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**Scott Aaronson Alex Arkhipov MIT**

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

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**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!

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Shor’s algorithm was a hardness result for one of the central computational problems of modern science: Quantum Simulation Use of DoE supercomputers by area (from a talk by Alán Aspuru-Guzik) Shor’s Theorem: Quantum Simulation is not in BPP, unless Factoring is also

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

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 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**

|0 C defines a distribution DC over n-bit output strings QSampling: Given C as input, sample a string x from any probability distribution D such that Certainly this problem is BQP-hard

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

Our Result: Suppose QSampling0.01 is in probabilistic polytime. Then P#P=BPPNP (so in particular, PH collapses to the third level) 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 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)

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**Warmup: Why Exact QSampling Is Hard**

Let f:{0,1}n{-1,1} be any efficiently computable function. Suppose we apply the following quantum circuit: H f |0 Then the probability of observing the all-0 string is

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**Conclusion: Suppose QSampling0 is easy. Then P#P=BPPNP**

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

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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 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

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

The Problem Suppose 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 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 Hmm … robust #P-complete problem … you mean like the Permanent?

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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 aijC be the amplitude for transitioning from initial state i to final state j 1 1 2 2 All I can say is, the bosons got the harder job… 3 Let 3 tinitial tfinal Then what’s the total amplitude for the above process? if the particles are bosons if they’re fermions

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**The BosonSampling Problem**

Input: 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 where AS is an nn matrix containing si copies of the ith row of A Neat Fact: The pS’s sum to 1

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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. Example:

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**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 form for some (i,j)

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**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.

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Outline of Proof Given 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

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**But Linear Interpolation Doesn’t Work!**

X A random line through XCnn “retains too much information” about X We need to redo Lipton/LFKN to work over the complex numbers rather than finite fields Solution: Choose a matrix Y(t) of random trigonometric polynomials, such that Y(0)=X

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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 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!

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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

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**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! 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

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**Experimental Prospects**

What would it take to implement BosonSampling 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 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

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**Open Problems Does 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)?

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**Conclusion For all intents and purposes**

I like to say that we have three choices: either The Extended Church-Turing Thesis is false, Textbook quantum mechanics is false, or QCs can be efficiently simulated classically. For all intents and purposes

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