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## Quantum Complexity in addition to Fundamental Physics A Personal Confession Rest of the Talk PART I. BQP-Infused Quantum Foundations Quantum Computing Is Not Analog

Ross, Woody, News Director has reference to this Academic Journal, PHwiki organized this Journal Quantum Complexity in addition to Fundamental Physics Scott Aaronson MIT RESOLVED: That the results of quantum complexity research over the last two decades have deepened our underst in addition to ing of physics. That this represents an intellectual payoff from quantum computing, whether or not scalable QCs are ever built. A Personal Confession While proving theorems about QCMA/qpoly in addition to QMAlog(2), sometimes even I wonder whether its all just an irrelevant mathematical game A quantum computer is obviously just a souped-up analog computer: continuous voltages, continuous amplitudes, whats the difference A quantum computer with 400 qubits would have ~2400 classical bits, so it would violate a cosmological entropy bound My classical cellular automaton model can explain everything about quantum mechanics! (How to account as long as , e.g., Schors algorithm as long as factoring prime numbers is a detail left as long as specialists) Who cares if my theory requires Nature to solve the Traveling Salesman Problem in an instant Nature solves hard problems all the timelike the Schrödinger equation! But then I meet distinguished physicists who say things like:

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The biggest implication of QC as long as fundamental physics is obvious: Shors Trilemma the Extended Church-Turing Thesisthe foundation of theoretical CS as long as decadesis wrong, textbook quantum mechanics is wrong, or theres a fast classical factoring algorithm. All three seem like crackpot speculations. At least one of them is true! Because of Shors factoring algorithm, either Ten of my favorite quantum complexity theorems in addition to their relevance as long as physics PART I. BQP-Infused Quantum Foundations BQP P P, BBBV lower bound, collision lower bound, limits of r in addition to om access codes PART II. BQP-Encrusted Many-Body Physics QMA-completeness, the limits of adiabatic computing, search by quantum walk PART III. Quantum Gravity With a Side of BQP TQFTs, postselection & closed timelike curves, black holes as mirrors Rest of the Talk PART I. BQP-Infused Quantum Foundations

Quantum Computing Is Not Analog The Fault-Tolerance Theorem Absurd precision in amplitudes is not necessary as long as scalable quantum computing is a linear equation, governing quantities (amplitudes) that are not directly observable This fact has many profound implications, such as I.e., if you want more than the N Grover speedup as long as solving an NP-complete problem, then youll need to exploit problem structure [Bennett, Bernstein, Brassard, Vazirani 1997] QCs Dont Provide Exponential Speedups as long as Black-Box Search The BBBV No SuperSearch Principle can even be applied in physics (e.g., to lower-bound tunneling times) Is it a historical accident that quantum mechanics courses teach the Uncertainty Principle but not the No SuperSearch Principle Computational Power of Hidden Variables Consider the problem of breaking a cryptographic hash function: given a black box that computes a 2-to-1 function f, find any x,y pair such that f(x)=f(y) Can also reduce graph isomorphism to this problem QCs can almost find collisions with just one query to f! Nevertheless, any quantum algorithm needs (N1/3) queries to find a collision [A.-Shi 2002] Conclusion [A. 2005]: If, in a hidden-variable theory like Bohmian mechanics, your whole life trajectory flashed be as long as e you at the moment of your death, you could solve problems that are (probably) intractable even as long as quantum computers (Probably not NP-complete problems though)

The Absent-Minded Advisor Problem Some consequences: BQP/qpoly PostBQP/poly [A. 2004] Any n-qubit state can be PAC-learned using O(n) sample measurementsexponentially better than tomography [A. 2006] One can give a local Hamiltonian H on poly(n) qubits, such that any ground state of H can be used to simulate on all yes/no measurements with small circuits [A.-Drucker 2009] Can you give your graduate student a state with poly(n) qubitssuch that by measuring in an appropriate basis, the student can learn your answer to any yes-or-no question of size n NO [Ambainis, Nayak, Ta-Shma, Vazirani 1999] PART II. BQP-Encrusted Many-Body Physics QMA-completeness Just one of many things we learned from this theory: In general, finding a ground state of a 1D nearest-neighbor Hamiltonian is just as hard as finding the ground state of any Hamiltonian [Aharonov, Gottesman, Irani, Kempe 2007] One of the great achievements of quantum complexity theory, initiated by Kitaev

The Quantum Adiabatic Algorithm Why do these two energy levels almost kiss An amazing quantum analogue of simulated annealing [Farhi, Goldstone, Gutmann et al. 2000] Seems to come tantalizingly close to solving NP-complete problems in polynomial time! But One answer: because NP-complete problems are hard! [Van Dam, Mosca, Vazirani 2001; Reichardt 2004] Quantum Walks To develop a quantum walk algorithm as long as spatial search, algorithmists essentially had to rediscover the Dirac equation [Childs, Goldstone 2004] A free particle in a 2D box To develop a quantum walk algorithm as long as game-tree search, they wouldve had to rediscover scattering theory [Farhi, Goldstone, Gutmann 2007] To develop a quantum walk algorithm as long as graph isomorphism, will we need to rediscover some more physics [Bacon] PART III. Quantum Gravity With a Side of BQP

Topological Quantum Field Theory Freedman, Kitaev, Larsen, Wang 2003 Aharonov, Jones, L in addition to au 2006 Witten 1980s TQFTs Jones Polynomial BQP Beyond Quantum Computing If QM were nonlinear, one could exploit that to solve NP-complete problems in polynomial time [Abrams & Lloyd 1998] Quantum computers with closed timelike curves (i.e. time travel) could solve PSPACE-complete problemsbut not more than that [A.-Watrous 2008] Quantum computers with postselected measurements could solve not only NP-complete problems, but even counting problems [A. 2005] Black Holes as Mirrors Against many physicists intuition, in as long as mation dropped into a black hole seems to come out as Hawking radiation almost immediatelyprovided you know the black holes state be as long as e the in as long as mation went in [Hayden & Preskill 2007] Their argument uses explicit constructions of approximate unitary 2-designs

For Even More Interdisciplinary Excitement, Heres What You Should Look For A plausible complexity-theoretic story as long as how quantum computing could fail (see A. 2004) Intermediate models of computation between P in addition to BQP (highly mixed states restricted sets of gates) Foil theories that lead to complexity classes slightly larger than BQP (only example I know of: hidden variables) A sane notion of quantum gravity polynomial time (first step: a sane notion of time) A bold (but true) hypothesis linking complexity in addition to fundamental physics Encompasses NPP, NPBQP, NPLHC My Prediction: Someday, this hypothesis will be about as canonical as the 2nd Law or no superluminal signalling

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