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

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In 1994, something big happened in our field, whose meaning is still debated today… Why exactly was Shors algorithm important? Boosters: Because it means well build QCs! Skeptics: Because it means we wont build QCs! Me: For reasons having nothing to do with building QCs!

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Shors algorithm was a hardness result for one of the central computational problems of modern science: Q UANTUM S IMULATION Shors Theorem: Q UANTUM S IMULATION is not in BPP, unless F ACTORING is also Use of DoE supercomputers by area (from a talk by Alán Aspuru-Guzik)

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Advantages of our result: Based on P #P BPP NP rather than F ACTORING BPP 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

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Let C be a quantum circuit, which acts on n qubits initialized to the all-0 state Certainly this problem is BQP-hard C |0 QS AMPLING : Given C as input, sample a string x from any probability distribution D such that C defines a distribution D C over n-bit output strings

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More generally: Suppose QS AMPLING 0.01 is in probabilistic polytime with A oracle. Then P #P BPP NP So QS AMPLING cant even be in BPP PH without collapsing PH! A Our Result: Suppose QS AMPLING 0.01 is in probabilistic polytime. Then P #P =BPP NP (so in particular, PH collapses to the third level) Extension to relational problems: Suppose FBQP=FBPP. Then P #P =BPP NP QS AMPLING is #P-hard under BPP NP -reductions (Provided the BPP NP machine gets to pick the random bits used by the QS AMPLING oracle)

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Warmup: Why Exact QS AMPLING Is Hard 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 Then the probability of observing the all-0 string is

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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 x f(x) using binary search and padding Claim 2: Suppose QS AMPLING was classically easy. Then we could estimate p in BPP NP Proof: Let M be a classical algorithm for QS AMPLING, and let r be its randomness. Use approximate counting to estimate Conclusion: Suppose QS AMPLING 0 is easy. Then P #P =BPP NP

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So Why Arent 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 noisemeaning we cant actually sample from D C, but only from some distribution D such that Could that be easy, even if sampling from D C itself was hard? To rule that out, we need to show that even a fast classical algorithm for QS AMPLING would imply P #P =BPP NP

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The Problem Suppose M knew that all we cared about was the final amplitude of |0 0 (i.e., thats 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 P ERMANENT ? Indeed. But to bring the permanent into quantum computing, we need a brief detour into particle physics (!) Well have to work harder … but as a bonus, well not only rule out approximate samplers, but approximate samplers for an extremely weak kind of QC

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Particle Physics In One Slide There are two types of particles in Nature… BOSONS Force-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…) FERMIONS Matter: quarks, electrons… Swap two identical fermions quantum state picks up -1 phase Pauli exclusion principle: no two fermions can occupy same state

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Consider a system of n identical, non-interacting particles… Let a ij C be the amplitude for transitioning from initial state i to final state j Then whats the total amplitude for the above process? if the particles are bosonsif theyre fermions Let All I can say is, the bosons got the harder job… t initial t final

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The B OSON S AMPLING Problem Input: An m n complex matrix A, whose n columns are orthonormal vectors in C m (here m n 2 ) Let a configuration be a list S=(s 1,…,s m ) of nonnegative integers with s 1 +…+s m =n Task: Sample each configuration S with probability Neat Fact: The p S s sum to 1 where A S is an n n matrix containing s i copies of the i th row of A

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Physical Interpretation: Were 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. Example:

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Theorem (implicit in Lloyd 1996) : B OSON S AMPLING QS AMPLING Proof Sketch: We need to simulate a system of n bosons on a conventional quantum computer The basis states |s 1,…,s m (s 1 +…+s m =n) just record the occupation number of each mode Given any scattering matrix U C m m on the m modes, we can decompose U as a product U 1 …U T, where T=O(m 2 ) and each U t acts only on 2-dimensional subspaces of the form for some (i,j)

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Theorem (Valiant 2001, Terhal-DiVincenzo 2002): F ERMION S AMPLING BPP In stark contrast, we prove the following: Suppose B OSON S AMPLING BPP. Then given an arbitrary matrix X C n n, one can approximate |Per(X)| 2 in BPP NP 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.

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Outline of Proof Given a matrix X C n n, with every entry satisfying |x ij | 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 C n n that passes through X, and let Y 1,…,Y k C n n be other matrices on V (where k n 2 ) If we can estimate |Per(Y i )| 2 for most i, then we can estimate |Per(X)| 2 using noisy polynomial interpolation

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But Linear Interpolation Doesnt Work! We need to redo Lipton/LFKN to work over the complex numbers rather than finite fields A random line through X C n n retains too much information about X X Solution: Choose a matrix Y(t) of random trigonometric polynomials, such that Y(0)=X

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Questions: How do we sample Y(t) and Y 1,…,Y k efficiently? How do we do the noisy polynomial interpolation? Lazy answer: Since were a BPP NP machine, just use rejection sampling! For sufficiently large L and t>>0, each y ij (t) will look like an independent Gaussian, uncorrelated with x ij : Furthermore, Per(Y(t)) is a univariate polynomial in e 2 it of degree at most Ln

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The problem reduces to estimating |Per(Y)| 2, for a matrix Y C n n of (essentially) independent N(0,1) Gaussians To do this, generate a random m n column-orthonormal matrix A that contains Y/m as an n n submatrix (i.e., such that A S =Y/m for some random configuration S) Let M be our BPP algorithm for approximate B OSON S AMPLING, and let r be Ms randomness Use approximate counting (in BPP NP ) to estimate Intuition: M has no way to determine which configuration S we care about. So if its right about most configurations, then w.h.p. we must have

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Problem: Bosons like to pile on top of each other! Call a configuration S=(s 1,…,s m ) good if every s i 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! rather than ½ as with classical particles Fortunately, we show that with n bosons and m kn 2 boxes, the probability of a collision is still at most (say) ½

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Experimental Prospects What would it take to implement B OSON S AMPLING with photonics? Reliable phase-shifters Reliable beamsplitters Reliable single-photon sources Reliable photodetectors But crucially, no nonlinear optics or postselected measurements! Problem: The output will be a collection of n n matrices B 1,…,B k with unusually large permanentsbut how would a classical skeptic verify that |Per(B i )| 2 was large? Our Proposal: Concentrate on (say) n=30 photons, so that classical simulation is difficult but not impossible

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Open Problems Does our result relativize? (Conjecture: No) Can we use B OSON S AMPLING to do universal QC? Can we use it to solve any decision problem outside BPP? Can you convince a skeptic (who isnt a BPP NP machine) that your QC is indeed doing B OSON S AMPLING ? 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 P ERMANENT #P-complete for +1/-1 matrices (with no 0s)?

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Conclusion I 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

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