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Computation, Quantum Theory, and You

Computation, Quantum Theory, and You. Scott Aaronson, UC Berkeley Qualifying Exam May 13, 2002. Talk Outline. Sermon 2. Quantum Computing Overview Collision Lower Bound Dynamical Models 5. Current and Future Work. 1. Sermon. The Computer Scientist’s Idea of Physics. + details.

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Computation, Quantum Theory, and You

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  1. Computation, Quantum Theory, and You Scott Aaronson, UC Berkeley Qualifying Exam May 13, 2002

  2. Talk Outline • Sermon 2. Quantum Computing Overview • Collision Lower Bound • Dynamical Models 5. Current and Future Work

  3. 1. Sermon

  4. The Computer Scientist’s Idea of Physics + details

  5. What Does Our World Have That Conway’s Doesn’t? • 3 or more spatial dimensions • Continuity? • Relativistic covariance Quantum theory • Quantum theory • And more?

  6. Quantum theory What we experience My Own View…

  7. Research Goal Prove complexity results, focusing on quantum computing, that are motivated by this gap between physics and what we experience. (Disclaimer: I will not bridge the gap in my thesis.)

  8. 2. Quantum Computing

  9. Some Milestones 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

  10. The Quantum Model • State of computer: superposition over binary strings • To each string Y, associate complex amplitudeY • Y |Y|2 = 1 • On measuring, see Y with probability |Y|2 • Dirac ket notation: State written | = Y Y |Y • Each |Y is called a basis state

  11. Unitary Evolution • Quantum state changes by multiplying amplitude vector with unitary matrix: |(t+1)= U|(t) • U is unitary iff U-1=U†,† conjugate transpose (Linear transformation that preserves norm=1) • Example: • Circuit model: U must be efficiently computable Black-box model: No such restriction (|0+ |1)/2 = |1

  12. By end: Quantum Query Model • State after t queries: : workbits i: index to query z: output • Query: |,i,z |xi,i,z • Arbitrary unitaries that don’t depend on X

  13. 3. Collision Lower Bound

  14. Collision Problem • Given • Promised: • (1) X is one-to-one (permutation) or • (2) X is two-to-one • Problem: Decide which w.h.p., using few queries to the xi • Randomized alg: (n)

  15. Result • Any quantum algorithm for the collision problem uses (n1/5) queries (A, STOC’2002) • Shi improved to (n1/4) • (n1/3) when |range| >> n • Previously no lower bound better than (1). Open since 1997

  16. Implications • Oracle A for which SZKA BQPA • SZK: Statistical Zero Knowledge • No “trivial” polytime quantum algorithms for • graph isomorphism • nonabelian hidden subgroup • breaking cryptographic hash functions

  17. Brassard-Høyer-Tapp (1997) (n1/3) quantum alg for collision problem Grover’s algorithm over n2/3 xi’s Do I collide with any of the pink xi’s? n1/3 xi’s, queried classically, sorted for fast lookup

  18. Previous Lower Bound Techniques • Block sensitivity (Beals et al. 1998): Q2(f) = (bs(f)) • Quantum adversary method (Ambainis 2000) • Problem: Every 1-1 input differs in at least n/2 places from every 2-1 input

  19. Proof: Let t,X,,i,z = amplitude of |,i,z after t queries. t,X,,i,z is poly of degt, by induction. Base case (t=0) trivial. Unitaries can’t increase degree. Query replaces t,X,,i,z by Lemma (follows Beals et al. 1998): Let (xi,h)=1 if xi=h, 0 otherwise. Then P(X) is poly of deg  2T over the (xi,h).

  20. Let Input Distribution • D(g): Uniform distribution over g-1 inputs • Technicality: g might not divide n • But assume for simplicity that it does • Exercise: Show that, if T=O(n), then P(g) is a polynomial of degree  2T in g for integers 1gn.

  21. Let • Then for some I, Monomials of P(X) • I(X) = product of r variables (xi,h)

  22. So • since Calculating (I,g): #1 • “Range” of I: Y. w=|Y|. • (I,g) = 0 unless YS (“range” of X)

  23. # of g-1 inputs X with range S s.t. I(X)=1: Calculating (I,g): #2 • Given an S containing Y, # of g-1 inputs of size n: n!/(g!)n/g • Let {y1,…,yw} be distinct values in Y • ri = # of times yi appears in Y • r1 + … + rw = r

  24. Polynomial in g of degree w + (r-w) = r  2T Becomes ~polynomial(g)

  25. Markov’s Inequality Let P(x) be a poly with b1P(x)b2 for all a1xa2 and |dP(x*)/dx|c for some a1x*a2. Then Large derivative Short Long

  26. Lower Bound • 0  P(g)  1 for all 0  g  n • P(1)  1/10 and P(2)  9/10 • So dP/dg  4/5 somewhere • (n1/4) lower bound would follow if g always divided n • Can fix to obtain an (n1/5) bound • Shi found a better way to fix

  27. 4. Dynamical Models

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

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

  30. Results (submitted to PRL, quant-ph/0205059) • What if you could examine an observer’s entire history? Defined class DQP • Showed SZK  DQP. Combined with collision bound, implies oracle A for which BQPA DQPA • Can search an N-element list in order N1/3 steps, though not fewer

  31. DQP BQP SZK BPP

  32. 5. Current and Future Work

  33. BQP versus PH • Almost-complete (?!) joint work with Umesh • Conjecture: BQPA PHA for an oracle A • (Best known: BQPA (2)A) • Use Recursive Fourier Sampling • Have reduced problem to generalizing the Razborov-Smolensky circuit lower bound • Need to show “replacer gates” don’t help us compute sum modulo 3

  34. BPPA vs. BQPA for random A • Conjecture: If BPP=BQP, then BPPA=BQPA with probability 1 • What I can show: If BPP=BQP then BPTime[polylog]=BQTime[polylog] • What’s missing: Extend the result of Beals et al. (1998) that D(f)=O(Q2(f)6) for all total f to almost-total f • Does the same hold for BPP vs. SZK, or even P vs. NPcoNP? (cf. Rudich’s thesis)

  35. Limitations of Shor-like algorithms • Defined a class BPPBQPshorBQP • Subclass of quantum algorithms that prepare a state x|x|f(x), then ignore |f(x) and do something “simple” to |x • Conjecture 1: BQPshorAM. Implies that if NPBQPshor then PH=2 • Conjecture 2: Shor-like query algorithms yield no asymptotic speedup for any total function

  36. Physics Modulo Complexity Assumptions • Can some version of M-theory decide SAT? (cf. Preskill’s talk) • If so, move on to the next version! • “Anthropic computer” for solving NP-complete problems efficiently • Stupid question: Why can’t I just “will” myself to solve NP-complete problems? (Or generate truly random sequences?)

  37. Postulate: No matter who you are, someone can give you a 3SAT instance that you can’t decide with probability ½+. What constraints does that impose?

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