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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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Problem: Given a flight map, is every airport reachable from every other in 5 flights or less? Any specific map is an instance of the problem The size of an instance, n, is the number of bits used 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 for which theres a deterministic, polynomial-time algorithm that correctly solves every instance Complexity Theory 101

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

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NP-hard: If you can solve it, then you can solve every NP problem NP-complete: NP-hard and in NP Can you send your passenger on a round-trip that visits each city 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 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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Q: Hey, wasnt PNP proved this summer? A: No. Q: Why is P NP so hard to prove? A: Basically because all known approaches, if they worked, would also prove other things that are false (e.g., 2SAT is hard) Q: How sure are you that P NP? A: If we were physicists, we wouldve long ago declared it a law of nature and been done with it… Q: If almost everyone believes P NP, why bother proving it? A: Its not the destination, its what else well learn along the way Q: Could it be that P NP is Gödel-undecidable? A: Maybe, but Fermats Last Theorem took 350 years! Weve only been at this 40 years and know way more than when we started Q: What if P=NP, but the algorithm took n steps? A: Then wed change the question Q: Even if P NP, cant there be clever heuristics that often work? A: Yes, but cleverness would be provably necessary P/NP FAQ

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Extended Church-Turing Thesis Any physically-realistic computing device can be simulated by a deterministic (or maybe 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? (you knew it was coming) Quantum algorithms: The power of 2 n complex numbers working for YOU 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) Shor 1994: A quantum computer, if built, could factor integers (and hence break RSA) in polynomial time

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But remember: factoring isnt thought to be NP-complete! To this day (and contrary to popular misconception), we dont know a fast quantum algorithm to solve NP-complete problems (though not surprisingly, we also cant prove there isnt one) 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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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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Anthropic Computing 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, youre 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!

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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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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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