the computational complexity of linear optics scott aaronson (mit) joint work with alex arkhipov vs
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The Computational Complexity of Linear Optics
Scott Aaronson (MIT)Joint work with Alex Arkhipov
vs
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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!
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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)
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Advantages:
Based on more “generic” complexity assumptions than the hardness of FACTORING
Gives evidence that QCs have capabilities outside the polynomial hierarchy
Only involves linear optics (With single-photon Fock state inputs, and nonadaptive multimode photon-detection measurements)
Today, a different kind of hardness result for simulating quantum mechanics
Disadvantages:
Applies to relational problems (problems with many possible outputs) or sampling problems, not decision problems
Harder to convince a skeptic that your QC is indeed solving the relevant hard problem
Less relevant for the NSA
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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”
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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 two plausible conjectures are true: the permanent of a Gaussian random matrix is(1) #P-hard to approximate, and(2) not too concentrated around 0.Then the output distribution of a linear-optics circuit can’t even be approximately sampled efficiently classically, unless the polynomial hierarchy collapses.
Our Results
If our conjectures hold, then even a noisy linear-optics experiment can
sample from a probability distribution that no classical
computer can feasibly sample from
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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 seem hard classically
Bremner, Jozsa, Shepherd 2010: Efficient simulation of instantaneous quantum computing would collapse PH
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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
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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
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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
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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
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Claim 1: p is #P-complete to estimate (up to a constant factor)
Idea: Valiant proved that the PERMANENT is #P-complete.
Can use a classical reduction 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
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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
Extensions:- Even simulation of QM in PH would imply P#P = PH - Can design a single hard linear-optics circuit for each n- Can let the inputs be coherent rather than Fock states
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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—one that was only 1/poly(n)-close to the true distribution in variation distance—still collapse the polynomial hierarchy?
Difficulty in showing this: The sampler might adversarially neglect to output the one submatrix whose permanent we care about! So we’ll need to “smuggle” the PERMANENT instance we care about into a random submatrix
Main Result: Yes, assuming two plausible conjectures about random permanents (the “PGC” and the “PCC”)
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There exist ε,δ ≥ 1/poly(n) for which the following problem is #P-hard. Given a Gaussian random matrix X drawn from N(0,1)C
n×n, output a complex number z such that
with probability at least 1- over X.
The Permanent-of-Gaussians Conjecture (PGC)
,1Per
X
z
We can prove the conjecture if =0 or =0! What makes it hard is the combination of average-case and approximation
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For all polynomials q, there exists a polynomial p such that for all n,
The Permanent Concentration Conjecture (PCC)
nqnp
nX
nnCNX
1!PerPr
1,0~
Empirically true!
Also, we can prove it with determinant in place of permanent
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U
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 assuming the PGC and PCC, BPPNP=P#P and hence PH collapses
Main Result
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Idea: 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 lemma 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
Assuming the PCC, the above lets us estimate Per(A) itself in BPPNP
And assuming the PGC, estimating Per(A) is #P-hard
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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
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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• Photodetector arrays that can reliably distinguish 0 vs. 1 photonBut crucially, no nonlinear optics or postselected measurements!
Our Proposal: Concentrate on (say)
n=20 photons and m=400 modes, so that classical simulation is
nontrivial but not impossible
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Main Open ProblemsProve the Permanent of Gaussians Conjecture! (That approximating the permanent of an N(0,1) Gaussian random matrix is #P-complete)
Do our linear-optics experiment!
Are there other (e.g., qubit-based) quantum systems for which approximate classical simulation would collapse PH?
Can our linear-optics model solve classically-intractable decision problems?
Prove the Permanent Concentration Conjecture! (That Pr[|Per(X)|2<n!/p(n)] < 1/q(n))