When Exactly Do Quantum Computers Provide A Speedup?

Scott Aaronson (MIT)Papers & slides at www.scottaaronson.com

“We all hear about the experimental progress toward building quantum

computers … but in the meantime, what about the applications? It’s been 20 years

since Peter Shor discovered his famous factoring algorithm. Where are all the

amazing new applications we were promised?”

Who promised you more quantum algorithms? Not me!

Genesis of This Talk

The Parallelism FallacyWhat’s the source of the popular belief that countless more quantum algorithms should exist?

But that’s not how quantum computing works! 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

To me, it seems tied to the idea that a quantum computer could just “try every possible answer in parallel”

An Inconvenient Truth

A problem has to be special even to be a plausible candidate for an exponential quantum speedup

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

III. Lemons into lemonade: implications for physics of the limitations of quantum computers

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

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

Breaks almost all public-key cryptosystems used todayBut theoretical public-key systems exist that are unaffected

Can we go further? Hidden Subgroup ProblemGeneralizes Shor to nonabelian groups. Captures e.g. Graph Isomorphism

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 classical walks, but for extremely “fine-tuned” graphs

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

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

Suppose we just want a quantum system for which there’s good 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)

For more: My complex quantum systems seminar tomorrow

“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

i xi

An algorithm can make query transformations, which map

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

wxaiwai iwaiwai ,,,, ,,,, (i=“query register,” a=“answer register,” w=“workspace”)

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

X“Query complexity” of f: The minimum

number of queries used by any algorithm that outputs f(X), with high

probability, for every X of interest to us

X=x1…xN

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 of f.

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 8

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 Result (Ben-David 2014): If f:SN{0,1} is any Boolean function of permutations, then D(f)=O(Q(f)12)

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?

“Forrelation”: 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

A.-Ambainis 2014: This problem is solvable using only 1 quantum query, but requires at least ~2n/2/n queries classically

Furthermore, this separation is essentially the largest possible! Any N-bit problem that’s solvable with k quantum queries, is also solvable with ~N1-1/2k classical queries

For details: My CS theory seminar on Friday

Can we turn the lemon of QCs’ limitations into the lemonade of physical insight?

Proposal: Adopt as a principle (conjecture?) that there’s no efficient way to solve NP-complete problems in the physical world, then investigate the implications for other issues

Example Implications:- No closed timelike curves (A.-Watrous 2009)- No postselected final state (probably rules out Horowitz-Maldacena)

- Something like the holographic entropy bound should hold- Metastable states must be unavoidable in spin glasses, protein folding, etc.- Many spectral gaps must decrease exponentially with number of particles

“Explanation” for the linearity of the Schrödinger equation

Abrams & Lloyd 1998: If quantum mechanics were nonlinear, one could generically exploit that to solve NP-complete problems in polynomial time

No solutions1 solution to NP-complete problem

A complexity-theoretic argument against hidden variables?

A. 2004: In theories like Bohmian mechanics, in order to sample the entire trajectory of the hidden variable, you’d need the ability to solve the collision problem—something I showed is generically hard even for a quantum computer

2

yx

N

x

xfxN 1

1Measure 2nd

register

xf

The Firewall Paradox (AMPS 2012): Refinement of Hawking’s information paradox that

challenges black hole complementarityIf the black hole interior is “built” out of the same qubits coming out as Hawking radiation, then why can’t we do something to those Hawking qubits, then dive into the black hole, and see that we’ve completely destroyed the spacetime geometry in the interior?

Entanglement among Hawking photons detected!

Harlow-Hayden 2013: Striking argument that doing the AMPS experiment would require solving a problem that’s

exponentially hard even for a quantum computer

nx

HBRHBRnRBHxgxxfx

1,01

11,00,2

1

R: “Old” Hawking photonsB: Hawking photon just now coming outH: Degrees of freedom still in black hole

MODEL SITUATION:

“So, long before you’ve made a dent in the problem, the black hole has already evaporated anyway, and there’s nowhere to jump to see a firewall!”

f,g: Two functions for which we want to know whether their ranges are equal or disjoint

If we could detect entanglement between R and B for any |RBH, then we could solve a close cousin of the collision problem!

A. 2014: Strengthened the Harlow-Hayden argument, to show that a general ability to

perform the AMPS experiment would imply the ability to invert any cryptographic one-way

function

Is the geometry of spacetime protected by an armor of computational complexity?

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…

The black-box model lets us develop a rich theory of what kinds of structure do or don’t suffice for exponential speedups

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

Understanding the limitations of quantum computers has given us new insights about seemingly-remote issues in physics