One Complexity Theorist’s View of Quantum Computing

1. One Complexity Theorist’s View of Quantum Computing Lance FortnowNEC Research Institute

2. Comp.Theory FAQ • 8. Complexity Theory • (a) Lower Bounds • (b) YACC (Yet Another Complexity Class) • Our ability to understand and handle new models of computation comes from our experience studying previous notions. • Case in Point: Quantum Computing

3. BQP: Yet AnotherComplexity Class Lance Fortnow NEC Research Institute

4. Quantum Computation • A computation model based on quantum principles of physics. • Ability to enter many parallel “states” and use interference to recover important information. • Transformations must be unitary.

5. Dephysicfying Quantum • To understand the computational powers of quantum computing, we should ignore the underlying physical model. • Nondeterministic computation has no known underlying physical model yet we have a good understanding of its computational power.

6. The Quantum Class BQP • The set of languages L such that there is a Polynomial-time Quantum Turing machine M such that for all strings x, • If x is in L then the measured probability of acceptance of M on input x is at least 2/3. • If x is not in L then the measured probability of acceptance of M on input x is at most 1/3.

7. Oddities of Quantum Computing • Many Parallel States • Similar to Probabilistic Computation. • Interference • Similar ideas in Counting Complexity. • Unitary Transformations • New and what makes quantum computing so hard to classify precisely.

8. A Product Machine • Traditional nondeterministic Turing machine has a transition function • Consider a generalized machine with transition function

9. The Computation Matrix • The function d imposes a linear function mapping configurations to themselves. • Consider the matrix Md capturing this linear function. The value of the computation after t steps is:

10. NP as Matrix Multiplication

11. #P as Matrix Multiplication

12. GapP as Matrix Multiplication

13. BPP as Matrix Multiplication

14. Small Changes

15. Small Changes

16. Small Changes

17. BQP as Matrix Multiplication

18. Questions • Where’s the Physics? • Where’s the <bra| and |ket>‘s? • Where’s the real/complex numbers? • Don’t we need reversibility? • What if there is more than one accepting configuration? • Where’s the measurements?

19. Where’s the Physics? • Car makers have given us a model from which we can drive a car. Details of how the car works are not necessary.

20. Where’s <bra| & |ket>’s? • Fancy way that physicists specify row and column vectors. • Don’t need to deal with them when studying quantum complexity. • Computer scientists like balance. • What’s wrong with braT and ket? • Scares away newcomers.

21. Where’s the complex numbers? • For BQP one can assume the transitions come from {-1,-4/5,-3/5,0, 3/5,4/5,1} instead of computable complex numbers. • Noncomputable numbers allow encoding of noncomputable functions. Similar problem in classical model.

22. Don’t we need reversibility? • The set of matrices M that preserve the L2 norm are unitary: M(M*)T is the identity. • In particular M is invertible so the computation could be reversed. • Reversibility is not a requirement of quantum computing but a consequence.

23. One accepting configuration? • In most models, can assume one accepting configuration by having machine erase work tape and moving to single accept state. • Not reversible process. • Can be simulated in quantum with negligible additional error by writing answer and reversing the rest of the computation.

24. Where’s the measurements? • Squaring value simulates process of measurement at end. • Taking measurements during computation does not give additional power.

25. BQP - A good definition • Simple and Robust. • Based on a physical model. • Contains interesting problems. • Other Quantum Classes not as robust: • EQP - Differences in set of allowable amplitudes may affect class. • BQL - When measurements are made may affect class.

26. BQP as Matrix Multiplication

28. The Class AWPP

29. The Class AWPP • “Almost-Wide Probabilistic Polynomial Time” • Previously Studied • Fenner-Fortnow-Kurtz-Li - 1993 • Lide Li’s Thesis - 1993 • AWPP contains BQP

30. Properties of AWPP • BQP Í AWPP Í PP Í PSPACE • AWPP is low for PP • PPAWPP = PP • For any L in AWPP, PPL = PP. • There exists a relativized world where AWPP = P and the polynomial-time hierarchy is infinite.

31. Properties of BQP • BQP Í PP Í PSPACE • BQP is low for PP • PPBQP = PP • For any L in BQP, PPL = PP. • There exists a relativized world where BQP = P and the polynomial-timehierarchy is infinite.

32. Diagram of Classes PSPACE PH PP NP AWPP PP-Low BPP BQP P

33. Diagram of Classes PSPACE PH PP NP AWPP PP-Low BPP BQP P

34. Diagram of Classes PSPACE PH PP NP AWPP PP-Low BPP BQP P

35. The Polynomial-Time Hierarchy • Nondeterministic Computation is a misleading title. Really Existential. • Similarly can have Universal Computation. • Alternating TM - Switches back and forth between Existential and Universal. • Unbounded Alternations - PSPACE • Constant Alternations - PH

36. BQP in PH? • Bernstein-Vazirani relativized language does not appear to sit in PH. • It would if we allowed slightly more than polynomial-time or constant alternations. • Suggestion: • Try to show that BQP can be solved in quasipolynomial time and/or polylogarithmic alternations.

37. Diagram of Classes PSPACE PH PP NP AWPP PP-Low BPP BQP P

38. NP in BQP? • Relative to a random oracle NP is in AWPP. • Two problems: • Random oracles do not give us a good view of the world. • Need unitary transformations to get NP in BQP. • Make it difficult to obtain bad consequences of NP in BQP.

39. Black Box Model

40. Black Box Model I N P U T

41. Black Box Model

42. Black Box Model N

43. Black Box Model N T • Count only number of queries made. • We do not care about computation time. • Also known as decision tree or oracle model. • Hard to define decision trees properly for quantum machines.

44. OR Function • The OR function requires all N queries on some input of N bits for a deterministic machine. • Adversary always answers zero on all queries. • OR has small nondeterministic black box complexity (1 query).

45. Black Box Classes • P – Polylogarithmic in N queries • NP – Nondeterministic polylogarithmic in N queries • The OR functions separates black box P from black box NP. • How about BQP?

46. Black Box BQP • The probability of acceptance of a black box BQP machine using t queries is a polynomial of degree at most 2t. • Easy to see from Matrix Multiplication view of BQP.

47. BQP as Matrix Multiplication

48. The OR function • The OR function has degree n. • However a BQP black box need only approximate the OR function. • Any polynomial that approximates the OR functions has degree (n).

49. Tightness of OR • Any black box BQP machine must use (n) queries. • OR function separates NP from BQP. • Grover shows that O(n) queries suffice to compute OR on a BQP machine.

50. General Result • Any function f:{0,1}n  {0,1} that can be approximated by a degree d polynomial has a deterministic black box algorithm using O(d6) queries. • Due to Nisan-Szegedy, Beals-Buhrman-Cleve-Mosca-de Wolf.