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Quantum Computing and the Limits of the Efficiently Computable Scott Aaronson ‘00 MIT 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… = And it’s conjectured that thousands of interesting problems are inherently intractable for Turing machines… Is there any feasible way to solve these problems, 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” What About Quantum Mechanics? “Like probability, but with minus signs” 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 Where we are:state A QC nowrequires factored into A general entangled of has n qubits ~2n21 amplitudes with high probability (Martín-López et al. 2012) to 37, specify: x of decoherence! But Scaling up is hard, because x0,1n x unless QM is wrong, there doesn’t seem to be any Presents an obvious practical problem when using fundamental obstacle conventional computers to simulate quantum mechanics 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 Quantum Adiabatic Algorithm (Farhi et al. 2000) Hi Hamiltonian with easilyprepared ground state Hf Ground state encodes solution to NP-complete problem 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 a one-way function Summary Quantum computing really is one of the most exciting things in science—just not for the reasons you usually hear Quantum computers are not known to provide any practically-important speedups for NP-complete problems (though they might provide modest ones, and they almost certainly provide speedups for problems like factoring and quantum simulation) 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 nontrivial insights in both directions