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BQP PSPACE NP P PostBQP Quantum Complexity and Fundamental Physics Scott Aaronson MIT

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RESOLVED: That the results of quantum computing research can deepen our understanding of physics. That this represents an intellectual payoff from quantum computing, whether or not scalable QCs are ever built. A Personal Confession When proving theorems about obscure quantum complexity classes, sometimes even I wonder whether its all just a mathematical game…

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A quantum computer is obviously just a souped-up analog computer: continuous voltages, continuous amplitudes, whats the difference? A quantum computer with 400 qubits would have ~2 400 classical bits, so it would violate a cosmological entropy bound My classical cellular automaton model can explain everything about quantum mechanics! (How to account for, e.g., Schors algorithm for factoring prime numbers is a detail left for specialists) Who cares if my theory requires Nature to solve the Traveling Salesman Problem in an instant? Nature solves hard problems all the timelike the Schrödinger equation! But then I meet distinguished physicists who say things like:

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The biggest implication of QC for fundamental physics is obvious: Shors Trilemma 1. the Extended Church-Turing Thesisthe foundation of theoretical CS for decadesis wrong, 2. textbook quantum mechanics is wrong, or 3. theres a fast classical factoring algorithm. All three seem like crackpot speculations. At least one of them is true! Thats why YOU should care about quantum computing Because of Shors factoring algorithm, either

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PART I. Classical Complexity Background Why computer scientists wont shut up about P vs. NP PART II. How QC Changes the Picture Physics invades Platonic heaven PART III. The NP Hardness Hypothesis A falsifiable prediction about complexity and physics Rest of the Talk

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PART I. Classical Complexity Background

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Problem: Given a graph, is it connected? Each particular graph is an instance The size of the instance, n, is the number of bits needed to specify it An algorithm is polynomial-time if it uses at most kn c steps, for some constants k,c P is the class of all problems that have polynomial-time algorithms CS Theory 101

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NP: Nondeterministic Polynomial Time Does have a prime factor ending in 7?

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NP-hard: If you can solve it, you can solve everything in NP NP-complete: NP-hard and in NP Is there a Hamilton cycle (tour that visits each vertex exactly once)?

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P NP NP- complete NP-hard Graph connectivity Primality testing Matrix determinant Linear programming … Matrix permanent Halting problem … Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique … Factoring Graph isomorphism …

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Does P=NP? The (literally) $1,000,000 question Q: What if P=NP, and the algorithm takes n steps? A: Then wed just change the question!

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What would the world actually be like if we could solve NP-complete problems efficiently? If there actually were a machine with [running time] ~Kn (or even only with ~Kn 2 ), this would have consequences of the greatest magnitude. Gödel to von Neumann, 1956 Proof of Riemann hypothesis with 10,000,000 symbols? Shortest efficient description of stock market data?

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PART II. How QC Changes the Picture

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BQP contains integer factoring [Shor 1994] But factoring isnt believed to be NP-complete. So the question remains: can quantum computers solve NP-complete problems efficiently? (Is NP BQP?) But quantum magic wont be enough [BBBV 1997] If we throw away the problem structure, and just consider a landscape of 2 n possible solutions, even a quantum computer needs ~2 n/2 steps to find a correct solution BQP: Bounded-Error Quantum Polynomial-Time Obviously we dont have a proof that they cant…

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QCs Dont Provide Exponential Speedups for Black-Box Search BBBV The BBBV No SuperSearch Principle can even be applied in physics (e.g., to lower-bound tunneling times) Is it a historical accident that quantum mechanics courses teach the Uncertainty Principle but not the No SuperSearch Principle?

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The Quantum Adiabatic Algorithm Why do these two energy levels almost kiss? An amazing quantum analogue of simulated annealing [Farhi, Goldstone, Gutmann et al. 2000] This algorithm seems to come tantalizingly close to solving NP-complete problems in polynomial time! But… Answer: Because otherwise wed be solving an NP-complete problem! [Van Dam, Mosca, Vazirani 2001; Reichardt 2004]

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Quantum Computing Is Not Analog The Fault-Tolerance Theorem Absurd precision in amplitudes is not necessary for scalable quantum computing is a linear equation, governing quantities (amplitudes) that are not directly observable This fact has many profound implications, such as… BQP EXP P #P

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Computational Power of Hidden Variables Measure 2 nd register Consider the problem of breaking a cryptographic hash function: given a black box that computes a 2-to-1 function f, find any x,y pair such that f(x)=f(y) Can also reduce graph isomorphism to this problem QCs can almost find collisions with just one query to f! Nevertheless, any quantum algorithm needs (N 1/3 ) queries to find a collision [A.-Shi 2002] Conclusion [A. 2005]: If, in a hidden-variable theory like Bohmian mechanics, your whole life trajectory flashed before you at the moment of your death, then you could solve problems that are presumably hard even for quantum computers (Probably not NP-complete problems though)

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The Absent-Minded Advisor Problem Some consequences: Not even quantum computers with magic initial states can do everything: BQP/qpoly PostBQP/poly An n-qubit state can be PAC-learned using only O(n) measurementsexponentially better than tomography [A. 2006] One can give a local Hamiltonian H on poly(n) qubits, such that any ground state of H can be used to simulate on all yes/no measurements with small circuits [A.-Drucker 2009] Can you give your graduate student a state | with poly(n) qubitssuch that by measuring | in an appropriate basis, the student can learn your answer to any yes-or-no question of size n? NO [Ambainis, Nayak, Ta-Shma, Vazirani 1999]

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PART III. The NP Hardness Hypothesis

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Things we never see… Warp drive Perpetuum mobile GOLDBACH CONJECTURE: TRUE NEXT QUESTION Übercomputer But does the absence of these devices have any scientific importance? YES

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A falsifiable hypothesis linking complexity and physics… There is no physical means to solve NP-complete problems in polynomial time. Encompasses NP P, NP BQP, NP LHC… Does this hypothesis deserve a similar status as (say) no-superluminal-signalling or the Second Law?

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Some alleged ways to solve NP-complete problems… Protein foldingDNA computing Can get stuck at local optima (e.g., Mad Cow Disease) A proposal for massively parallel classical computing

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My Personal Favorite Dip two glass plates with pegs between them into soapy water; let the soap bubbles form a minimum Steiner tree connecting the pegs (thereby solving a known NP-complete problem)

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Relativity Computing DONE

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Topological Quantum Field Theories Freedman, Kitaev, Larsen, Wang 2003 Aharonov, Jones, Landau 2006 Witten 1980s TQFTs Jones Polynomial BQP

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Quantum Gravity Computing? Example: Against many physicists intuition, information dropped into a black hole seems to come out as Hawking radiation almost immediatelyprovided you know the black holes state before the information went in [Hayden & Preskill 2007] Their argument uses explicit constructions of approximate unitary 2-designs We know almost nothingbut there are hints of a nontrivial connection between complexity and QG

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Do the first step of a computation in 1 second, the next in ½ second, the next in ¼ second, etc. Problem: Quantum foaminess Zeno Computing Below the Planck scale ( cm or sec), our usual picture of space and time breaks down in not-yet-understood ways

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Nonlinear variants of the Schrödinger equation Abrams & Lloyd 1998: If quantum mechanics were nonlinear, one could exploit that to solve NP-complete problems in polynomial time No solutions 1 solution to NP-complete problem Can take as an additional argument for why QM is linear

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Closed Timelike Curve Computing Quantum computers with closed timelike curves could solve PSPACE-complete problemsthough not more than that [A.-Watrous 2008] R CTC R CR C 000 Answer Causality- Respecting Register CTC Register Polynomial Size Circuit

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Anthropic Principle Foolproof way to solve NP-complete problems in polynomial time (at least in the Many-Worlds Interpretation): First guess a random solution. Then, if its wrong, kill yourself Technicality: If there are no solutions, youd seem to be out of luck! Solution: With tiny probability dont do anything. Then, if you find yourself in a universe where you didnt do anything, there probably were no solutions, since otherwise you wouldve found one Again, I interpret these results as providing additional evidence that nonlinear QM, closed timelike curves, postselection, etc. arent possible. Why? Because Im an optimist.

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For Even More Interdisciplinary Excitement, Heres What You Should Look For A plausible complexity-theoretic story for how quantum computing could fail (see A. 2004) Intermediate models of computation between P and BQP (highly mixed states? restricted sets of gates?) Foil theories that lead to complexity classes slightly larger than BQP (only example I know of: hidden variables) A sane notion of quantum gravity polynomial-time (first step: a sane notion of time in quantum gravity?)

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Scientific American, March 2008:

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