mechanics ofEntanglement in Random Unitary Circuits, Phys. Rev. B99, 174205 (2019).

[81] F. G. S. L. Brandao and M. Horodecki, Exponential Quan-tum Speed-ups Are Generic, Quantum Inf. Comput. 13,0901 (2013).

[82] S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N.Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H.Neven, Characterizing Quantum Supremacy in Near-TermDevices, Nat. Phys. 14, 595 (2018).

[83] A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, Onthe Complexity and Verification of Quantum RandomCircuit Sampling, Nat. Phys. 15, 159 (2019).

[84] R. Movassagh, Quantum Supremacy and RandomCircuits, arXiv:1909.06210.

[85] J. M. Deutsch, Quantum Statistical Mechanics in a ClosedSystem, Phys. Rev. A 43, 2046 (1991).

[86] M. Srednicki, Entropy and Area, Phys. Rev. Lett. 71, 666(1993).

[87] R. Nandkishore and D. A. Huse, Many-Body Localizationand Thermalization in Quantum Statistical Mechanics,Annu. Rev. Condens. Matter Phys. 6, 15 (2015).

[88] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol,From Quantum Chaos and Eigenstate Thermalization toStatistical Mechanics and Thermodynamics, Adv. Phys.65, 239 (2016).

[89] Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno Effectand the Many-Body Entanglement Transition, Phys. Rev.B 98, 205136 (2018).

[90] B. Skinner, J. Ruhman, and A. Nahum, Measurement-Induced Phase Transitions in the Dynamics of Entangle-ment, Phys. Rev. X 9, 031009 (2019).

[91] Y. Li, X. Chen, and M. P. A. Fisher, Measurement-DrivenEntanglement Transition in Hybrid Quantum Circuits,Phys. Rev. B 100, 134306 (2019).

[92] A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith,Unitary-Projective Entanglement Dynamics, Phys. Rev. B99, 224307 (2019).

[93] M. J. Gullans and D. A. Huse, Dynamical PurificationPhase Transition Induced by Quantum Measurements,Phys. Rev. X 10, 041020 (2020).

[94] S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quantum ErrorCorrection in Scrambling Dynamics and Measurement-Induced Phase Transition, Phys. Rev. Lett. 125, 030505(2020).

[95] X. Cao, A. Tilloy, and A. D. Luca, Entanglement in aFermion Chain under Continuous Monitoring, SciPostPhys. 7, 24 (2019).

[96] M. Szyniszewski, A. Romito, and H. Schomerus,Entanglement Transition from Variable-Strength WeakMeasurements, Phys. Rev. B 100, 064204 (2019).

[97] Y. Bao, S. Choi, and E. Altman, Theory of the PhaseTransition in Random Unitary Circuits with Measure-ments, Phys. Rev. B 101, 104301 (2020).

[98] C.-M. Jian, Y.-Z. You, R. Vasseur, and A.W.W. Ludwig,Measurement-Induced Criticality in Random QuantumCircuits, Phys. Rev. B 101, 104302 (2020).

[99] M. J. Gullans and D. A. Huse, Scalable Probes of Meas-urement-Induced Criticality, Phys. Rev. Lett. 125, 070606(2020).

[100] A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrishnan,D. A. Huse, and J. H. Pixley, Critical Properties of theMeasurement-Induced Transition in Random QuantumCircuits, Phys. Rev. B 101, 060301(R) (2020).

[101] L. Zhang, J. A. Reyes, S. Kourtis, C. Chamon, E. R.Mucciolo, and A. E. Ruckenstein, Nonuniversal Entangle-ment Level Statistics in Projection-Driven QuantumCircuits, Phys. Rev. B 101, 235104 (2020).

[102] Q. Tang and W. Zhu, Measurement-Induced Phase Tran-sition: A Case Study in the Nonintegrable Model byDensity-Matrix Renormalization Group Calculations,Phys. Rev. Research 2, 013022 (2020).

[103] Y. Li, X. Chen, A. W.W. Ludwig, and M. P. A. Fisher,Conformal Invariance and Quantum Non-locality inHybrid Quantum Circuits, arXiv:2003.12721.

[104] Y. Fuji and Y. Ashida, Measurement-Induced QuantumCriticality under Continuous Monitoring, Phys. Rev. B102, 054302 (2020).

[105] A. Lavasani, Y. Alavirad, and M. Barkeshli,Measurement-Induced Topological Entanglement Transitions in Sym-metric Random Quantum Circuits, Nat. Phys. 17, 342(2021).

[106] S. Sang and T. H. Hsieh,Measurement Protected QuantumPhases, Phys. Rev. Research 3, 023200 (2021).

[107] M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A.Huse, and V. Khemani, Entanglement Phase Transitionsin Measurement-Only Dynamics, Phys. Rev. X 11, 011030(2021).

[108] O. Alberton, M. Buchhold, and S. Diehl, TrajectoryDependent Entanglement Transition in a Free FermionChain—From Extended Criticality to Area Law, Phys.Rev. Lett. 126, 170602 (2021).

[109] O. Lunt and A. Pal, Measurement-Induced EntanglementTransitions in Many-Body Localized Systems, Phys. Rev.Research 2, 043072 (2020).

[110] D. Rossini and E. Vicari,Measurement-Induced Dynamicsof Many-Body Systems at Quantum Criticality, Phys. Rev.B 102, 035119 (2020).

[111] S. Goto and I. Danshita, Measurement-Induced Transi-tions of the Entanglement Scaling Law in Ultracold Gaseswith Controllable Dissipation, Phys. Rev. A 102, 033316(2020).

[112] J. Lopez-Piqueres, B. Ware, and R. Vasseur, Mean-FieldEntanglement Transitions in Random Tree TensorNetworks, Phys. Rev. B 102, 064202 (2020).


031066-22

https://doi.org/10.1103/PhysRevB.98.035118









https://doi.org/10.26421/QIC13.11-12-1

https://doi.org/10.26421/QIC13.11-12-1

https://doi.org/10.1038/s41567-018-0124-x

https://doi.org/10.1038/s41567-018-0318-2





https://doi.org/10.1146/annurev-conmatphys-031214-014726

https://doi.org/10.1080/00018732.2016.1198134

https://doi.org/10.1080/00018732.2016.1198134










https://doi.org/10.21468/SciPostPhys.7.2.024

https://doi.org/10.21468/SciPostPhys.7.2.024












https://doi.org/10.1038/s41567-020-01112-z

https://doi.org/10.1038/s41567-020-01112-z













[113] O. Shtanko, Y. A. Kharkov, L. P. García-Pintos, and A. V.Gorshkov, Classical Models of Entanglement in MonitoredRandom Circuits, arXiv:2004.06736.

[114] S. Vijay, Measurement-Driven Phase Transition within aVolume-Law Entangled Phase, arXiv:2005.03052.

[115] N. Lang and H. P. Büchler, Entanglement Transition in theProjective Transverse Field Ising Model, Phys. Rev. B102, 094204 (2020).

[116] A. Nahum, S. Roy, B. Skinner, and J. Ruhman, Measure-ment and Entanglement Phase Transitions in All-to-AllQuantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory, PRX Quantum 2, 010352 (2021).

[117] R. Fan, S. Vijay, A. Vishwanath, and Y.-Z. You, Self-Organized Error Correction in Random Unitary Circuitswith Measurement, Phys. Rev. B 103, 174309 (2021).

[118] Y. Li and M. P. A. Fisher, Statistical Mechanics ofQuantum Error-Correcting Codes, Phys. Rev. B 103,104306 (2021).

[119] L. Fidkowski, J. Haah, and M. B. Hastings, How Dynami-cal Quantum Memories Forget, Quantum 5, 382 (2021).

[120] J. Napp, R. L. La Placa, A.M. Dalzell, F. G. S. L. Brandao,and A.W. Harrow, Efficient Classical Simulation of Ran-dom Shallow 2D Quantum Circuits, arXiv:2001.00021.

[121] P. Hayden, M. Horodecki, A. Winter, and J. Yard, ADecoupling Approach to the Quantum Capacity, OpenSyst. Inf. Dyn. 15, 7 (2008).

[122] C. T. Chubb, V. Y. F. Tan, and M. Tomamichel, ModerateDeviation Analysis for Classical Communication overQuantum Channels, Commun. Math. Phys. 355, 1283(2017).

[123] M. Tomamichel, M. Berta, and J. M. Renes, QuantumCoding with Finite Resources, Nat. Commun. 7, 11419(2016).

[124] D. P. DiVincenzo, P. W. Shor, and J. A. Smolin, Quantum-Channel Capacity of Very Noisy Channels, Phys. Rev. A57, 830 (1998).

[125] G. Smith and J. A. Smolin, Degenerate Quantum Codesfor Pauli Channels, Phys. Rev. Lett. 98, 030501 (2007).

[126] The failure probability PIIDðFÞ for IID random erasurescan be found from the fixed-fraction failure probabilityPðFÞ and the probability pIIDðneÞ of erasure number ne inthe IID model. Since ne is known to the decoder, PðFÞ ¼PIIDðFjneÞ is effectively a conditional distribution, i.e.,PIIDðFÞ ¼

PnePðFÞpIIDðneÞ.

[127] C. H. Bennett, D. P. DiVincenzo, and J. A. Smolin,Capacities of Quantum Erasure Channels, Phys. Rev.Lett. 78, 3217 (1997).

[128] A. W. Harrow and R. A. Low, Random Quantum CircuitsAre Approximate 2-Designs, Commun. Math. Phys. 291,257 (2009).

[129] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information, 10th ed. (CambridgeUniversity Press, New York, NY, 2011).

[130] B. Schumacher, Sending Entanglement through NoisyQuantum Channels, Phys. Rev. A 54, 2614 (1996).

[131] M. Horodecki, P. Horodecki, and R. Horodecki, GeneralTeleportation Channel, Singlet Fraction, and Quasidistil-lation, Phys. Rev. A 60, 1888 (1999).

[132] M. A. Nielsen, A Simple Formula for the Average GateFidelity of a Quantum Dynamical Operation, Phys. Lett. A303, 249 (2002).

[133] B. Schumacher and M. A. Nielsen, Quantum DataProcessing and Error Correction, Phys. Rev. A 54,2629 (1996).

[134] B. Schumacher and M. D. Westmoreland, ApproximateQuantum Error Correction, arXiv:quant-ph/0112106.

[135] S. Lloyd, Capacity of the Noisy Quantum Channel, Phys.Rev. A 55, 1613 (1997).

[136] I. Devetak and P.W. Shor, The Capacity of a QuantumChannel for Simultaneous Transmission of Classical andQuantum Information, Commun. Math. Phys. 256, 287(2005).

[137] M. M. Wilde, Quantum Information Theory (CambridgeUniversity Press, Cambridge, England, 2017).

[138] D. Gottesman, The Heisenberg Representation of Quan-tum Computers, arXiv:quant-ph/9807006.

[139] S. Aaronson and D. Gottesman, Improved Simulation ofStabilizer Circuits, Phys. Rev. A 70, 052328 (2004).

[140] See Ch. 7 of lecture notes by J. Preskill, http://theory.caltech.edu/~preskill/ph229/.

[141] K. A. S. Abdel-Ghaffar, Counting Matrices over FiniteFields Having a Given Number of Rows of Unit Weight,Linear Algebra Appl. 436, 2665 (2012).

[142] A. M. Dalzell, N. Hunter-Jones, and F. G. S. L. Brandao,Random Quantum Circuits Anti-concentrate in Log Depth,arXiv:2011.12277.

[143] M. E. Fisher, Walks, Walls, Wetting, and Melting, J. Stat.Phys. (N.Y.) 34, 667 (1984).

[144] R. Raussendorf, S. Bravyi, and J. Harrington, Long-RangeQuantum Entanglement in Noisy Cluster States, Phys. Rev.A 71, 062313 (2005).

[145] Y.-J. Han, R. Raussendorf, and L.-M. Duan, Scheme forDemonstration of Fractional Statistics of Anyons in anExactly Solvable Model, Phys. Rev. Lett. 98, 150404(2007).


031066-23





https://doi.org/10.1103/PRXQuantum.2.010352




https://doi.org/10.22331/q-2021-01-17-382


https://doi.org/10.1142/S1230161208000043

https://doi.org/10.1142/S1230161208000043

https://doi.org/10.1007/s00220-017-2971-1

https://doi.org/10.1007/s00220-017-2971-1








https://doi.org/10.1007/s00220-009-0873-6

https://doi.org/10.1007/s00220-009-0873-6



https://doi.org/10.1016/S0375-9601(02)01272-0

https://doi.org/10.1016/S0375-9601(02)01272-0






https://doi.org/10.1007/s00220-005-1317-6

https://doi.org/10.1007/s00220-005-1317-6



http://theory.caltech.edu/%7Epreskill/ph229/



https://doi.org/10.1016/j.laa.2011.08.049


https://doi.org/10.1007/BF01009436

https://doi.org/10.1007/BF01009436





Quantum Coding with Low-Depth Random Circuits

Documents