efficient simulation of quantum mechanics collapses the polynomial hierarchy
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Efficient Simulation of Quantum Mechanics Collapses the Polynomial Hierarchy. (yes, really). Scott Aaronson Alex Arkhipov MIT. In 1994, something big happened in our field, whose meaning is still debated today…. Why exactly was Shor’s algorithm important? - PowerPoint PPT PresentationTRANSCRIPT
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
1,0221:
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
1,0221:
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 CDDCould 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
12
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
aaA
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.
2/12/12/12/1
2/12/12/12/1
A
81
2/12/12/12/1
Per!2
13 mode togo bosonsboth Pr2
Example:
41
2/12/12/12/1
Perare they stay where bosonsPr2
0
2/12/12/12/1
Permode oneshift bosonsPr2
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
2 Per1 outputs Pr Ym
SrM 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! ,
32box 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