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Quantum Computing and the Limits of the Efficiently Computable Scott Aaronson MIT Things we never see… GOLDBACH CONJECTURE: TRUE NEXT QUESTION Warp drive Perpetuum mobile Übercomputer The (seeming) impossibility of the first two machines reflects fundamental principles of physics—Special Relativity and the Second Law respectively So what about the third one? Moore’s Law Extrapolating: Robot uprising? But even a killer robot would still be “merely” a Turing machine, operating on principles laid down in the 1930s… = But Turing machines have fundamental limits—even more so, if you need the answer in a reasonable amount of time! P: Polynomial Time Class of all “decision problems” (infinite sets of yes-or-no questions) solvable by a Turing machine, using a number of steps that scales at most like the size of the question raised to some fixed power Example: Given this map, is there a route from Charlottesville to Bartow? NP: Nondeterministic Polynomial Time Class of all decision problems for which a “yes” answer can be verified in polynomial time, if you’re given a witness or proof for it Example: Does 37976595177176695379702491479374117272627593301950462 68899636749366507845369942177663592040922984159043233 98509069628960404170720961978805136508024164948216028 85927126968629464313047353426395204881920475456129163 30509384696811968391223240543368805156786230378533714 9184281196967743805800830815442679903720933 have a divisor ending in 7? NP-hard: If you can solve it, then you can solve every NP problem NP-complete: NP-hard and in NP Example: Is there a tour that visits each city once? Does P=NP? The (literally) $1,000,000 question If there actually were a machine with [running time] ~Kn (or even only with ~Kn2), this would have consequences of the greatest magnitude. —Gödel to von Neumann, 1956 Most computer scientists believe that PNP But if so, there’s a further question: is there any way to solve NP-complete problems in polynomial time, consistent with the laws of physics? Old proposal: 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-hard problem “instantaneously” Relativity Computer DONE Zeno’s Computer Time (seconds) STEP 1 STEP 2 STEP 3 STEP 4 STEP 5 Time Travel Computer S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent, Proceedings of the Royal Society A 465:631-647, 2009. arXiv:0808.2669. Answer Polynomial Size Circuit C “Closed Timelike Curve Register” R CTC R CR 0 0 0 “CausalityRespecting Register” Ah, but what about quantum computing? (you knew it was coming) Quantum mechanics: “Probability theory with minus signs” (Nature seems to prefer it that way) The Famous Double-Slit Experiment Probability of landing in “dark patch” = |amplitude|2 = |amplitudeSlit1 + amplitudeSlit2|2 = 0 Yet if you close one of the slits, the photon can appear in that previously dark patch!! A bit more precisely: the key claim of quantum mechanics is that, if an object can be in two distinguishable states, call them |0 or |1, then it can also be in a superposition a|0 + b|1 1 Here a and b are complex numbers called amplitudes satisfying |a|2+|b|2=1 If we observe, we see |0 with probability |a|2 |1 with probability |b|2 Also, the object collapses to whichever outcome we see 0 1 2 0 To modify a state n a i 1 i i we can multiply the vector of amplitudes by a unitary matrix—one that preserves n a i 1 i 2 1 0 1 1 2 1 2 1 1 1 2 12 20 1 01 11 2 2 2 2 1 0 1 2 0 We’re seeing interference of amplitudes—the source of “quantum weirdness” Quantum Computing A general entangled state of n qubits requires ~2n amplitudes Where we are: A QC has now factored 21 into to specify: 37, with high probability x(Martín-López et al. 2012) n x 0 , 1 Scaling up is hard, because of decoherence! But a x Presents obvious practical problem when usingto be any unlessan QM is wrong, there doesn’t seem conventional computers to simulate quantum mechanics fundamental obstacle Interesting Feynman 1981: So then why not turn things around, and build computers that themselves exploit superposition? Shor 1994: Such a computer could do more than simulate QM—e.g., it could factor integers in polynomial time But factoring is not believed to be NP-complete! And today, we don’t believe quantum computers can solve NP-complete problems in polynomial time in general (though not surprisingly, we can’t prove it) Bennett et al. 1997: “Quantum magic” won’t be enough If you throw away the problem structure, and just consider an abstract “landscape” of 2n possible solutions, then even a quantum computer needs ~2n/2 steps to find the correct one (That bound is actually achievable, using Grover’s algorithm!) If there’s a fast quantum algorithm for NP-complete problems, it will have to exploit their structure somehow The “Adiabatic Optimization” Approach to Solving NP-Hard Problems with a Quantum Computer Hi Operation with easilyprepared lowest energy state Hf Operation whose lowest-energy state encodes solution to NP-hard problem Hope: “Quantum tunneling” could give speedups over classical optimization methods for finding local optima Remains unclear whether you can get a practical speedup this way over the best classical algorithms. We might just have to build QCs and test it! Problem: “Eigenvalue gap” can be exponentially small Some Examples of My Research on Computational Complexity and Physics BosonSampling (with Alex Arkhipov): A proposal for a rudimentary photonic quantum computer, which doesn’t seem useful for anything (e.g. breaking codes), but does seem hard to simulate using classical computers (We showed that a fast, exact classical simulation would “collapse the polynomial hierarchy to the third level”) Experimentally demonstrated (with 3-4 photons…) by groups in Brisbane, Oxford, Vienna, and Rome! Computational Complexity of Decoding Hawking Radiation Firewall Paradox (2012): Hypothetical experiment that involves waiting outside a black hole for ~1070 years, collecting all the Hawking photons it emits, doing a quantum computation on them, then jumping into the black hole to observe that your computation “nonlocally destroyed” the structure of spacetime inside the black hole Harlow-Hayden (2013): Argument that the requisite computation would take exponential time (~210^70 years) even for a QC—by which time the black hole has already fully evaporated! Recently, I strengthened Harlow and Hayden’s argument, to show that performing the computation is generically at least as hard as inverting cryptographic “one-way functions” Summary Quantum computing is one of the most exciting things in science—but the reasons are a little different from what the press says Even a quantum computers couldn’t solve all problems in an instant (though they’d provide amazing speedups for a few problems, like factoring and quantum simulation, and maybe broader speedups) And building them is hard (though the real shock for physics would be if they weren’t someday possible) On the other hand, one thing quantum computing has already done, is create a bridge between computer science and physics, carrying amazing insights in both directions