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Quantum Computing and the Limits of the Efficiently Computable Scott Aaronson MIT

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Things we never see… Warp drive Perpetuum mobile GOLDBACH CONJECTURE: TRUE NEXT QUESTION Übercomputer The (seeming) impossibility of the first two machines reflects fundamental principles of physicsSpecial Relativity and the Second Law respectively So what about the third one?

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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 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

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Extended Church-Turing Thesis Any physically-realistic computing device can be simulated by a deterministic or probabilistic Turing machine, with at most polynomial overhead in time and memory An important presupposition underlying P vs. NP is the... So how sure are we of this thesis? Have there been serious challenges to it?

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Old proposal: Dip two glass plates with pegs between them into soapy water. Let the soap bubbles form a minimum Steiner tree connecting the pegsthereby solving a known NP-hard problem instantaneously

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Protein folding: Can also get stuck at local optima (e.g., Mad Cow Disease) DNA computers: Just massively parallel classical computers Other Approaches

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Ah, but what about quantum computing? In the 1980s, Feynman, Deutsch, and others noticed that quantum systems with n particles seemed to take ~2 n time to simulateand had the amazing idea of building a quantum computer to overcome that problem Quantum mechanics: Probability theory with minus signs (Nature seems to prefer it that way) Actually building a QC: Damn hard, because of decoherence. (But seems possible in principle!)

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Science Popularizers Beware: A quantum computer is NOT a massively-parallel classical computer! Exponentially-many basis states, but you only get to observe one of them Any hope for a speedup rides on the magic of quantum interference

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BQP (Bounded-Error Quantum Polynomial-Time): The class of problems solvable efficiently by a quantum computer, defined by Bernstein and Vazirani in 1993 Shor 1994: Factoring integers is in BQP NP NP-complete P Factoring BQP Interesting

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Remember: factoring isnt thought to be NP-complete! Today, we dont believe BQP contains all of NP (though not surprisingly, we cant prove that it doesnt) Bennett et al. 1997: Quantum magic wont be enough If you throw away the problem structure, and just consider an abstract landscape of 2 n possible solutions, then even a quantum computer needs ~2 n/2 steps to find the correct one (That bound is actually achievable, using Grovers algorithm!) So, is there any quantum algorithm for NP-complete problems that would exploit their structure?

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Quantum Adiabatic Algorithm (Farhi et al. 2000) HiHi Hamiltonian with easily- prepared ground state HfHf Ground state encodes solution to NP-complete problem Problem: Eigenvalue gap can be exponentially small

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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

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Relativity Computer DONE

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Zenos Computer STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 Time (seconds)

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Heres a polynomial-time algorithm to solve NP-complete problems (only drawback is that it requires time travel): Read an integer x {0,…,2 n -1} from the future If x encodes a valid solution, then output x Otherwise, output (x+1) mod 2 n Closed Timelike Curves (CTCs) If valid solutions exist, then the only fixed-points of the above program input and output them Building on work of Deutsch, [A.-Watrous 2008] defined a formal model of CTC computation, and showed that in both the classical and quantum cases, it has exactly the power of PSPACE (believed to be even larger than NP)

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Includes P NP as a special case, but is stronger No longer a purely mathematical conjecture, but also a claim about the laws of physics If true, would explain why adiabatic systems have small spectral gaps, the Schrödinger equation is linear, CTCs dont exist... The No-SuperSearch Postulate There is no physical means to solve NP-complete problems in polynomial time.

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Question: What exactly does it mean to solve an NP- complete problem? Example: Its been known for decades that, if you send n identical photons through a network of beamsplitters, the amplitude for the photons to reach some final state is given by the permanent of an n n matrix of complex numbers But the permanent is #P-complete (believed even harder than NP- complete)! So how can Nature do such a thing? Resolution: The amplitudes arent directly observable, and require exponentially-many probabilistic trials to estimate Lesson: If you cant observe the answer, it doesnt count! Recently, Alex Arkhipov and I gave the first evidence that even the observed output distribution of such a linear-optical network would be hard to simulate on a classical computer but the argument was necessarily more subtle

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One could imagine worse research agendas than the following: Prove PNP (better yet, prove factoring is classically hard, implying PBQP) Prove NP BQPi.e., that not even quantum computers can solve NP-complete problems Build a scalable quantum computer (or even more interesting, show that its impossible) Determine whether all of physics can be simulated by a quantum computer Derive as much physics as one can from No-SuperSearch and other impossibility principles Conclusion

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Papers, talk slides, blog:

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