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Computational Complexity and Fundamental Physics Scott Aaronson (MIT) www.scottaaronson.com 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? What are the ultimate physical limits on what can be feasibly computed? And do those limits have any implications for physics? NP-hard All NP problems are efficiently reducible to these Hamilton cycle Steiner tree Graph 3-coloring Satisfiability Maximum clique … NPcomplete NP Efficiently verifiable Graph connectivity Primality testing Matrix determinant Linear programming … P Efficiently solvable Matrix permanent Halting problem … Factoring Graph isomorphism … 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 An important presupposition underlying P vs. NP is the The Extended Church-Turing Thesis (ECT) “Any physically-realistic computing device can be simulated by a deterministic or probabilistic Turing machine, with at most polynomial overhead in time and memory” But how sure are we of this thesis? What would a challenge to it look like? 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 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) In the 1980s, Feynman, Deutsch, and others noticed that quantum systems with n particles seemed to take ~2n time to simulate—and had the amazing idea of building a “quantum computer” to overcome that problem Quantum computing: “The power of 2n complex numbers working for YOU” Quantum Mechanics in One Slide 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 Journalists Beware: A quantum computer is NOT like a massively-parallel classical computer! x x x1,, 2 n Exponentially many possible measurement outcomes, but you only see one, probabilistically! Any hope for a speedup rides on the magic of interference between positive and negative contributions to an amplitude BQP (Bounded-Error Quantum Polynomial-Time): The class of problems solvable efficiently by aInteresting quantum computer, defined by Bernstein and Vazirani in 1993 Shor 1994: Factoring integers is in BQP NP-complete NP BQP Factoring P Can QCs Actually Be Built? Where we are now: A quantum computer has factored 21 into 37, with high probability (Martín-López et al. 2012) Why is scaling up so hard? Because of decoherence: unwanted interaction between a QC and its external environment, “prematurely measuring” the quantum state A few skeptics, in CS and physics, even argue that building a QC will be fundamentally impossible I don’t expect them to be right, but I hope they are! If so, it would be a revolution in physics And for me, putting quantum mechanics to the test is the biggest reason to build QCs—the applications are icing! Key point: 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 Most striking application so far of complexity to fundamental physics? Hawking 1970s: Black holes radiate The radiation seems thermal (uncorrelated with whatever fell in)—but if quantum mechanics is true, then it can’t be Susskind et al. 1990s: “Black-hole complementarity.” In string theory / quantum gravity, the Hawking radiation should just be a scrambled re-encoding of the same quantum states that are also inside the black hole The Firewall Paradox [Almheiri et al. 2012] If the black hole interior is “built” out of the same qubits coming out as Hawking radiation, then why can’t we do something to those Hawking qubits (after waiting ~1067 years for enough to come out), then dive into the black hole, and see that we’ve completely destroyed the spacetime geometry in the interior? Entanglement among Hawking photons detected! Harlow-Hayden 2013: Sure, there’s some unitary transformation that Alice could apply to the Hawking radiation, that would generate a “firewall” inside the event horizon. But how long would it take her to apply it? They gave evidence that this problem would take exponential time, even for a quantum computer—in which case, long before one had made a dent in the problem, the black hole would’ve already evaporated anyway! Their evidence relied on a lower bound on the number of steps needed by a quantum computer to find collisions (i.e., duplicates in a list of numbers), which I proved in 2002 Recently I improved their argument, so that it no longer needs the collision lower bound, just one-way functions Conclusion My suggested research agenda: Prove P≠NP Prove that not even quantum computers can solve NPcomplete problems Build a scalable quantum computer (or even more interesting, show that it’s impossible) Clarify whether all of known physics can be simulated by a quantum computer Use computational complexity ideas to make progress toward a quantum theory of gravity