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Computational Complexity and Physics Scott Aaronson (MIT) New Insights Into Computational Intractability Oxford University, October 3, 2013

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Title is too broad! So, Ill just give two tales from the trenches: -One about BosonSampling, one about black holes What you should take away: 1990s: Computational Complexity Quantum Physics Shor & Grover Today: Computational Complexity Quantum Physics

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BosonSampling (A.-Arkhipov 2011) A rudimentary type of quantum computing, involving only non-interacting photons Classical counterpart: Galtons Board Replacing the balls by photons leads to famously counterintuitive phenomena, like the Hong-Ou-Mandel dip

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In general, we consider a network of beamsplitters, with n input modes (locations) and m>>n output modes n identical photons enter, one per input mode Assume for simplicity they all leave in different modesthere are possibilities The beamsplitter network defines a column-orthonormal matrix A C m n, such that where is the matrix permanent n n submatrix of A corresponding to S For simplicity, Im ignoring outputs with 2 photons per mode

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Example For Hong-Ou-Mandel experiment, In general, an n n complex permanent is a sum of n! terms, almost all of which cancel How hard is it to estimate the tiny residue left over? Answer: #P-complete, even for constant-factor approx (Contrast with nonnegative permanents!)

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So, Can We Use Quantum Optics to Solve a #P-Complete Problem? Explanation: If X is sub-unitary, then |Per(X)| 2 will usually be exponentially small. So to get a reasonable estimate of |Per(X)| 2 for a given X, wed generally need to repeat the optical experiment exponentially many times That sounds way too good to be true…

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Better idea: Given A C m n as input, let BosonSampling be the problem of merely sampling from the same distribution D A that the beamsplitter network samples fromthe one defined by Pr[S]=|Per(A S )| 2 Upshot: Compared to (say) Shors factoring algorithm, we get different/stronger evidence that a weaker system can do something classically hard Theorem (A.-Arkhipov 2011): Suppose BosonSampling is solvable in classical polynomial time. Then P #P =BPP NP Better Theorem: Suppose we can sample D A even approximately in classical polynomial time. Then in BPP NP, its possible to estimate Per(X), with high probability over a Gaussian random matrix

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Experiments # of experiments > # of photons! Last year, groups in Brisbane, Oxford, Rome, and Vienna reported the first 3-photon BosonSampling experiments, confirming that the amplitudes were given by 3x3 permanents

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Goal (in our view): Scale to 10-30 photons Dont want to scale much beyond thatboth because (1)you probably cant without fault-tolerance, and (2)a classical computer probably couldnt even verify the results! Obvious Challenges for Scaling Up: -Reliable single-photon sources (optical multiplexing?) -Minimizing losses -Getting high probability of n-photon coincidence Theoretical Challenge: Argue that, even with photon losses and messier initial states, youre still solving a classically-intractable sampling problem

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Recent Criticisms of Gogolin et al. ( arXiv:1306.3995 ) Suppose you ignore which actual photodetectors light up, and count only the number of times each output configuration occurs. In that case, the BosonSampling distribution D A is exponentially-close to the uniform distribution U Response: Why would you ignore which detectors light up?? The output of almost any algorithm is also gobbledygook if you ignore the order of the output bits…

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Recent Criticisms of Gogolin et al. ( arXiv:1306.3995 ) OK, so maybe D A isnt close to uniform. Still, the very same arguments we gave for why polynomial-time classical algorithms cant sample D A, suggest that they cant even distinguish D A from U! Response: Thats why we said to focus on 10-30 photonsa range where a classical computer can verify a BosonSampling devices output, but the BosonSampling device might be faster! (And 10-30 photons is probably the best you can do anyway, without quantum fault-tolerance)

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More Decisive Responses (A.-Arkhipov, arXiv:1309.7460) Theorem: Let A C m n be a Haar-random BosonSampling matrix, where mn 5.1 /. Then with 1-O( ) probability over A, the BosonSampling distribution D A has (1) variation distance from the uniform distribution U Under U Histogram of (normalized) probabilities under D A Necessary, though not sufficient, for approximately sampling D A to be hard

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Theorem (A. 2013): Let A C m n be Haar-random, where m>>n. Then there is a classical polynomial-time algorithm C(A) that distinguishes D A from U (with high probability over A and constant bias, and using only O(1) samples) Strategy: Let A S be the n n submatrix of A corresponding to output S. Let P be the product of squared 2-norms of A S s rows. If P>E[P], then guess S was drawn from D A ; otherwise guess S was drawn from U P under uniform distribution (a lognormal random variable) P under a BosonSampling distribution A ASAS

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Using Quantum Optics to Prove that the Permanent is #P-Complete [A., Proc. Roy. Soc. 2011] Valiant showed that the permanent is #P-completebut his proof required strange, custom-made gadgets We gave a new, arguably more transparent proof by combining three facts: (1)n-photon amplitudes correspond to n n permanents (2) Postselected quantum optics can simulate universal quantum computation [Knill-Laflamme-Milburn 2001] (3) Quantum computations can encode #P-complete quantities in their amplitudes

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Black Holes and Computational Complexity?? YES!! Amazing connection made this year by Harlow & Hayden But first, we need to review 40 years of black hole history SZK QSZK BPP BQP AM QAM

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Bekenstein 1972, Hawking 1975: Black holes have entropy and temperature! They emit radiation The Information Loss Problem: Calculations suggest that Hawking radiation is thermaluncorrelated with whatever fell in. So, is infalling information lost forever? Would violate the unitarity / reversibility of QM OK then, assume the information somehow gets out! The Xeroxing Problem: How could the same qubit | fall inexorably toward the singularity, and emerge in Hawking radiation? Would violate the No-Cloning Theorem Black Hole Complementarity (Susskind, t Hooft): An external observer can describe everything unitarily without including the interior at all! Interior should be seen as just a scrambled re-encoding of the exterior degrees of freedom

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Violates monogamy of entanglement! The same qubit cant be maximally entangled with 2 things The Firewall Paradox (AMPS 2012) B = Interior of Old Black Hole R = Faraway Hawking Radiation H = Just-Emitted Hawking Radiation Near-maximal entanglement Also near-maximal entanglement

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Harlow-Hayden 2013 (arXiv:1301.4504) : Striking argument that Alices decoding task would require exponential time Complexity theory to the rescue of quantum field theory?? The HH decoding problem: Given an n-qubit pure state | BHR produced by a known, poly-size quantum circuit. Promised that, by acting only on R (the Hawking radiation), its possible to distill an EPR pair (|00 +|11 )/2 between R and B (the black hole interior). Distill such a pair, by applying a unitary transformation U R to the qubits in R. Theorem (HH): This decoding problem is QSZK-hard. So presumably intractable, unless QSZK=BQP (which we have oracle evidence against)!

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Improvement (A. 2013): Suppose there are poly-size quantum circuits for the HH decoding problem. Then any OWF f:{0,1} n {0,1} p(n) can be inverted in quantum polynomial time. Proof: Assume for simplicity that f is injective. Consider Suppose applying U R to R decodes an EPR pair between R and B. Then for some {| x } x, {| x } x, we must have Furthermore, to get perfect entanglement, we need | x =| x for all x! So from U R, we can get unitaries V,W such that Generalizing to arbitrary OWFs requires techniques similar to those used in HILL Theorem!

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Other Exciting Recent Complexity/Physics Connections Quantum PCP Conjecture. Is approximate quantum MAX-k-SAT hard for quantum NP? If so, will require the construction exotic condensed matter systems, exhibiting room-temperature entanglement! Beautiful recent survey by Aharonov et al. (arXiv:1309.7495) Using Entanglement to Steer Quantum Systems. Can use Bell inequality violation for guaranteed randomness expansion (Vazirani-Vidick), blind and authenticated QC with a classical polynomial-time verifier, QMIP=MIP* (Reichardt-Unger-Vazirani)… I hope for, and expect, many more such striking connections! Indeed, making these connections possible might be the most important result of quantum computing researcheven if useful QCs eventually get built!

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