1. BQP PSPACE NP P PostBQP Quantum Complexity and Fundamental Physics Scott Aaronson MIT

2. RESOLVED: That the results of quantum complexity research over the last two decades have deepened our understanding 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 and QMA log (2), sometimes even I wonder whether it’s all just an irrelevant mathematical game

3. “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 ~2 400 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!” But then I meet distinguished physicists who say things like:

4. The biggest implication of QC for fundamental physics is obvious: “Shor’s Trilemma” 1. the Extended Church-Turing Thesis—the foundation of theoretical CS for decades—is wrong, 2. textbook quantum mechanics is wrong, or 3. there’s a fast classical factoring algorithm. All three seem like crackpot speculations. At least one of them is true!  That’s why YOU should care about quantum computing Because of Shor’s factoring algorithm, either

5. Ten 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, the limits of adiabatic computing, search by quantum walk PART III. Quantum Gravity With a Side of BQP TQFT’s, postselection & closed timelike curves, black holes as mirrors Rest of the Talk

6. PART I. BQP-Infused Quantum Foundations BQP

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

8. 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] QC’s Don’t Provide Exponential Speedups for Black-Box Search BBBV 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”?

9. Computational Power of Hidden Variables Measure 2 nd register 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  (N 1/3 ) 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 before you at the moment of your death, you could solve problems that are (probably) intractable even for quantum computers (Probably not NP-complete problems though)

10. 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 measurements—exponentially 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) 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]

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

12. 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

13. 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]

14. Quantum Walks To develop a quantum walk algorithm for 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 for game-tree search, they would’ve had to rediscover scattering theory [Farhi, Goldstone, Gutmann 2007] To develop a quantum walk algorithm for graph isomorphism, will we need to rediscover some more physics? [Bacon]

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

16. Topological Quantum Field Theory Freedman, Kitaev, Larsen, Wang 2003 Aharonov, Jones, Landau 2006 Witten 1980’s TQFTs Jones Polynomial BQP

17. 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 problems—but 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] R CTC R CR C 000 Answer

18. 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

19. 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”?)

20. A bold (but true) hypothesis linking complexity and fundamental physics… GOLDBACH CONJECTURE: TRUE NEXT QUESTION There is no physical means to solve NP-complete problems in polynomial time. Encompasses NP  P, NP  BQP, NP  LHC… My Prediction: Someday, this hypothesis will be about as canonical as the 2 nd Law or no superluminal signalling