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

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 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?

P Efficiently solvable NP Efficiently verifiable NP- complete NP-hard All NP problems are efficiently reducible to these 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 … “OUR STANDARD MODEL”

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

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? An important presupposition underlying P vs. NP is the

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

Time Travel Computer S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent, Proceedings of the Royal Society A 465: , arXiv:

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

Ah, but what about quantum computing? (you knew it was coming) Quantum computing: “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 simulate—and 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)

Quantum Mechanics in One Slide Quantum Mechanics: Linear transformations that conserve 2-norm of amplitude vectors: Unitary matrices Probability Theory: Linear transformations that conserve 1-norm of probability vectors: Stochastic matrices

Journalists Beware: A quantum computer is NOT like 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

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

Can QCs Actually Be Built? Where we are now: A quantum computer has factored 21 into 3  7, 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 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 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) HiHi Hamiltonian with easily- prepared ground state HfHf Ground state encodes solution to NP-complete problem Problem: “Eigenvalue gap” can be exponentially small

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 Could be invoked to “explain” why adiabatic systems have small spectral gaps, why protein folding gets stuck in metastable states, why the Schrödinger equation is linear, why time only flows in one direction… “The No-SuperSearch Postulate” There is no physical means to solve NP-complete problems in polynomial time.

BosonSampling (with Alex Arkhipov): A proposal for a rudimentary optical quantum computer, which doesn’t seem useful for anything (e.g. breaking codes), but does seem hard to simulate using classical computers Some of My Recent Research Computational Complexity of Decoding Hawking Radiation: Building on a striking recent proposal by Harlow and Hayden—that part of the resolution of the black hole information problem might be that reconstructing the infalling information from the Hawking radiation would require an exponentially long computation

My suggested research agenda: Prove P≠NP Prove that not even quantum computers can solve NP- complete 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 No-SuperSearch or related impossibility principles to make progress in quantum gravity Conclusion