Quantum Computation

1. Stephen Jordan Quantum Computation

2. Church-Turing Thesis • Weak Form: Anything we would regard as “computable” can be computed by a Turing machine. • Strong Form: Anything we would regard as efficiently computable can be computed in polynomial time by a Turing machine.

3. Models of Computation • Turing machines • multiple tapes • multiple read/write heads • Logic Circuits • Parallel Computation • All have been shown polynomially equivalent to Turing machines

4. Thesis Revised? • “Computers are physical objects and computations are physical processes. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics.” -David Deutsch

5. What Quantum Computers Are • A reasonable model of computation based on currently known physics • Apparently more powerful than the Turing machine • can do prime factorization in polynomial time • The first challenge to the strong Church-Turing thesis.

6. What Quantum Computers Aren't • Extant • A challenge to the weak Church-Turing thesis • Just like classical computers except smaller and faster • Analog

7. Relation To Other Models

8. Quantum Church-Turing Thesis? • Many models of quantum computation: • quantum turing machines • quantum circuits • adiabatic quantum computation • measurement based quantum computation • nonabelian anyons • All have equivalent power (BQP) • One exception: one clean qubit model

10. State of The Art • Quantum Computers • many approaches • still in the laboratory • Quantum Cryptography • fundamentally unbreakable • commercialized

11. Earliest Inklings • At small scales the laws of classical mechanics break down and quantum mechanics takes over. • Can computers still work when their components reach this scale? • Yes: any computation can be made reversible with minimal overhead. [1973] • Quantum computers can do reversible computation. C. Bennett

12. Advantages? • “The full description of quantum mechanics for a large system with R particles...has too many variables. It cannot be simulated with a normal computer with a number of elements proportional to R.” [1982] • An n-bit number can be factored in time on a quantum computer. [1994] R. Feynman P. Shor

13. More Advantages • An unstructured database with N items can be searched in time. L. Grover • Quantum computers can efficiently simulate quantum systems. • Quantum computers cannot speed up all problems.

14. Quantum Mechanics • The state of a system is represented by a normalized complex vector. • Example: a bit

15. Dirac Notation

16. Inner Product

17. Two Bits

18. Dynamics!

19. Example

20. Measurement

21. Quantum Computing • Start with some state encoding your problem. • Example: factoring 9 = 1001 • Apply some sequence of unitary time evolutions. • Measure, and with high probability obtain a desired result, e.g. 3 = 0011

22. Quantum Computing • 2 questions about quantum computing • How can we build a quantum computer? • We'll ignore this. • What can we do with them? • We'll turn this into a precise question: For a problem of size n, how many computational steps do we need to solve it on a quantum computer?

23. Computational Problems • Examples • Find the prime factors of an n-digit number. • Find the shortest route visiting n cities. • Compute for given f. • Which problems can be solved with fewer steps on quantum computers than on classical computers for large n?

24. Model of Computation: Quantum Circuits • Use only k-body interactions, “gates” • k=2 suffices • CNOT + one qubit gates suffice • only finite precision required

25. Family of Quantum Circuits • One quantum circuit for each input size • Trivial Example: bitwise XOR

26. Circuit Complexity • Return to our original question: For a problem of size n, how many computational steps do we need to solve it on a quantum computer? • We can now make it precise: What is the minimum number of gates needed, as a function of n, in a family of quantum circuits which solves the problem?

27. Problems with Circuit Complexity • Circuit complexity is notoriously difficult to evaluate • Explicit circuit families (algorithms) provide upper bounds • Lower bounds are very difficult, even classically (e.g. P vs. NP)

28. Query Complexity • Many problems are naturally formulated in in terms of a blackbox f • Find • Find x s.t. f(x)=1 • Find x which minimizes f • Classical blackboxes can be made reversible, hence unitary

29. An Easier Question For a given problem, how many black box queries do we need to solve it on a quantum computer, as a function of problem size? • Algorithms provide upper bounds. • Information arguments provide lower bounds. • Quantum speedups for several black box problems are known. • In many cases matching quantum lower bounds are known.

30. Bernstein-Vazirani Problem

31. Classical Algorithm

33. Phase Kickback

34. Bernstein-Vazirani Algorithm

35. Classical Gradient Estimation • Classically, you need at least d+1 queries • Otherwise the system is underdetermined • Quantumly, one query suffices

37. Transforms • Hadamard transform on n bits uses n Hadamard gates • Quantum Fourier Transform on n bits can be done using gates • The transforms are on amplitudes! • Inverse transforms are easy. Just take the adjoint.

39. Minimizing a Quadratic Form

40. Further Reading • Michael Nielsen and Isaac Chuang, Quantum Computation and Quantum Information (2000)

41. An Optical Analogy

42. An Optical Analogy

43. Lower Bounds

44. Lower Bounds by Polynomials

45. Lower Bounds by Polynomials • After q queries, the amplitudes are polynomials of degree at most q, hence the p(1) is of degree 2q • Recall that desired result is some boolean function of the blackbox values • There is a minimal degree for a polynomial to match this function

46. Paturi's Theorem

47. Specific Lower Bounds

48. Other Techniques • Quantum adversary methods • Reductions

49. Further Reading • E. Bernstein and U. Vazirani, “Quantum complexity theory,” proceedings of STOC 1993 • S. Jordan, “Fast quantum algorithm for numerical gradient estimation,” Phys. Rev. Lett. 95, 050501 (2005) [quant-ph/0405146] • R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. De Wolf. “Quantum lower bounds by polynomials,” Journal of the ACM, Vol. 48, No. 4 (2001) [quant-ph/9802049]