Experimental Comparison of Two Quantum Computing …used to e ect gates between arbitrary qubits using the connections available. For a general circuit, reducing a fully-connected

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Experimental Comparison of Two Quantum Computing Architectures

N. M. Linke,1 D. Maslov,2, 3 M. Roetteler,4 S. Debnath,1 C.

Figgatt,1 K. A. Landsman,1 K. Wright,1 and C. Monroe1, 3, 5

1Joint Quantum Institute and Department of Physics,University of Maryland, College Park, MD 20742

2National Science Foundation, Arlington, VA 222303Joint Center for Quantum Information and Computer Science,

University of Maryland, College Park, MD 207424Microsoft Research, Redmond, WA 98052

5IonQ, Inc., College Park, MD 20742

We run a selection of algorithms on two state-of-the-art 5-qubit quantum computers that are basedon different technology platforms. One is a publicly accessible superconducting transmon device [1]with limited connectivity, and the other is a fully connected trapped-ion system [2]. Even though thetwo systems have different native quantum interactions, both can be programmed in a way that isblind to the underlying hardware, thus allowing the first comparison of identical quantum algorithmsbetween different physical systems. We show that quantum algorithms and circuits that employ moreconnectivity clearly benefit from a better connected system of qubits. While the quantum systemshere are not yet large enough to eclipse classical computers, this experiment exposes critical factorsof scaling quantum computers, such as qubit connectivity and gate expressivity. In addition, theresults suggest that co-designing particular quantum applications with the hardware itself will beparamount in successfully using quantum computers in the future.

Inspired by the vast computing power a universal quan-tum computer could offer, several candidate systemsare being explored. They have allowed experimentaldemonstrations of quantum gates, operations, and algo-rithms of ever increasing sophistication. Recently, two ar-chitectures, superconducting transmon qubits [3–7] andtrapped ions [2, 8], have reached a new level of matu-rity. They have become fully programmable multi-qubitmachines that provide the user with the flexibility to im-plement arbitrary quantum circuits from a high-level in-terface. This makes it possible for the first time to testquantum computers irrespective of their particular phys-ical implementation.

While the quantum computers considered here are stillsmall scale and their capabilities do not currently reachbeyond small demonstration algorithms, this line of in-quiry can still provide useful insights into the perfor-mance of existing systems and the role of architecturein quantum computer design. These will be crucial forthe realization of more advanced future incarnations ofthe present technologies.

The standard abstract model of quantum computa-tion assumes that interactions between arbitrary pairsof qubits are available. However, physical architectureswill in general have certain constraints on qubit connec-tivity, such as nearest-neighbor couplings only. These re-strictions do not in principle limit the ability to performarbitrary computations, since SWAP operations may beused to effect gates between arbitrary qubits using theconnections available. For a general circuit, reducing afully-connected system to the more sparse star-shaped orlinear nearest-neighbor connectivity requires an increasein the number of gates of O(n), where n is the number

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FIG. 1. Graphic representations of the two systems: (a) thesuperconducting qubits connected by microwave resonators(Credit: IBM Research), and (b) the linear chain of trappedions connected by laser-mediated interactions. Insets: Qubitconnectivity graphs, (a) star-shaped and (b) fully connected.

of qubits [9]. How much overhead is incurred in practicedepends on the connections used in a particular circuitand how efficiently they can be matched to the physicalqubit-to-qubit interaction graph.

In this article, we make use of the public access re-cently granted by IBM to a 5-qubit superconducting de-vice (illustrated in fig.1(a)) via their “Quantum Experi-ence” cloud service [1]. This allows us to repeat algo-rithms that we perform in our own ion trap experimenton an independent quantum computer of identical sizeand comparable capability but with a different physicalimplementation at its core.

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PHYSICAL SYSTEMS

The ion trap system consists of five 171Yb+ ions whichare confined in a linear Paul trap and laser-cooled closeto their motional ground state (see fig.1(b)) [2]. Thequbits are magnetic field-insensitive pairs of states in thehyperfine-split 2S1/2 ground-level of each atom, whichgives a qubit frequency of 12.642821 GHz. All controland measurement is performed optically. State prepara-tion and readout are accomplished by optical pumpingand state-dependent fluorescence detection [10]. Qubitoperations are realized via pairs of Raman beams, de-rived from a single 355 nm mode-locked laser. These op-tical controllers consist of an array of individual address-ing beams and a counter-propagating global beam thatilluminates the entire chain [2]. Single-qubit rotationsare driven by a Raman beat-note of defined amplitude,phase, and duration resonant with the qubit frequency.Two-qubit operations are produced by applying Ramanbeams to a pair of ions, with beat-note frequencies nearthe motional sidebands. This creates an effective XX-Ising interaction between the spins mediated by all modesof motion [11–13]. We use a pulse-shaping scheme toensure spin and motion are disentangled at the end ofthe operation [14, 15]. Since all ions partake in the col-lective motion of the chain, gates between any pair canbe invoked in this way (see inset of fig.1(b)). The ad-dressing during operations and the distinction betweenqubits during readout are both achieved by spatially re-solving the ions. The fidelities for single- and two-qubitgates are typically 99.1(5)% and 97(1)%, respectively.The single-qubit readout fidelity is 99.7(1)% for state |0〉,and 99.1(1)% for state |1〉. The latter is lower since off-resonant excitation during readout predominantly causes|1〉 → |0〉 pumping. The average readout fidelity for anentire 5-qubit state is 95.7(1)%. This is lower than onewould expect from the average single-qubit readout fi-delity, since there is crosstalk that leads to |0〉 → |1〉errors on adjacent channels. Typical gate times are 20µsfor single- and 250 µs for two-qubit gates. Spin depo-larization is negligible for hyperfine ground level qubits(T1 ∼ ∞). The spin-dephasing time (T ∗2 ) is ∼ 0.5s in thecurrent setup, and can be easily extended by suppressingmagnetic field noise.

In analogy to atoms given by nature, the man-madesuperconducting circuits in the IBM quantum computercan be thought of as “artificial atoms” [17]. They aretransmon qubits [18], or superconducting islands con-nected by Josephson junctions and shunt capacitors thatprovide superpositions of charge states which are insen-sitive to charge fluctuations. The device used here has arange of qubit frequencies between 5 and 5.4 GHz [19].The qubits are connected to each other and the classicalcontrol system by microwave resonators. State prepa-ration [20] and readout, as well as single- [21] and two-

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FIG. 2. High level circuits of the implemented example com-putations (gates defined in [16]): Margolus gate (a), Tof-foli gate (b), Bernstein-Vazirani (c), and hidden shift (d).The Bernstein-Vazirani algorithm is shown for the oraclec = (1111), where all CNOTs are present. The hidden shiftdiagram represents the shift pattern s = (1011), where X-operations are present on qubits 1, 3 and 4.

qubit gates [22], are achieved by applying tailored mi-crowave signals to this network and measuring the re-sponse. Qubits are resolved in the frequency domainduring addressing and readout. In the Quantum Ex-perience hardware, the qubits are connected in a star-shaped pattern that provides four two-qubit interactions(see inset fig.1(a)), which are CNOT gates targeting thecentral qubit. Single-qubit readout fidelities are typically∼ 96% [1], and the average readout fidelity for an arbi-trary 5-qubit state is ∼ 80% [19]. Typical gate fidelitiesare 99.7% and 96.5% for single- and two-qubit gates, re-spectively. Typical gate times are 130 ns for single- and250 − 450 ns for two-qubit gates, while coherence timesare ∼ 60 µs for both depolarization (T1) and spin de-phasing (T2). The publicly accessible system runs au-tonomously, not requiring any human intervention overmany weeks [19]. This level of reliability may come at a

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TABLE I. Single- and two-qubit gate counts for the circuitson the superconducting (star-shaped) and the ion trap (fullyconnected) system after mapping to the respective hardwareusing the respective gate libraries. For comparison, the gatecounts for a linear nearest-neighbor (LNN) architecture asimplemented in [3] are included. We also note the gate countfor the Quantum Fourier Transfrom (QFT) for 3 and 5 qubits.The latter was implemented in [2] using a sequence of modulargates that was not optimized for gate count. The QFT-5cannot be implemented exactly using the current IBM gatelibrary. If we assume Za operations are possible, the countsshown as ∗ are 47 for single- and 29 for two-qubit gates.

connectivity star-shaped LNN fully conn.hardware supercond. ion trap

gate library Clifford+T Clifford+Za R/XXgate type single two single two single two

Margolus 20 3 20 3 11 3Toffoli 17 10 9 10 9 5

Bernstein-Vazirani 10 0-4 10 0-10 14-26 0-4Hidden Shift 28-34 10 20-26 4 42-50 4

QFT-3 42 19 11 7 8 3QFT-5 ∗ ∗ 35 28 22 10

cost due to drifts between periodic calibrations. Higherconnectivity can in general be achieved by coupling 3-4transmons to one resonator, limited by spectral resolu-tion. The present layout could be modified to provideconnections from qubit 1 to 5, and 2 to 4 [19]. Fur-thermore, other superconducting architectures involvingmulti-mode resonators [5] can offer higher connectivity.

On these two machines, we compare a selection of com-posite gates and algorithms that represent a variety ofcircuit connectivities. In each case, we map the algo-rithms to the device by breaking them down into circuitsmade up of gates native to the specific hardware. We relyon an optimization protocol [23] to accomplish this taskfor the trapped ions, and CNOT+T/Za algebra [24] withfurther manual optimization to compose the experimentsfor the IBM machine. The available gate set for the iontrap system consists of the two-qubit XX gate, as wellas arbitrary single-qubit Rθ

α gate rotations by an angle θabout any axis (given by α) on the equator of the Blochsphere. We call this the R/XX library. The IBM systemmakes available the family of gates (X, Y, Z, H, S, CNOTand T [16]), known as the Clifford+T library. Since eachgate is subject to errors, the circuits are optimized tominimize the number of operations used. The resultinggate numbers are optimal for two-qubit gates, and eitheroptimal or close to optimal for single-qubit gates. Thetotal number of single- and two-qubit gates for each al-gorithm is shown in table I. The R/XX library offers abetter overall expressive power. However, we note thatthe Clifford+T library was likely chosen for didactic rea-sons and is not native to superconducting systems, whichdo in principle offer continuous parameters for single- andtwo-qubit gates.

In addition to the two systems considered here, the ta-ble also gives the numbers for a linear nearest-neighbor(LNN) connectivity architecture as used, e.g., in super-conducting qubits [3] as well as semiconductor gatedquantum dots [25]. The numbers in table I show thatthe two-qubit gate count strongly depends on the match-ing between the circuit and the qubit connectivity graph.The LNN architecture is as efficient as the fully connectedsystem for the hidden shift algorithm, while the star-shaped system incurs overheads; the reverse is true forthe Bernstein-Vazirani algorithm (see fig.2).

ALGORITHMS

Margolus and Toffoli Gate

The Toffoli gate is a 3-qubit controlled-controlled-NOTgate that requires 6 CNOT gates [26, 27]. It is possible toimplement a Toffoli with 5 entangling gates if the square-root of the CNOT operation is available [16], which is thecase with the trapped ion XX gate. The Margolus gate isa simplified version of the Toffoli operation, which intro-duces an additional phase on the state corresponding to|100〉. It can be realized with just 3 CNOT gates [28, 29].The circuits are shown in figure 2(a,b). Note that forthe Margolus gate, all entangling operations connect tothe same qubit, which means that this circuit can berealized efficiently with star-shaped qubit connectivity.The systems perform this circuit at success probability74.1(7)% for superconductors and 90.1(2)% for ions (seefigure 3(a1,b1)).

The full Toffoli circuit uses the same three qubits as theMargolus implementation so that preparation and mea-surement errors remain the same. The optimized circuitfor the fully connected ion trap system contains 5 two-qubit gates and the additional operations lower the fi-delity to 85.0(2)% (see figure 3(b2)). For the star-shapedsystem, an additional 7 two-qubit gates are needed toeffect the SWAP operations necessary to go from theMargolus to the full Toffoli gate. This leads to a re-duced success rate of 52.6(8)% for the superconductingsystem (3(a2)). Note that the transformation |a, b, c〉 →|c ⊕ ab, b, a〉 may be obtained with the Clifford+T li-brary on a star-shaped graph with the provably mini-mal number of 7 CNOT gates. We do not consider suchinput-to-output mappings of the composite gates in thiswork. However, we always choose the most favorableinput-to-output mapping for the IBM star and LNN ar-chitectures when executing entire quantum algorithms,which is merely a classical swap between physically mea-sured signals.

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(a1) Margolus: Supercond. (b1) Margolus: Ion Trap

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FIG. 3. Margolus gate results from the star-shaped super-conductor (a1) and the fully connected ion trap system (b1).The fidelities are 74.1(7)% and 90.1(2)%, respectively. Thefull Toffoli gate results give success probabilities of 52.6(8)%for the superconducting (a2) and 85.0(2)% for the ion trap(b2) system. The axes represent states as 3-bit binary num-bers. For each input state, the probabilities of detecting eachstate are shown.

Bernstein-Vazirani and Hidden Shift Algorithms

In the Bernstein-Vazirani algorithm, an oracle imple-ments the function fc(x) = x ·c. The algorithm finds theunknown bit string c in a single shot. In the oracle, c isencoded in a pattern of CNOT gates, all of which targetthe ancilla qubit [30]. As can be seen from the circuit infigure 2(c), the entire algorithm maps well onto a star-shaped architecture. This algorithm is very similar to aparity check circuit used in error correction applications,and indeed the IBM system was laid out with this appli-cation in mind [7]. The single-shot success probabilitiesare 72.8(5)% for the star-shaped superconducting systemand 85.1(1)% for the fully connected ion trap system (fig-ure 4(a1,b1)).

To compare this to a similar algorithm with differ-ent connectivity requirements, we implement the hid-den shift algorithm [31] for a black box bent function[32, 33]. An oracle implements the shifted version f(x+s)of the known Boolean function f . We want to determinethe n-bit string s that constitutes the “hidden shift”.For a subset of Boolean functions, there exists a quan-tum algorithm that can solve this problem in a singleoracle query, while classical algorithms require Ω(

√2n)

queries. This subset contains functions which have a flatFourier spectrum and whose dual f∼ can be calculatedefficiently, i.e. so-called bent functions of the Maiorana-

McFarland class [33]. Here we choose the 4-bit functionf(x) = x1x2 ⊕ x3x4 for which f = f∼. We implementall possible 4-bit shift patterns s using the circuit shownin figure 2(d). The algorithm output state directly cor-responds to the hidden shift s. The circuit involves gatesbetween two disconnected pairs of qubits, which createsan overhead of 6 two-qubit gates for a star-shaped archi-tecture. The results are shown in figure 4(a2,b2). Thefidelity of the fully connected ion trap implementation is77.1(2)%, compared to 35.1(6)% for the superconductingdevice. The numerical values of the data plotted in figure3 and 4 are reproduced as tables in the Appendix.

The errors in both devices appear concentrated in cer-tain sets of states, leading to patterns in the off-diagonalelements of the result plots (see figure 4). These highlystructured signatures suggest that systematic errors dom-inate, especially readout errors. The grouped patternssuch as in figure 4(a1) indicate flips of the least-significantbits, while parallel lines correspond to the most signifi-cant bits changing their state. In the trapped ion re-sults, these lines can be modulated in height due toread-out crosstalk and are more pronounced on the lower-numbered state side due to 1 → 0 being the dominantdetection error channel. Finally, we stress that compar-ing quantum computations across systems depends onthe specifics of error propagation, which will vary be-tween different hardware implementations, through theirparticular connectivity and physical errors. We summa-rize the success probabilities for the implemented cir-cuits on both machines in Table II. We also show theexpected values for two simple error propagation mod-els based on the errors of the individual gates εg and ofM -qubit single-shot readout εM for both systems. Thefirst model assumes random error propagation per oper-

ation with overall error (1 − εM )M (1 − εg)√N , where N

is the number of gates. Since the errors for each stepare independent and comparable to a random walk, theoverall error involves

√N factors. The second model is

based on systematic (coherent) over- or under-rotationswith overall error (1−εM )M (1−εg)N , which accumulateswith N factors. We see that the numbers are broadlyconsistent, with systematic errors better predicting thesuperconducting system while the ion trap performancefalls in between the two. The superconducting HiddenShift algorithm is the only example with a significantlylower experimental result, perhaps from inhomogeneouserrors in the device.

OUTLOOK

Comparing quantum computing architectures involvesmany interrelated factors. Quantum gate operation fi-delities, qubit numbers, primitive gate speeds, and co-herence times are obviously important low-level metricsin a large scale quantum computer. The results pre-

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(a2) Hidden shift: Superconductor (b2) Hidden shift: Ion Trap

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FIG. 4. Results from the Bernstein-Vazirani algorithm implementing the oracle function fc(x) = x0c0⊕c1x1⊕c2x2⊕c3x3 for allpossible 4-bit oracles c performed on the star-shaped (a1) and the fully connected (b1) systems. The average success probabilitiesare 72.8(5)% for the superconductor and 85.1(1)% for the ion trap system. Hidden shift algorithm for f(x) = x0x1 ⊕ x2x3. Allpossible 4-bit shifted oracle functions are implemented on the superconducting system (a2) as well as the ion trap (b2). Theaverage success probabilities are 35.1(6)% and 77.1(2)%, respectively. The axes represent states and oracle parameters as 4-bitbinary numbers.

sented here show higher absolute fidelities and coher-ence times in the trapped ion system, with higher clockspeeds for the superconducting system. However, thesemetrics are moving targets: while these systems are themost advanced and versatile quantum computing plat-forms built to date, both technologies are currently ad-vancing rapidly.

In any case, such metrics should not be considered inisolation. Our comparison points to important higherlevel considerations in scaling a quantum computer. Theoverall performance of a quantum circuit and the “timeto solution” will depend critically on architectural re-strictions, qubit connectivity, gate reconfigurability, andgate expressivity, and these attributes will become evermore important as the system is scaled up. Even with 5-

qubit systems, we find that the qubit connectivity graphis best co-designed to mirror the structure of the par-ticular quantum circuit and that the choice of a moreexpressive gate library affects the efficiency of the com-putations.

The physical scaling of each of these leading technolo-gies has many challenges, and how they will be connectedand reconfigured at large scales is an open question. Oneof the biggest challenges is the management of the con-trol complexity in larger systems and potential cross-talkfrom overlapping qubit interactions or control buses. Inmost superconducting designs, there are many current-carrying wires necessary for control and biasing the indi-vidual qubits, and this may be difficult to route througha large superconducting chip [3–7]. It will likely become

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TABLE II. Summary of the achieved success probabilities forthe implemented circuits, in percent. The observed probabil-ities (“obs”) are tabulated alongside two simple error prop-agation models given the gate number N and the individ-ual gate and readout errors of the two systems encapsulatedin the parameters εg and εM , respectively (see main text).The first estimate assumes random (“rand”) error propaga-

tion with overall error (1 − εg)√

N while the second is basedon systematic (“sys”) coherent over- or under-rotations withoverall error (1 − εg)N , where N is the number of gates. Thereadout error for M qubits is (1 − εM )M in both cases.

connectivity star-shaped fully conn.hardware supercond. ion trap

success prob/% obs rand sys obs rand sys

Margolus 74.1(7) 82 75 90.1(2) 91 81Toffoli 52.6(8) 78 59 85.0(2) 89 78

Bernstein-Vazirani 72.8(5) 80 74 85.1(1) 90 77Hidden Shift 35.1(6) 75 52 77.1(2) 86 57

a great challenge to manage the dilution refrigerator heatbudget with such circuitry. Alternative modular super-conducting architectures improve connectivity by inte-grating qubits with microwave cavity modes, at the ex-pense of significant added volume per qubit [34]. Ion trapdesigns will hinge upon the stable and accurate deliveryof laser beams (or near-field microwave sources) to ad-dress each qubit individually in a vacuum chamber. Thefully connected nature of the ion trap architecture maynot scale to arbitrarily large numbers of qubits, owingto the spectral overlap of collective normal modes of mo-tion. However, full connectivity between 20−100 trappedion qubits appears possible [2] and a modular approachfor scaling to much larger systems with high connectiv-ity and distance-independent operations seems promising[35, 36]. In any hardware, an automated calibration pro-cedure and powerful user interface will likely provide ahigher level of integration. Such system-level attributeswill become even more important as quantum circuitsgrow in complexity, regardless of physical platform.

ACKNOWLEDGEMENTS

We thank D. L. Moehring, J. Kim, and K. R. Brown forkey discussions, and J. Gambetta and J. Chow at IBM fortheir assistance in interfacing with the IBM Quantum Ex-perience project. This work was supported by the AROwith funds from the IARPA LogiQ program, the AFOSRMURI program on Optimal Quantum Circuits, and theNSF Physics Frontier Center at JQI. DM acknowledgessupport by the NSF. Any opinion, finding, and conclu-sions or recommendations expressed in this material arethose of the authors and do not necessarily reflect theviews of the NSF, IBM, or any of their employees.

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APPENDIX

The detailed results from the algorithms presented inFigs. 3-4 are shown below as tables containing numericprobabilities. The target populations, with nominal unitprobabilities, are highlighted in yellow. The others, rep-resenting errors, show a bar graph scaled from 0 to 0.1to emphasize the systematic error patterns.

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MargolusSuperconductor 000 001 010 011 100 101 110 111

000 0.8252 0.0859 0.0713 0.0107 0.1230 0.0186 0.0059 0.0293001 0.0791 0.8086 0.0146 0.0801 0.0156 0.1006 0.0254 0.0029010 0.0322 0.0039 0.7725 0.0947 0.0156 0.0078 0.0215 0.0771011 0.0029 0.0420 0.0576 0.7148 0.0010 0.0205 0.0752 0.0225100 0.0459 0.0088 0.0127 0.0039 0.6953 0.0850 0.0117 0.0771101 0.0039 0.0352 0.0020 0.0254 0.0762 0.7031 0.0605 0.0039110 0.0098 0.0020 0.0576 0.0176 0.0605 0.0088 0.1113 0.7197111 0.0010 0.0137 0.0117 0.0527 0.0127 0.0557 0.6885 0.0674

MargolusIon Trap 000 001 010 011 100 101 110 111

000 0.9180 0.0406 0.0219 0.0110 0.0292 0.0018 0.0010 0.0010001 0.0318 0.9156 0.0067 0.0245 0.0016 0.0288 0.0000 0.0012010 0.0146 0.0052 0.9151 0.0421 0.0004 0.0012 0.0124 0.0192011 0.0066 0.0150 0.0333 0.8954 0.0002 0.0016 0.0228 0.0112100 0.0278 0.0014 0.0007 0.0009 0.9274 0.0274 0.0090 0.0218101 0.0006 0.0210 0.0004 0.0009 0.0192 0.9158 0.0166 0.0124110 0.0006 0.0006 0.0120 0.0119 0.0114 0.0098 0.0376 0.9050111 0.0000 0.0006 0.0099 0.0134 0.0106 0.0136 0.9006 0.0282

ToffoliSuperconductor 000 001 010 011 100 101 110 111

000 0.5107 0.2832 0.0830 0.0215 0.1172 0.0693 0.0430 0.0381001 0.2490 0.4316 0.0156 0.0635 0.1475 0.1182 0.0322 0.0352010 0.0420 0.0264 0.5996 0.1250 0.0127 0.0146 0.0889 0.0654011 0.0234 0.0303 0.1162 0.5996 0.0205 0.0205 0.0762 0.0732100 0.0723 0.1084 0.0469 0.0264 0.4912 0.1152 0.0303 0.0381101 0.0693 0.0674 0.0156 0.0420 0.1260 0.5947 0.0283 0.0322110 0.0176 0.0166 0.0820 0.0625 0.0537 0.0176 0.2197 0.4990111 0.0156 0.0361 0.0410 0.0596 0.0313 0.0498 0.4814 0.2188

ToffoliIon Trap 000 001 010 011 100 101 110 111

000 0.8878 0.0298 0.0497 0.0111 0.0395 0.0089 0.0035 0.0110001 0.0224 0.8762 0.0101 0.0525 0.0091 0.0358 0.0133 0.0029010 0.0370 0.0090 0.8571 0.0385 0.0152 0.0014 0.0265 0.0224011 0.0075 0.0373 0.0374 0.8521 0.0015 0.0149 0.0206 0.0259100 0.0210 0.0082 0.0117 0.0019 0.8486 0.0332 0.0420 0.0436101 0.0083 0.0229 0.0017 0.0126 0.0237 0.8460 0.0350 0.0374110 0.0149 0.0008 0.0154 0.0171 0.0334 0.0247 0.0377 0.8101111 0.0013 0.0158 0.0169 0.0142 0.0290 0.0351 0.8214 0.0467

input state

detected state

input state

detected state

input state

detected state

input state

detected state

FIG. 5. Numerical quantum computer 3-qubit input/output matrix for the Margolis gate (top two panels) and Toffoli gate(bottom two panels), corresponding to Fig. 3 of the main text. For each gate, the results from both superconductor and iontrap quantum computers are displayed.

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FIG. 6. Numerical quantum computer 4-qubit input/output matrix for the Berstein-Vazirani algorithm for the superconductorsystem (top) and ion trap system (bottom), corresponding to Figs. 4(a1) and 4(b1) of the main text.

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13

0.0

13

70

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30

0.0

04

90

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54

0.0

17

60

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37

10

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72

0.0

66

40

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57

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14

60

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31

30

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91

11

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90

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76

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50

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98

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58

60

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36

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10

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61

11

01

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60

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96

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54

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74

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80

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54

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70

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09

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20

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49

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46

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77

10

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56

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20

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76

20

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78

70

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71

50

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93

11

11

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12

70

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73

0.0

16

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13

70

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70

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23

40

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32

20

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89

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18

Hid

de

n s

hif

tIo

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rap

00

00

00

01

00

10

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01

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01

01

10

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11

10

00

10

01

10

10

10

11

11

00

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01

11

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11

11

00

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0.7

73

00

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42

0.0

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60

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82

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75

20

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86

00

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32

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02

60

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16

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21

00

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10

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00

40

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17

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01

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.77

18

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80

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12

0.0

02

00

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06

0.0

00

20

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30

0.0

03

20

.09

16

0.0

00

40

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40

0.0

00

60

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94

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60

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89

20

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46

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01

80

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14

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79

00

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26

0.0

02

70

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08

0.0

86

80

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36

0.0

01

00

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04

0.0

20

80

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28

00

11

0.0

20

00

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26

0.0

18

80

.76

90

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01

80

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20

0.0

02

30

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20

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28

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01

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26

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01

20

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03

60

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82

01

00

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71

60

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18

0.0

02

20

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0.7

63

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22

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48

0.0

24

50

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0.0

01

40

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14

0.0

55

00

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12

0.0

03

20

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20

01

01

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00

60

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30

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64

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46

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80

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33

10

10

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02

40

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26

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20

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25

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22

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50

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64

0.7

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27

10

11

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40

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88

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14

80

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60

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98

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80

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18

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40

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10

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04

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49

60

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18

0.0

02

00

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06

0.0

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30

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12

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01

40

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20

0.7

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34

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20

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01

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01

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0.0

00

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26

0.0

01

40

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18

0.0

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00

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0.0

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50

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56

0.0

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20

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38

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27

40

.77

78

0.0

05

80

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08

11

10

0.0

01

40

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04

0.0

21

80

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14

0.0

01

80

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08

0.0

51

80

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10

0.0

02

70

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08

0.0

58

40

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32

0.0

46

80

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08

0.8

00

60

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62

11

11

0.0

00

40

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22

0.0

01

20

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94

0.0

01

20

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38

0.0

02

80

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94

0.0

01

50

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36

0.0

02

60

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94

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21

60

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16

0.0

30

40

.76

85

ora

cle

s

det

ecte

d

stat

e

ora

cle

s

det

ecte

d

stat

e

FIG. 7. Numerical quantum computer 4-qubit input/output matrix for the Hidden Shift algorithm for the superconductorsystem (top) and ion trap system (bottom), corresponding to Figs. 4(a2) and 4(b3) of the main text.