When Exactly Do Quantum Computers Provide A Speedup? Scott Aaronson MIT.

When Exactly Do Quantum Computers Provide A Speedup?

Scott AaronsonMIT

“It’s been 20 years since Shor’s factoring algorithm. Where are all the amazing new quantum algorithms we were promised?”

Who promised you more quantum algorithms? Not me!

…Is that all? What else is there?

Quantum simulationFactoring

Grover search Adiabatic alg / quantum walks

+ A few other things…

The Parallelism Fallacy

Fueling the belief that countless more quantum algorithms should exist (or that are not finding them is a failure), seems to be the idea that a quantum computer could just “try every possible answer in parallel” (modulo some technical details)

But we’ve understood since the early 90s that that’s not how quantum algorithms work! You need to choreograph an interference pattern, where the unwanted paths cancel

The miracle, I’d say, is that this trick yields a speedup for any classical problems, not that it doesn’t work for more of them

Underappreciated challenge of quantum algorithms research: beating 60 years of classical algorithms research

An Inconvenient Truth

If we set aside NP-complete problems, there just aren’t that many compelling candidates left for exponential quantum speedups! (And for many of those, we do

have exponential speedups, and for many of the rest we have polynomial ones)

P

NP

NP-completeNP-hard

BQP(Quantum P)

Facto

ring

Graph Iso

Quant

um Si

m

3SAT

Lattice Problems

P≠BQP, NPBQP: Plausible conjectures,

which we have no hope of proving given

the current state of complexity theory

Rest of the Talk

I. Survey of the main families of quantum algorithms that have been discovered (and their limitations)

II. Results in the black-box model, which aim toward a general theory of when quantum speedups are possible

Quantum Simulation“What a QC does in its sleep”

The “original” application of QCs!

My personal view: still the most important one

Major applications (high-Tc superconductivity, protein folding, nanofabrication, photovoltaics…)

High confidence in possibility of a quantum speedup

Can plausibly realize even before universal QCs are available

Suppose we just want a quantum system for which there’s good complexity-theoretic evidence that it’s hard to simulate classically—we don’t care what it’s useful for

BosonSampling

Our proposal: Identical single photons sent through network of interferometers, then measured at output modes

A.-Arkhipov 2011, Bremner-Jozsa-Shepherd 2011: In that case, we can plausibly improve both the hardware requirements and the evidence for classical hardness, compared to Shor’s factoring algorithm

We showed: if a fast, classical exact simulation of

BosonSampling is possible, then the polynomial hierarchy

collapses to the third level.

Experimental demonstrations with 3-4

photons achieved (by groups in Oxford,

Brisbane, Rome, Vienna)

“The magic of the Fourier transform”

Shor-like AlgorithmsInteresting

In BQP: Pretty much anything you can think of that reduces to finding hidden structure in abelian groups

Factoring, discrete log, elliptic curve problems, Pell’s equation, unit groups, class groups, Simon’s problem…

Breaks almost all public-key cryptosystems used today

But theoretical public-key systems exist that are unaffected

Attempt to go further: Hidden Subgroup Framework

Can Shor’s algorithm be generalized to nonabelian groups?

Hidden Subgroup ProblemGiven: A finite group G, a function f:GZ such that f(x)=f(y) iff x,y belong to the same coset of a “hidden subgroup” H≤G

Problem: Find generators for H

But if G is nonabelian, “interpreting the results” of those queries could still be extremely hard!

Classically, this problem could require queries to f

Ettinger-Høyer-Knill 1997: Quantumly, logO(1)(|G|) queries always suffice

G~

Example Application:

Graph Isomorphism ≤HSP over the symmetric group

Alas, nonabelian HSP has been the Afghanistan of quantum algorithms!

Grover-like Algorithms

Bennett et al. 1997: For black-box searching, the square-root speedup of Grover’s algorithm is the best possible

Quadratic speedup for any problem involving searching an unordered list, provided the list elements can be queried in superposition

Implies subquadratic speedups for many other basic problems

Quantum Walk AlgorithmsChilds et al. 2003: Quantum walks can achieve provable exponential speedups over any classical algorithm (in query complexity), but for extremely “fine-tuned” graphs

Also lots of polynomial

speedups—e.g., Element Distinctness

in O(N2/3) time (Ambainis 2003)

THE GLUED TREES

Quantum Adiabatic Algorithm(Farhi et al. 2000)

HiHamiltonian with easily-prepared ground state

HfGround state encodes solution

to NP-complete problem

Problem: “Eigenvalue gap” can be exponentially small

LandscapeologyAdiabatic algorithm can find global minimum exponentially faster than simulated annealing (though maybe other classical algorithms do better)

Simulated annealing can find global minimum exponentially faster than adiabatic algorithm (!)

Simulated annealing and adiabatic algorithm both need exponential time to find global minimum

Quantum Machine Learning Algorithms

THE FINE PRINT:

1.Don’t get solution vector explicitly, but only as vector of amplitudes. Need to measure to learn anything!

2.Dependence on condition number could kill exponential speedup

3.Need a way of loading huge amounts of data into quantum state (which, again, could kill exponential speedup)

4.Not ruled out that there are fast randomized algorithms for the same problems (even just considering query complexity)

‘Exponential quantum speedups’ for solving linear systems, support vector machines, Google PageRank, computing Betti numbers…

“But you just listed a bunch of examples where you know a quantum speedup, and

other examples where you don’t! What you guys need is a theory, which would tell

you from first principles when quantum speedups are possible.”

The Quantum Black-Box ModelThe setting for much of what we know about the power of

quantum algorithms

fx f(x)

An algorithm can make query transformations, which map

as well as arbitrary unitary transformations that don’t depend on f (we won’t worry about their computational cost).

wxfaxwax waxwax ,,,, ,,,, (x=“query register,” a=“answer register,” w=“workspace”)

Its goal is to learn some property F(f) (for example: is f 1-to-1?)

“Query complexity” of F: The minimum number of queries used by any

algorithm that outputs F(f), with high probability, for every f of interest to us

Total Boolean Functions

1,01,0: NF

Example: NORQNORRORD NNN ~,

Theorem (Beals et al. 1998): For all Boolean functions F,

6FQOFD

How to reconcile with the exponential speedup of Shor’s algorithm? Totality.

Longstanding Open Problem: Is there any Boolean function with a quantum quantum/classical gap better than quadratic?

D(F): Deterministic query complexity of FR(F): Randomized query complexityQ(F): Quantum query complexity

Almost-Total Functions?

Conjecture (A.-Ambainis 2011): Let Q be any quantum algorithm that makes T queries to an input X{0,1}N.

Then there’s a classical randomized that makes poly(T,1/,1/) queries to X, and that approximates Pr[Q accepts X] to within on a ≥1- fraction of X’s

Theorem (A.-Ambainis): This would follow from an extremely natural conjecture in discrete Fourier analysis (“every bounded low-degree polynomial p:{0,1}N[0,1] has a highly influential variable”)

The Collision ProblemGiven a 2-to-1 function f:{1,…,N}{1,…,N}, find a collision (i.e., two inputs x,y such that f(x)=f(y))

Variant: Promised that f is either 2-to-1 or 1-to-1, decide which

Models the breaking of collision-resistant hash functions—a central problem in cryptanalysis

“More structured than Grover search, but less structured than Shor’s period-finding problem”

10 4 1 8 7 9 11 5 6 4 2 10 3 2 7 9 11 5 1 6 3 8Interesting

Birthday Paradox: Classically, ~N queries are necessary and sufficient to find a collision with high probability

Brassard-Høyer-Tapp 1997: Quantumly, ~N1/3 queries suffice

Grover on N2/3 f(x) values

N1/3 f(x) values queried classically

A. 2002: First quantum lower bound for the collision problem (~N1/5 queries are needed; no exponential speedup possible)

Shi 2002: Improved lower bound of ~N1/3. Brassard-Høyer-Tapp’s algorithm is the best possible

Symmetric Problems

New Results (Ben-David 2014): If F:SN{0,1} is any Boolean function of permutations, then D(F)=O(Q(F)12). If F is any function with a symmetric promise, and at most M possible results of each query, then R(F)=O(Q(F)12(M-1)).

A.-Ambainis 2011: Massive generalization of collision lower bound. If F is any function whatsoever that’s symmetric under permuting the inputs and outputs, and has sufficiently many outputs (like collision, element distinctness, etc.), then

FQFQOFR logpoly7

Upshot: Need a “structured” promise if you want an exponential quantum speedup

What’s the largest possible quantum speedup?

Period-finding: O(1) quantum queries, ~N1/4 classical queries

Simon’s problem, the glued-trees problem: O(log N) quantum queries, ~N classical queries

Forrelation (A. 2009): Given two Boolean functions f,g:{0,1}n{-1,1}, estimate how correlated g is with the Fourier transform of f:

?3/2

?3/11

2

1

1,0,2/3

nyx

yx

nygxf

Can we do even better?

Examplef(0000)=-1f(0001)=+1f(0010)=+1f(0011)=+1f(0100)=-1f(0101)=+1f(0110)=+1f(0111)=-1f(1000)=+1f(1001)=-1f(1010)=+1f(1011)=-1f(1100)=+1f(1101)=-1f(1110)=-1f(1111)=+1

g(0000)=+1g(0001)=+1g(0010)=-1g(0011)=-1g(0100)=+1g(0101)=+1g(0110)=-1g(0111)=-1g(1000)=+1g(1001)=-1g(1010)=-1g(1011)=-1g(1100)=+1g(1101)=-1g(1110)=-1g(1111)=+1

H

H

H

H

H

H

f

|0

|0

|0

g

H

H

H

Trivial 2-query quantum algorithm for Forrelation!

(Can even improve to 1 query using standard tricks)

A.-Ambainis 2014: By contrast, any classical randomized algorithm to solve Forrelation needs at least N / logN queries

Furthermore, this separation is optimal: any problem solvable with k quantum queries, is also solvable with ~N1-1/2k classical randomized queries

Our Conjecture: The above is tight for all k. A generalization of Forrelation involving k Boolean functions achieves it.

Summary

Single most important application of QC (in my opinion): Disproving the people who said QC was impossible!

Exponential quantum speedups depend on structure

For example, abelian group structure, glued-trees structure, forrelational structure…

After 20 years of quantum algorithms research, we know a lot about which kinds of structure sufficeThe black-box model lets us make formal statements about what kinds of structure don’t suffice for exponential speedups

In both cases, of course, many open problems remain

Sometimes we can even find such structure in real, non-black-box problems of practical interest (e.g., factoring)