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**Quantum Complexity and Fundamental Physics**

BQP PSPACE NP P PostBQP Scott Aaronson MIT

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**RESOLVED: That the results of quantum complexity research can deepen our understanding of physics.**

That this represents an intellectual payoff from quantum computing, whether or not scalable QCs are ever built. A Personal Confession When proving theorems about QCMA/qpoly and QMAlog(2), sometimes even I wonder whether it’s all just an irrelevant mathematical game…

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**But then I meet distinguished physicists who say things like:**

“A quantum computer is obviously just a souped-up analog computer: continuous voltages, continuous amplitudes, what’s 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 for, e.g., Schor’s algorithm for factoring prime numbers is a detail left for specialists)” “Who cares if my theory requires Nature to solve the Traveling Salesman Problem in an instant? Nature solves hard problems all the time—like the Schrödinger equation!”

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**That’s why YOU should care about quantum computing**

The biggest implication of QC for fundamental physics is obvious: “Shor’s Trilemma” Because of Shor’s factoring algorithm, either the Extended Church-Turing Thesis—the foundation of theoretical CS for decades—is wrong, textbook quantum mechanics is wrong, or there’s a fast classical factoring algorithm. That’s why YOU should care about quantum computing All three seem like crackpot speculations. At least one of them is true!

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Rest of the Talk Eleven of my favorite quantum complexity theorems … and their relevance for physics PART I. BQP-Infused Quantum Foundations BQP P#P, BBBV lower bound, collision lower bound, limits of random access codes PART II. BQP-Encrusted Many-Body Physics QMA-completeness and the limits of adiabatic computing PART III. Quantum Gravity With a Side of BQP Black holes as mirrors, topological QFTs, computational power of nonlinearities, postselection, and CTCs

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**PART I. BQP-Infused Quantum Foundations**

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**Quantum Computing Is Not Analog**

is a linear equation, governing quantities (amplitudes) that are not directly observable This fact has many profound implications, such as… BQP EXP P#P The Fault-Tolerance Theorem Absurd precision in amplitudes is not necessary for scalable quantum computing

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**QCs Don’t Provide Exponential Speedups for Black-Box Search**

BBBV I.e., if you want more than the N Grover speedup for solving an NP-complete problem, then you’ll need to exploit problem structure [Bennett, Bernstein, Brassard, Vazirani 1997] 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?

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**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) Conclusion [A. 2005]: If, in a hidden-variable theory like Bohmian mechanics, your whole life trajectory flashed before you at the moment of your death, then you could solve problems that are presumably hard even for quantum computers (Probably not NP-complete problems though) Can also reduce graph isomorphism to this problem QCs can “almost” find collisions with just one query to f! Measure 2nd register Nevertheless, any quantum algorithm needs (N1/3) queries to find a collision [A.-Shi 2002]

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**The Absent-Minded Advisor Problem**

Can you give your graduate student a state | with poly(n) qubits—such 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] Some consequences: Any n-qubit state can be “PAC-learned” using O(n) sample measurements—exponentially better than quantum state 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]

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**PART II. BQP-Encrusted Many-Body Physics**

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**Just one of many things we learned from this theory:**

QMA-completeness One of the great achievements of quantum complexity theory, initiated by Kitaev Just one of many things we learned from this theory: In general, finding the ground state of a 1D nearest-neighbor Hamiltonian is just as hard as finding the ground state of any physical Hamiltonian [Aharonov, Gottesman, Irani, Kempe 2007]

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**The Quantum Adiabatic Algorithm**

An amazing quantum analogue of simulated annealing [Farhi, Goldstone, Gutmann et al. 2000] This algorithm seems to come tantalizingly close to solving NP-complete problems in polynomial time! But… Why do these two energy levels almost “kiss”? Answer: Because otherwise we’d be solving an NP-complete problem! [Van Dam, Mosca, Vazirani 2001; Reichardt 2004]

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**PART III. Quantum Gravity With a Side of BQP**

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Black Holes as Mirrors Against many physicists’ intuition, information dropped into a black hole seems to come out as Hawking radiation almost immediately—provided you know the black hole’s state before the information went in [Hayden & Preskill 2007] Their argument uses explicit constructions of approximate unitary 2-designs

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**Topological Quantum Field Theories**

TQFTs Witten 1980’s Freedman, Kitaev, Larsen, Wang 2003 Jones Polynomial BQP Aharonov, Jones, Landau 2006

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**Beyond Quantum Computing?**

If QM were nonlinear, one could exploit that to solve NP-complete problems in polynomial time [Abrams & Lloyd 1998] I interpret these results as providing additional evidence that nonlinear QM, postselection, and closed timelike curves are physically impossible. Why? Because I’m an optimist. Quantum computers with postselected measurement outcomes could solve not only NP-complete problems, but even counting problems [A. 2005] R CTC R CR C Answer Quantum computers with closed timelike curves (i.e. time travel) could solve PSPACE-complete problems—but not more than that [A.-Watrous 2008]

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**For Even More Interdisciplinary Excitement, Here’s What You Should Look For**

A plausible complexity-theoretic story for how quantum computing could fail (see A. 2004) Intermediate models of computation between P and 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 in quantum gravity?)

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**A bold (but true) hypothesis linking complexity and fundamental physics…**

There is no physical means to solve NP-complete problems in polynomial time. Encompasses NPP, NPBQP, NPLHC… Prediction: Someday, this hypothesis will be as canonical as no-superluminal-signalling or the Second Law GOLDBACH CONJECTURE: TRUE NEXT QUESTION

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