The Complexity of Sampling Histories

1. The Complexity of Sampling Histories Scott Aaronson, UC Berkeley http://www.cs.berkeley.edu/~aaronson August 5, 2003

2. Words You Should Stop Me If I Use Words I’ll Stop You If You Use Words To Be Careful With polysize oracle relativizing zero-knowledge #P-complete nonuniform holonomy gauge SU(2) intertwinor kinematical Lagrangian loop string

3. Outline • Why you should stay up at night worrying about quantum mechanics • Dynamical quantum theories • Solving Graph Isomorphism by sampling histories • Search in N1/3 queries (but not fewer)

4. Quantum mechanics What we experience

5. You Assumption Time Quantum state of the universe

6. What is the probability that you see the dot change color? A Puzzle • Let |OR = you seeing a red dot • |OB = you seeing a blue dot

7. The Goal Quantum state Quantum state Unitary matrix Probability distribution Probability distribution Stochastic matrix

8. Why Look for This? • Quantum theory says nothing about multiple-time or transition probabilities • Reply: • “But we have no direct knowledge of the past anyway, just records” • Then what is a “prediction,” or the “output of a computation,” or the “utility of a decision”?

9. Bohm’s Theory • Gives a deterministic evolution rule for particle positions and momenta • But doesn’t make sense for discrete observables: • Mathematicianly approach: Study the set of all discrete dynamical rules, without presupposing one of them is “true”

10. Our Results • We define dynamical theories for obtaining classical histories, and investigate what axioms they can satisfy • We give evidence that by examining a history, one could solve problems that are intractable even for a quantum computer • Graph Isomorphism and Approximate Shortest Lattice Vector in polynomial time • Unordered search in N1/3 steps instead of N1/2 • We obtain the first model of computation “slightly” more powerful than quantum computing

11. Given an NN unitary U and state acted on, returns a stochastic matrix Dynamical Theory • Fix an N-dimensional Hilbert space (N finite) and orthogonal basis • Must marginalize to single-time probabilities of quantum mechanics: diagonal entries of  and UU-1

12. Axiom: Symmetry D is invariant under relabeling of basis states:

13. Axiom: Indifference If U acts on and is the identity on H2, then S should also be the identity on H2 Can formalize without tensor products: partition U into minimal blocks of nonzero entries Not the same as commutativity:

14. If UA applied first: If UB applied first: Theorem: No dynamical theory satisfies both indifference and commutativity Proof: Suppose A and B share an EPR pair UA applies /8 rotation to first qubit, UB applies -/8 to second qubit. Consider probability p of being at |00 initially and |10 at the end

15. Axiom: Robustness • Small (1/poly(N)) change to  or U •  Small (1/poly(N)) change to joint probabilities matrix, S·diag() Arguably that’s needed for any physical theory or model of computation

16. Example 1: Product Dynamics Take probabilities at any two times to be independent of each other Symmetric, robust, commutative, but not indifferent

17. Example 2: Dieks Dynamics Partition U into minimal blocks, then apply product dynamics separately to each Symmetric, indifferent, but not commutative or robust

18. Theorem: Suppose Then there is a weight-1 “flow” through the network where flow through an edge can’t exceed the edge’s capacity

19. Proof Idea: By the Max-Flow-Min-Cut Theorem (Ford-Fulkerson 1956), it suffices to show that any set of edges separating s from t (a cut) has total capacity at least 1. Let A,B be right, left edges respectively not in cut C. Then the capacity of C is so we need to show Fix U and consider maximum of right-hand side. Equals the max eigenvalue of a positive semidefinite matrix, which we can analyze using some linear algebra…

20. Example 3: Flow Dynamics Using the previous theorem, we construct a dynamical theory that satisfies the symmetry, indifference, and robustnessaxioms Not obvious a priori that any such theory exists

21. Model of Computation • Polynomial-time classical computation, with one query to a history oracle • Oracle takes as input descriptions of quantum circuits U1,…,UT • Any dynamical theory D induces a distribution D over classical histories for • Oracle chooses a symmetric robust indifferent theory D “adversarially,” then returns a sample from D • At least as powerful as standard quantum computing

22. The Graph Isomorphism Problem • Decide whether two graphs G and H are isomorphic • The best known algorithm takes about time n = number of vertices • But we don’t think Graph Isomorphism is NP-complete • Intuitively, it’s “only” as hard as counting collisions in • Could be easier than finding a needle in a haystack! 

23. The Collision Problem 3 6 1 5 4 2 vs. 6 2 2 5 6 5 • Given a list of N numbers x1,…,xN, you’re promised that either every number occurs once, or every number occurs twice. Decide which. • Best classical algorithm makes ~ queries (“birthday paradox”) • Brassard, Høyer, Tapp 1997 gave a quantum algorithm that makes ~N1/3 queries • Is there a faster quantum algorithm—say, log N queries? If so, we’d get a polynomial-time quantum algorithm for Graph Isomorphism!

24. The Collision Problem (con’t) • Aaronson 2002: Any quantum algorithm needs at least ~N1/5 queries • Improved by Shi to ~N1/3 queries • Previously, couldn’t even rule out constant number of queries! • Proofs use multivariate polynomials • Implications: • No “dumb” quantum algorithm for Graph Isomorphism • “Oracle separation” between the complexity classes BQP (Bounded-Error Quantum Polynomial-Time) and DQP (Dynamical Quantum Polynomial-Time)

25. ConjecturedWorld Map NP Satisfiability, Traveling Salesman, etc. DQPMy New Class Graph Isomorphism Approximate Shortest Vector Factoring BQPQuantumPolynomialTime PPolynomial Time

26. Two bitwise Fourier transforms GOAL: When we inspect the classical history, see both |i and |j with high probability Solving the Collision Problem by Sampling Histories Suppose every number occurs twice. Then “Measurement” of 2nd register

27. Solving the Collision Problem by Sampling Histories (con’t) Theorem:Under any dynamical theory satisfying the symmetry and indifference axioms, the first Fourier transform makes the hidden variable “forget” whether it was at |i or |j. So after the second Fourier transform, it goes to |i half the time and |j half the time; thus with ½ probability we see both |i and |j in the history Proof Idea: Use symmetry axiom, together with automorphisms of Indifference axiom needed to “trace out” second register

28. Hidden variable Finding a Marked Item in N1/3 Queries Probability of observing the marked item after T iterations is ~T2/N N1/3 iterations of Grover’s quantum search algorithm

29. N1/3 Search Algorithm Is Optimal • Bennett, Bernstein, Brassard, Vazirani 1996: If a quantum computer searches a list of N items for a single randomly-placed marked item, the probability of observing the marked item after T steps is at most • So probability of observing it in a history of the first T steps is at most

30. Summary: If “your whole life flashed before you in an instant,” and if you’d prepared for this by putting your brain in certain superpositions, then (under reasonable axioms) you could solve Graph Isomorphism in polynomial time • But probably still not Satisfiability • Contrast: Nonlinear quantum mechanics would put Satisfiability and even harder problems in polynomial time (Abrams and Lloyd 1998)

31. What does the postulate imply? (under plausible complexity assumptions) • Postulate: NP-complete problems can’t be efficiently solved in physical reality • Justification for the postulate: Maybe I’m wrong, but then I’d be too busy solving NP-complete problems to care that I was wrong (1) Quantum states evolve linearly (2) We can’t make unlimited-precision measurements (3) The “self-sampling” anthropic principle (Bostrom 2000) is false (4) Constraints on quantum gravity? • The postulate does not imply your whole life couldn’t flash before you in an instant