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

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

Scott Aaronson

Parts based on 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: For reasons having nothing to do with building QCs!

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)

Advantages of the new results:

Based on more “generic” complexity assumptions than classical hardness of FACTORING

Give evidence that QCs have capabilities outside the entire polynomial hierarchy

Use only extremely weak kinds of QC (e.g. nonadaptive linear optics)

Today: New kinds of hardness results for simulating quantum mechanics

Disadvantages:

Most apply to sampling problems (or problems with many possible valid outputs), rather than decision problems

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

Problems not “useful” (?)

Example of a PH problem:

“For all n-bit strings x, does there exist an n-bit string y such that for all n-bit strings z, (x,y,z) holds?”

What Is The Polynomial Hierarchy?

NPNPNP NPNPNP

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

is complexity-ese for

“such-and-such would be almost as insane as P=NP”

BQP vs. PH: A Timeline

Bernstein and Vazirani define BQP

They construct an oracle problem, RECURSIVE FOURIER SAMPLING, that has quantum query

complexity n but classical query complexity n(log n) First example where quantum is superpolynomially better!

A simple extension yields RFSMA

Natural conjecture: RFSPH

Alas, we can’t even prove RFSAM!

19901995

20002005

2010

There exist oracle sampling and relational problems in BQP that are not in BPPPH

Unconditionally, there exists an oracle decision problem that requires (N1/4) queries classically (even using postselection), but only 1 query quantumly

Results In The Oracle WorldFrom arXiv:0910.4698

Assuming the “Generalized Linial-Nisan Conjecture,” there exists an oracle decision problem in BQP but not in PH

Original Linial-Nisan Conjecture was recently proved by Braverman, after being open for 20 years

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

Indeed, even if such a distribution can be sampled in BPPPH, still PH collapses.

Suppose the output distribution of any linear-optics circuit can even be approximately sampled in BPP. Then a BPPNP machine can additively approximate Per(X), with high probability over a matrix X of i.i.d. N(0,1) Gaussians.

“Permanent-of-Gaussians Conjecture”: The above problem is#P-complete.

Results In The “Real” WorldFrom not-yet-arXived joint work with Alex Arkhipov

FOURIER FISHING ProblemGiven oracle access to a random Boolean function

1,11,0: nf

The Task:

Output strings z1,…,zn, at least 75% of which satisfy

and at least 25% of which satisfy

nx

xz

nxfzf

1,02/

12

1:ˆwhere

1ˆ izf

2ˆ izf

FOURIER FISHING Is In BQP

Algorithm:

H

H

H

H

H

H

f

|0

|0

|0

Repeat n times; output whatever

you see

Distribution over Fourier coefficients

Distribution over Fourier coefficients output by quantum algorithm

FOURIER FISHING Is Not In PHBasic Strategy: Suppose an oracle problem is in PH. Then by reinterpreting every quantifier as an OR gate, and every quantifier as an AND gate, we can get an AC0 (constant-depth, unbounded-fanin, quasipolynomial-size) circuit for an “exponentially-scaled down” version of the problem

And AC0 circuits are one of the few things in complexity theory that we can actually lower-bound! In particular, it was proved in the 1980s that any AC0 circuit for MAJORITY (or for computing a Fourier coefficient) must have exponential size

Problem: In our case, the AC0 circuit C doesn’t have to compute the Fourier coefficients—it just has to sample from some probability distribution defined in terms of them!

To deal with that, we use a nondeterministic reduction (which adds more layers to the circuit), to show that C would nevertheless lead to an AC0 circuit for MAJORITY

Decision Version: FOURIER CHECKINGGiven oracle access to two Boolean functions

1,11,0:, ngf

Decide whether

if,g are drawn from the uniform distribution U, or

iif,g are drawn from the following “forrelated” distribution F: pick a random unit vector ,2nv

then let

xx vxgvxf ˆsgn:,sgn:

FOURIER CHECKING Is In BQP

H

H

H

H

H

H

f

|0

|0

|0

g

H

H

H

Probability of observing |0n:

forrelated are if1

random are if21

2

12

1,0,3 f,g

f,gygxf

n

yx

yx

nn

Evidence That FOURIER CHECKING PHWe can prove that, even after you condition on any

setting for any polynomial number of f(x)’s and g(y)’s, you still have “almost” no information about whether f and g are independent or forrelated

We conjecture that this property, by itself, is enough to imply an oracle problem is not in PH. We call this the Generalized Linial-Nisan Conjecture

The original Linial-Nisan Conjecture—the same statement, but without the “almost”—was proved last year by Braverman, in a major breakthrough in complexity theory (indirectly inspired by this work )

Coming back to the first result, what’s surprising is that we showed hardness of a BQP sampling problem, by using a nondeterministic reduction from MAJORITY—a “#P” problem!

This raises a question: is something similar possible in the unrelativized (non-black-box) world?Indeed it is. Consider the following problem:

QSAMPLING: Given a quantum circuit C, which acts on n qubits initialized to the all-0 state. Sample from C’s output distribution.

Suppose QSAMPLINGBPP. Then P#P=BPPNP

(so in particular, PH collapses to the third level)

Result/Observation:

Why QSAMPLING Is Hard

2

1,022

1:

nxn

xfp

Let f:{0,1}n{-1,1} be any efficiently computable function. Suppose we apply the following quantum circuit:

H

H

H

H

H

H

f

|0

|0

|0

Then the probability of observing the all-0 string is

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

Proof: If we can estimate p, then we can also compute xf(x) using binary search and padding

Claim 2: Suppose QSAMPLINGBPP. Then we could estimate p in BPPNP

Proof: Let M be a classical algorithm for QSAMPLING, and let r be its randomness. Use approximate counting to estimate

Conclusion: Suppose QSAMPLINGBPP. Then P#P=BPPNP

nr

rM 0 outputs Pr

2

1,022

1:

nxn

xfp

Related ResultsA. 2004: PostBQP=PP

Bremner, Jozsa, Shepherd (poster #1): PostIQP=PP, hence efficient simulation of IQP collapses PH

Fenner, Green, Homer, Pruim 1999: Determining whether a quantum circuit accepts with nonzero probability is hard for PH

Ideally, we want a simple, explicit quantum system Q, such that any classical algorithm that even approximately simulates Q would have dramatic consequences for classical complexity theory

We believe this is possible, using non-interacting bosons

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…

Starting from a fixed basis state (like |=|1,…,1,0,…0), you get to choose an arbitrary mm unitary U to apply

U induces an unitary V 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 photons m = # of “modes” (boxes) that each photon can be in

!!!!

PerTVS

11

,

mm

TS

ttss

U

n

nm

n

nm 11

Then you get to measure V| in the computational basis

where US,T is an nn submatrix of U indexed by S,T(containing an sitj block of uij’s for each i,j)

Theorem (Lloyd 1996 et al.): BosonP BQPProof Idea: Decompose U into a product of

O(m2) “elementary linear-optics gates” (beamsplitters and phase-shifters), then

simulate each gate using standard qubit gates

Theorem (Knill, Laflamme, Milburn 2001): Linear optics with adaptive measurements

can do all of BQPBy contrast, we’ll use just a single (nonadaptive) measurement of the photon numbers at the end

U

Our 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 BPP 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 X of i.i.d. N(0,1) complex Gaussians, to additive error with high probability over X.

,!1On

n

Permanent-of-Gaussians Conjecture: This problem

is #P-complete

PGCHardness of BOSONSAMPLINGIdea: Given a Gaussian random matrix X, we’ll “smuggle” X into the unitary transition matrix U for m=O(n2) photons—in such a way that S|V|=Per(X), 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 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(X)|2

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

Difficulty: The “bosonic birthday paradox”! Identical bosons like to pile on top of each other, and that’s bad for us

SO WE DEAL WITH IT

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

• Reliable photodetector arraysBut crucially, no nonlinear optics or postselected measurements!

Our Proposal: Concentrate on (say) n=30 photons, so that classical simulation is difficult but not impossible

Prize ProblemsProve the Generalized Linial-Nisan Conjecture!Yields an oracle A such that BQPAPHA

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

$200

140Fr

Do a linear optics experiment that overthrows the Polynomial-Time Church-Turing Thesis ?