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Exploring the Limits of the Efficiently Computable Research Directions in Computational Complexity and Physics That I Find Exciting Scott Aaronson (MIT) Papers & slides at www.scottaaronson.com Quantum Mechanics in One Slide Probability Theory: Quantum Mechanics: s11 s1n p1 q1 s s p q nn n n1 n u11 u1n 1 1 u u nn n n1 n pi 0, n p i 1 i 1 Linear transformations that conserve 1-norm of probability vectors: Stochastic matrices i C, n i 1 2 i 1 Linear transformations that conserve 2-norm of amplitude vectors: Unitary matrices Quantum Computing A general entangled state of n qubits requires ~2n amplitudes to specify: x x0,1n x Presents an obvious practical problem when using conventional computers to simulate quantum mechanics Feynman 1981: So then why not turn things around, and build computers that themselves exploit superposition? Could such a machine get any advantage over a classical computer with a random number generator? If so, it would have to come from interference between amplitudes BQP (Bounded-Error Quantum Polynomial-Time): The class of problems solvable efficiently by aInteresting quantum computer, defined by Bernstein and Vazirani in 1993 Shor 1994: Factoring integers is in BQP NP-complete NP BQP Factoring P Examples of My Past Work Quantum lower bound for the collision problem [A. 2002] Quantum (+classical!) lower bound for local search [A. 2004] First direct product theorem for quantum search [A. 2004] PostBQP = PP [A. 2004] BQP vs. the polynomial hierarchy: black-box relation problems in BQP but not BPPPH [A. 2009] Publicly-verifiable quantum money [A.-Christiano 2012] BQP/qpoly QMA/poly, learnability of quantum states [A.Drucker 2010, A. 2004, A. 2006] Algebrization [A.-Wigderson 2008] This Talk: Three Recent Directions 1. Meeting Experimentalists Halfway Using complexity theory to find quantum advantage in systems of current experimental interest (e.g. linear-optical networks), which fall short of universal quantum computers 2. Computational Complexity and Black Holes An amazing role for complexity theory in the recent “firewall” debate and the AdS/CFT correspondence 3. Physical Universality When Turing-universality isn’t enough: the complexity and realizability of physical transformations 1. Meeting Experimentalists Halfway BosonSampling (A.-Arkhipov 2011) A rudimentary type of quantum computing, involving only non-interacting photons Classical counterpart: Galton’s Board Replacing the balls by photons leads to famously counterintuitive phenomena, like the Hong-Ou-Mandel dip 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 modes—there are m possibilities n The beamsplitter network defines a column-orthonormal matrix ACmn, such that Pr outcome S Per A 2 where Per X n x S n i 1 i, i S nn submatrix of A corresponding to S Amazing connection between permanents and physics, which even leads to a simpler proof of Valiant’s famous result that the permanent is #P-complete [A. 2011] So, Can We Use Quantum Optics to Solve a #P-Complete Problem? That sounds way too good to be true… 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, we’d generally need to repeat the optical experiment exponentially many times Better idea: Given ACmn as input, let BosonSampling be the problem of merely sampling from the same distribution DA that the beamsplitter network samples from—the one defined by Pr[S]=|Per(AS)|2 Theorem (A.-Arkhipov 2011): Suppose BosonSampling is #P=BPPNP solvable in classical polynomial time. Then P Upshot: Compared to (say) Shor’s factoring algorithm, we get different/stronger evidence Better Theorem: Suppose we can sample DA eventhat a weaker system can dopolynomial somethingtime. classically approximately in classical Thenhard in BPPNP, it’s possible to estimate Per(X), with high nn probability over a Gaussian random matrix X ~ Ν 0,1C We conjecture that the above problem is already #P-complete. If it is, then a fast classical algorithm for approximate BosonSampling would already have the consequence that P#P=BPPNP Challenges Prove #P-completeness for natural average-case approximation problems (like permanents of Gaussians)— thereby yielding hardness for approximate BosonSampling As a first step, understand the distribution of Per(X), X Gaussian Early experimental implementations have been done (Rome, Brisbane, Vienna, Oxford)! But so far with just 3-4 photons. For scaling, will be crucial to understand the complexity of BosonSampling when a constant fraction of photons are lost Can the BosonSampling model solve hard “conventional” problems? How do we verify that a BosonSampling device is working correctly? [A.-Arkhipov 2014, A.-Nguyen 2014] BosonSampling with thermal states: fast classical algorithm to approximate Per(X) when X is positive semidefinite? 2. Computational Complexity and Black Holes Most striking application so far of complexity to fundamental physics? Hawking 1970s: Black holes radiate The radiation seems thermal (uncorrelated with whatever fell in)—but if quantum mechanics is true, then it can’t be Susskind et al. 1990s: “Black-hole complementarity.” In string theory / quantum gravity, the Hawking radiation should just be a scrambled re-encoding of the same quantum states that are also inside the black hole The Firewall Paradox [Almheiri et al. 2012] If the black hole interior is “built” out of the same qubits coming out as Hawking radiation, then why can’t we do something to those Hawking qubits (after waiting ~1070 years for enough to come out), then dive into the black hole, and see that we’ve completely destroyed the spacetime geometry in the interior? Entanglement among Hawking photons detected! Harlow-Hayden 2013: Sure, there’s some unitary transformation that Alice could apply to the Hawking radiation, that would generate a “firewall” inside the event horizon. But how long would it take her to apply it? They showed: A natural formalization of Alice’s decoding task is QSZK-hard (QSZK = Quantum Statistical Zero Knowledge) My 2002 collision lower bound suggests that QSZKBQP. In that case, decoding would presumably take time exponential in the number of qubits of the black hole—so the black hole would’ve evaporated before Alice had even made a dent! R = Faraway Hawking Radiation B = Just-Emitted Hawking Radiation H = Interior of “Old” Black Hole The HH Decoding Problem Given a description of a quantum circuit C, such that C0 n RBH Promised that, by acting only on R (the “Hawking radiation part”), it’s possible to distill an EPR pair 0 0 11 2 between R and B Problem: Distill such an EPR pair, by applying a unitary transformation UR to the qubits in R My strengthening: Harlow-Hayden decoding is as hard as inverting an arbitrary one-way function RBH 1 2 2 n 1 f x, s, a x s a x , s0,1 , a0,1 n R B x, s H R: “old” Hawking photons / B: photons just coming out / H: still in black hole B is maximally entangled with the last qubit of R. But in order to see that B and R are even classically correlated, one would need to learn xs (a “hardcore bit” of f), and therefore invert f With realistic dynamics, the decoding task seems like it should only be “harder” than in this model case (though open how to formalize that) Is computational intractability the only “armor” protecting the geometry of spacetime inside the black hole? Quantum Circuit Complexity and Wormholes [A.-Susskind, in progress] The AdS/CFT correspondence relates antideSitter quantum gravity in D spacetime dimensions to conformal field theories (without gravity) in D-1 dimensions Conjecture: The But the mapping is Susskind’s extremely nonlocal! quantum circuit complexity of a CFT It was recently found that an expanding wormhole, on the AdS state can encode information about side, maps to a collection of qubits on the CFT side that just the geometry of the dual AdS. n seems to get more and more “complex”: 00 11 Not clear if it’s true, but thas t U tests survived some nontrivial 2 Theorem: Suppose U implements (say) a computationally-universal cellular automaton. Then after t=exp(n) iterations, |t has superpolynomial quantum circuit complexity unless PSPACEPP/poly 3. Physical Universality Four Related Questions For every n-qubit unitary U, is there a Boolean function f such that U can be implemented in BQPf? Which n-qubit unitaries could we efficiently implement if P=PSPACE? Can every n-qubit unitary be implemented by a quantum circuit with poly(n) depth (but maybe exp(n) ancilla qubits)? Could we prove—unconditionally, with today’s technology—that exponentially many gates are needed to implement some n-qubit unitary U? Generalizations of the Natural Proofs barrier? A Grand Challenge Can we classify all possible sets of quantum gates acting on qubits, in terms of which unitary transformations they approximately generate? “Quantum Computing’s Classification of Finite Simple Groups” Warmup: Classify all the possible Hamiltonians / Lie algebras. Even just on 1 and 2 qubits! A.-Bouland 2014: Every nontrivial two-mode beamsplitter is universal Baby case that already took lots of representation theory… The Classical Case A.-Grier-Schaefer 2015: Classified all sets of reversible gates in terms of which reversible transformations F:{0,1}n{0,1}n they generate (assuming swaps and ancilla bits are free) CNOT Toffoli Fredkin Cellular Automata Beyond Turing-Universality Schaeffer 2014: The first known “physically-universal” cellular automaton (able to implement any transformation in any bounded region, by suitably initializing the complement of that region) The Coffee Automaton A., Carroll, Mohan, Ouellette, Werness 2015: Detailed study of the rise and fall of “complex organization,” in a reversible cellular automaton that models the thermodynamic mixing of cream into coffee Compressed File Size We prove that, under coarsegraining, the Kolmogorov complexity of this image has a rising-falling pattern 500 450 400 350 300 250 200 150 100 50 0 -100 100 300 500 Time Steps 700 900 Summary Quantum computing established a remarkable intellectual bridge between computer science and physics That’s always been why I’ve cared! Actual devices would be a bonus My research agenda: to see just how much weight this bridge can carry Rebuilding physics in the language of computation won’t be nearly as easy as Stephen Wolfram thought! Not only does it require engaging our actual understanding of physics (QM, QFT, AdS/CFT…); it requires hard mathematical work, often making new demands on theoretical computer science But sure, I think it’s ultimately possible