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New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov
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New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Mar 26, 2015

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Page 1: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers

Scott Aaronson (MIT)

Joint work with Alex Arkhipov

Page 2: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Computer Scientist / Physicist Nonaggression Pact

You tolerate these complexity classes:

P NP BPP BQP #P PH

And I don’t inflict these on you:

AM AWPP BQP/qpoly MA P/poly PSPACE QCMA QIP QMA SZK YQP

Page 3: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

In 1994, something big happened in the foundations of computer science, whose meaning

is still debated today…

Why exactly was Shor’s algorithm important?

Boosters: Because it means we’ll build QCs!

Skeptics: Because it means we won’t build QCs!

Me: Even for reasons having nothing to do with building QCs!

Page 4: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Shor’s algorithm was a hardness result for one of the central computational problems

of modern science: QUANTUM SIMULATION

Shor’s Theorem:

QUANTUM SIMULATION is not in

probabilistic polynomial time,

unless FACTORING is also

Use of DoE supercomputers by area (from a talk by Alán Aspuru-Guzik)

Page 5: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Advantages:

Based on a more “generic” complexity assumption than the hardness of FACTORING

Gives evidence that QCs have capabilities outside the entire polynomial hierarchy

Only involves linear optics! (With single-photon Fock state inputs, and nonadaptive multimode photon-detection measurements)

Today: A new kind of hardness result for simulating quantum mechanics

Disadvantages:

Applies to relational problems (problems with many possible valid outputs) or sampling problems, not to decision problems

Harder to convince a skeptic that your QC is indeed solving the relevant hard problem

Less relevant for the NSA

Page 6: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Before We Go Further, A Bestiary of Complexity Classes…

BQP

P#P

BPP

P

NP

PH

FACTORIN

G

PERMANENT

COUNTING

3SAT

XYZ…

How complexity theorists say “such-and-such is damn unlikely”:

“If such-and-such is true, then PH collapses to a finite level”

Page 7: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Suppose the output distribution of any linear-optics circuit can be efficiently sampled classically (e.g., by Monte Carlo). Then the polynomial hierarchy collapses (indeed P#P=BPPNP).

Indeed, even if such a distribution can be sampled by a classical computer with an oracle for the polynomial hierarchy, still the polynomial hierarchy collapses.

Suppose the output distribution of any linear-optics circuit can even be approximately sampled efficiently classically. Then in BPPNP, one can nontrivially approximate the permanent of a matrix of independent N(0,1) Gaussian entries (with high probability over the choice of matrix).

“Permanent-of-Gaussians Conjecture” (PGC): The above problem is #P-complete (i.e., as hard as worst-case PERMANENT)

Our Results

If the PGC is true, then even a noisy linear-optics experiment can sample from a probability distribution that no classical computer can feasibly

sample from, unless the polynomial hierarchy collapses

Page 8: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Related WorkKnill, Laflamme, Milburn 2001: Linear optics with adaptive measurements yields universal QC

Valiant 2002, Terhal-DiVincenzo 2002: Noninteracting fermions can be simulated in P

A. 2004: Quantum computers with postselection on unlikely measurement outcomes can solve hard counting problems (PostBQP=PP)

Shepherd, Bremner 2009: “Instantaneous quantum computing” can solve sampling problems that might be hard classically

Bremner, Jozsa, Shepherd 2010: Efficient simulation of instantaneous quantum computing would collapse PH

Page 9: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

nS

n

iiiaA

1,Per

BOSONS

nS

n

iiiaA

1,

sgn1Det

FERMIONS

There are two basic types of particle in the universe…

Their transition amplitudes are given respectively by…

All I can say is, the bosons got the harder job

Particle Physics In One Slide

Page 10: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Starting from a fixed initial state—say, |I=|1,…,1,0,…0— you get to choose any mm mode-mixing unitary U

U induces an unitary (U) on n-photon

states, defined by

Linear Optics for Dummies (or computer scientists)

Computational basis states have the form |S=|s1,…,sm, where s1,…,sm are nonnegative integers such that s1+…+sm=n

n = # of identical photons m = # of modes For us, m>n

!!!!

PerTUS

11

,

mm

TS

ttss

U

n

nm

n

nm 11

Then you get to measure (U)|I in the computational basis

Here US,T is an nn matrix obtained by taking si copies of the ith row of U and tj copies of the jth column, for all i,j

Page 11: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Theorem (Feynman 1982, Abrams-Lloyd 1996): Linear-optics computation can be simulated in BQPProof Idea: Decompose the mm unitary U into a product of O(m2) elementary “linear-optics gates” (beamsplitters and phaseshifters), then simulate each gate using polylog(n) standard qubit gates

Theorem (Gurvits): There exist classical algorithms to approximate S|(U)|T to additive error in randomized poly(n,1/) time, and to compute the marginal distribution on photon numbers in k modes in nO(k) time

Theorem (Bartlett-Sanders et al.): If the inputs are Gaussian states and the measurements are homodyne, then linear-optics computation can be simulated in P

Upper Bounds on the Power of Linear Optics

Page 12: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

By contrast, exactly sampling the distribution over all n photons is extremely hard! Here’s why …

222Per: AIUIp n

Given any matrix ACnn, we can construct an mm mode-mixing unitary U (where m2n) as follows:

Suppose we start with |I=|1,…,1,0,…,0 (one photon in each of the first n modes), apply (U), and measure.

Then the probability of observing |I again is

DC

BAU

Page 13: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Claim 1: p is #P-complete to estimate (up to a constant factor)

Idea: Valiant proved that the PERMANENT is #P-complete.

Can use known (classical) reductions to go from a multiplicative approximation of |Per(A)|2 to Per(A) itself.

Claim 2: Suppose we had a fast classical algorithm for linear-optics sampling. Then we could estimate p in BPPNP

Idea: Let M be our classical sampling algorithm, and let r be its randomness. Use approximate counting to estimate

Conclusion: Suppose we had a fast classical algorithm for linear-optics sampling. Then P#P=BPPNP.

IrMr

outputs Pr

222Per: AIUIp n

Page 14: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

High-Level IdeaEstimating a sum of exponentially many positive or negative numbers: #P-complete

Estimating a sum of exponentially many nonnegative numbers: Still hard, but known to be in BPPNP PH

If quantum mechanics could be efficiently simulated classically, then these two problems would become equivalent—thereby placing #P in PH, and collapsing PH

So why aren’t we done?

Because real quantum experiments are subject to noise

Would an efficient classical algorithm that sampled from a noisy distribution still collapse the polynomial hierarchy?

Page 15: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

U

Main Result: Take a system of n identical photons with m=O(n2) modes. Put each photon in a known mode, then apply a Haar-random mm unitary transformation U:

Let D be the distribution that results from measuring the photons. Suppose there’s a fast classical algorithm that takes U as input, and samples any distribution even 1/poly(n)-close to D in variation distance. Then in BPPNP, one can estimate the permanent of a matrix A of i.i.d. N(0,1) complex Gaussians, to additive error with high probability over A. ,

!1On

n

Permanent-of-Gaussians Conjecture (PGC): This

problem is #P-complete

Page 16: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

PGC Hardness of Linear-Optics SamplingIdea: Given a Gaussian random matrix A, we’ll “smuggle” A into the unitary transition matrix U for m=O(n2) photons—in such a way that S|(U)|I=Per(A), for some basis state |S

Useful fact we rely on: given a Haar-random mm unitary matrix, an nn submatrix looks approximately Gaussian

Then the classical sampler has “no way of knowing” which submatrix of U we care about—so even if it has 1/poly(n) error, with high probability it will return |S with probability |Per(A)|2

Then, just like before, we can use approximate counting to estimate Pr[|S]|Per(A)|2 in BPPNP, and thereby solve a #P-complete problem

Page 17: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Problem: Bosons like to pile on top of each other!

Call a basis state S=(s1,…,sm) good if every si is 0 or 1 (i.e.,

there are no collisions between photons), and bad otherwise

If bad basis states dominated, then our sampling algorithm might “work,” without ever having to solve a hard PERMANENT instance

Furthermore, the “bosonic birthday paradox” is even worse than the classical one!

,3

2box same in the land particlesboth Pr

rather than ½ as with classical particles

Fortunately, we show that with n bosons and mkn2 modes, the probability of a collision is still at most (say) ½

Page 18: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Experimental ProspectsWhat would it take to implement the requisite experiment?• Reliable phase-shifters and beamsplitters, to implement an arbitrary unitary on m photon modes• Reliable single-photon sources

Fock states, not coherent states

• Photodetector arrays that can reliably distinguish 0 vs. 1 photonBut crucially, no nonlinear optics or postselected measurements!

Our Proposal: Concentrate on (say) n30 photons and

m1000 modes, so that classical simulation is

difficult but not impossible

Page 19: New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson (MIT) Joint work with Alex Arkhipov.

Open Problems

Prove the Permanent of Gaussians Conjecture!Would imply that even approximate classical simulation of linear-optics circuits would collapse PH

140Fr

Do a linear-optics experiment that solves a classically-intractable sampling problem! ?

What are the exact resource requirements? E.g., can our experiment be done using a log(n)-depth linear-optics circuit?

Are there other quantum systems for which approximate classical simulation would collapse PH?