Efficient Simulation of Quantum Mechanics Collapses the Polynomial Hierarchy Scott Aaronson Alex Arkhipov MIT (yes, really)

Efficient Simulation of Quantum Mechanics Collapses the

Polynomial Hierarchy

Scott Aaronson Alex ArkhipovMIT

(yes, really)

In 1994, something big happened in our field, 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

BPP, unless FACTORING is also

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

Advantages of our result:

Based on P#PBPPNP rather than FACTORINGBPP

Applies to an extremely weak subset of QC(“Non-interacting bosons,” or linear optics with a single nonadaptive measurement at the end)

Even gives evidence that QCs have capabilities outside PH

Today: A completely different kind of hardness result for simulating quantum mechanics

Disadvantages:

Applies to distributional and relation problems, not to decision problems

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

Let C be a quantum circuit, which acts on n qubits initialized to the all-0 state

CDD

Certainly this problem is BQP-hard

C

|0

|0

|0

QSAMPLING: Given C as input, sample a string x from any probability distribution D such that

C defines a distribution DC over n-bit output strings

More generally:Suppose QSAMPLING0.01 is in probabilistic polytime with A oracle. Then P#PBPPNP So QSAMPLING can’t even be in BPPPH without collapsing PH!

A

Our Result: Suppose QSAMPLING0.01 is in probabilistic polytime. Then P#P=BPPNP

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

Extension to relational problems:Suppose FBQP=FBPP. Then P#P=BPPNP

“QSAMPLING is #P-hard under BPPNP-reductions”(Provided the BPPNP machine gets to pick the random bits used by the QSAMPLING oracle)

Warmup: Why Exact 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)

Related to my result that PostBQP=PP

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

Claim 2: Suppose QSAMPLING was classically easy. 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 QSAMPLING0 is easy. Then P#P=BPPNP

nr

rM 0 outputs Pr

2

1,022

1:

nxn

xfp

So Why Aren’t We Done?Ultimately, our goal is to show that Nature can actually perform computations that are hard to simulate classically, thereby overthrowing the Extended Church-Turing Thesis

But any real quantum system is subject to noise—meaning we can’t actually sample from DC, but only from some distribution D such that CDD

Could that be easy, even if sampling from DC itself was hard?

To rule that out, we need to show that even a fast classical algorithm for QSAMPLING would imply P#P=BPPNP

The ProblemSuppose M “knew” that all we cared about was the final amplitude of |00

(i.e., that’s where we shoehorned a hard #P-complete instance)

Then it could adversarially choose to be wrong about that one, exponentially-small amplitude and still be a good sampler

So we need a quantum computation that more “robustly” encodes a #P-complete problem

Hmm … robust #P-complete problem … you mean like the PERMANENT?

Indeed. But to bring the permanent into quantum computing, we need a brief detour

into particle physics (!)

We’ll have to work harder … but as a bonus, we’ll not only rule out approximate

samplers, but approximate samplers for an extremely weak kind of QC

Particle Physics In One SlideThere are two types of particles in Nature…

BOSONSForce-carriers: photons, gluons…

Swap two identical bosons quantum state | is unchanged

Bosons can “pile on top of each other” (and do: lasers, Bose-

Einstein condensates…)

FERMIONSMatter: quarks, electrons…

Swap two identical fermions quantum state picks up -1 phase

Pauli exclusion principle: no two fermions can occupy same state

Consider a system of n identical, non-interacting particles…

1

2

3

1

2

3

Let aijC be the amplitude for transitioning from initial state i to final state j

Then what’s the total amplitude for the above process?

APer ADet

nnn

n

aa

aa

A

1

111

:

if the particles are bosons if they’re fermions

Let

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

tinitial tfinal

The BOSONSAMPLING ProblemInput: An mn complex matrix A, whose n columns are orthonormal vectors in Cm (here mn2)

Let a configuration be a list S=(s1,…,sm) of nonnegative integers with s1+…+sm=n

Task: Sample each configuration S with probability

!!

Per:

1

2

m

SS ss

Ap

Neat Fact: The pS’s sum to 1

where AS is an nn matrix containing si copies of the ith row of A

Physical Interpretation: We’re simulating a unitary evolution of n identical bosons, each of which can be in m=poly(n) “modes.” Initially, modes 1 to n have one boson each and modes n+1 to m are unoccupied. After applying the unitary, we measure the number of bosons in each mode.

2/12/1

2/12/1

2/12/1

2/12/1

A

8

1

2/12/1

2/12/1Per

!2

13 mode togo bosonsboth Pr

2

Example:

4

1

2/12/1

2/12/1Perare they stay where bosonsPr

2

0

2/12/1

2/12/1Permode oneshift bosonsPr

2

Theorem (implicit in Lloyd 1996): BOSONSAMPLING QSAMPLING

Proof Sketch: We need to simulate a system of n bosons on a conventional quantum computer

The basis states |s1,…,sm (s1+…+sm=n) just record the occupation number of each mode

Given any “scattering matrix” UCmm on the m modes, we can decompose U as a product U1…UT, where T=O(m2) and each Ut acts only on 2-dimensional subspaces of the formmjim ssssss ,,1,,1,,,,, 11

for some (i,j)

Theorem (Valiant 2001, Terhal-DiVincenzo 2002): FERMIONSAMPLINGBPP

In stark contrast, we prove the following:

Suppose BOSONSAMPLINGBPP. Then given an arbitrary matrix XCnn, one can approximate |Per(X)|2 in BPPNP

But I thought we could approximate the permanent in BPP

anyway, by Jerrum-Sinclair-Vigoda!

Yes, for nonnegative matrices.

For general matrices, approximating |Per(X)|2 is #P-complete.

Outline of ProofGiven a matrix XCnn , with every entry satisfying |xij|1, we want to approximate |Per(X)|2 to within n!

This is already #P-complete (proof: standard padding tricks)

Notice that |Per(X)|2 is a degree-2n polynomial in the entries of X (as well as their complex conjugates)

As in Lipton/LFKN, we can let V be some random curve in Cnn that passes through X, and let Y1,…,YkCnn be other matrices on V (where kn2)

If we can estimate |Per(Yi)|2 for most i, then we can estimate |Per(X)|2 using noisy polynomial interpolation

But Linear Interpolation Doesn’t Work!

We need to redo Lipton/LFKN to work over the complex numbers rather than finite fields

A random line through XCnn “retains too much information” about X

X

Solution: Choose a matrix Y(t) of random trigonometric polynomials, such that Y(0)=X

ijij

Lti

ijij xyety

0,:1

2

Questions: How do we sample Y(t) and Y1,…,Yk efficiently? How do we do the noisy polynomial interpolation?

Lazy answer: Since we’re a BPPNP machine, just use rejection sampling!

For sufficiently large L and t>>0, each yij(t) will look like an independent Gaussian, uncorrelated with xij:

Furthermore, Per(Y(t)) is a univariate polynomial in e2it of degree at most Ln

The problem reduces to estimating |Per(Y)|2, for a matrix YCnn of (essentially) independent N(0,1) Gaussians

To do this, generate a random mn column-orthonormal matrix A that contains Y/m as an nn submatrix

(i.e., such that AS=Y/m for some random configuration S)

Let M be our BPP algorithm for approximate BOSONSAMPLING, and let r be M’s randomness

Use approximate counting (in BPPNP) to estimate

Intuition: M has no way to determine which configuration S we care about. So if it’s right about most configurations, then w.h.p. we must have

SrMr

outputs Pr

2

2Per

1 outputs Pr Y

mSrM

nr

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

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

there are no collisions between bosons), and bad otherwise

We assumed for simplicity that all configurations were good

But suppose bad configurations dominated. Then M could be wrong on all good configurations, yet still “work”

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 boxes, the probability of a collision is still at most (say) ½

Experimental ProspectsWhat would it take to implement BOSONSAMPLING with photonics?• Reliable phase-shifters• Reliable beamsplitters• Reliable single-photon sources• Reliable photodetectorsBut crucially, no nonlinear optics or postselected measurements!Problem: The output will be a collection of nn matrices B1,…,Bk with “unusually large permanents”—but how would a classical skeptic verify that |Per(Bi)|2 was large?

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

Open ProblemsDoes our result relativize? (Conjecture: No)

Can we use BOSONSAMPLING to do universal QC? Can we use it to solve any decision problem outside BPP?

Can you convince a skeptic (who isn’t a BPPNP machine) that your QC is indeed doing BOSONSAMPLING?

Can we get unlikely complexity collapses from P=BQP or PromiseP=PromiseBQP?

Would a nonuniform sampling algorithm (one that was different for each scattering matrix A) have unlikely complexity consequences?

Is PERMANENT #P-complete for +1/-1 matrices (with no 0’s)?

ConclusionI like to say that we have three choices: either

(1)The Extended Church-Turing Thesis is false,

(2)Textbook quantum mechanics is false, or

(3)QCs can be efficiently simulated classically.For all intents and purposes