Short course on quantum computing

1. Short course on quantum computing Andris Ambainis University of Latvia

2. Lecture 2 Quantum algorithms and factoring

3. Factoring • Input: composite N. • Output: p, q  {2, …, N-1} s.t. pq=N. • Hard for classical computers. • Factoring large integers would break RSA.

4. Factoring • Quantum computers can factor integers in polynomial (quadratic) time [Shor’94]. • Similar approach also solves discrete logarithm by quantum algorithm. • Today: Shor’s algorithm.

5. Outline 1) Computational model. 2) Quantum parallelism and quantum interference. 3) Simon’s algorithm. 4) Shor’s algorithm.

6. Basic ideas • State space consisting of n (quantum) bits. • Elementary gates on 1 or 2 (qu)bits. • Efficiently computable = poly-size circuits.

7. Classical circuits X1 X2 X3 X5  ^ ^  Result

8. Quantum circuit H H H H Gates on quantum bits

9. Elementary gates (1) • Hadamard gate • Phase shift

10. Elementary gates (2) • Rotation by angle  • Controlled NOT

11. Universality • Any quantum computation can be performed by a circuit consisting of Hadamard, phase, rotation by /8 and controlled NOT gates.

12. Classical vs. quantum circuits • We have a classical circuit. • Can we construct a quantum circuit that computes the same function?

13. Reversibility • Assume f(x)=f(y)=z. • If then • U not unitary.

14. Reversibility We can transform a classical circuit for F to quantum circuit. |x> |x> F |0> |F(x)> Add extra input initialized to 0.

15. |a> |a(xy)> Example Quantum Classical x y |x> |x> |y> |y> ^ |0> |xy> Toffoli gate.

16. Quantum parallelism • By linearity, • Many evaluations of f in unit time. |x> |x> |0> |f(x)>  |x> |f(x)>  |x> |0> x x

17. Quantum parallelism • Once we measure we get one particular x and f(x). • Same as if we evaluated f on a random x.  |x> |f(x)> x

18. Quantum parallelism • Is it useful? • We cannot obtain all values f(x) from because quantum states cannot be measured completely. • We can obtain quantities that depend on many f(x).  |x> |f(x)> x

19. Quantum interference • Hadamard transform:

20. Quantum interference • Negative interference: |1> and -|1> cancel out one another. • Positive interference: |0> and |0> add up to a higher probability.

21. Parallelism+interference • Use quantum parallelism to compute many f(x). • Use interference to obtain information that depends on many values f(x). • Requires algebraic structure. • Ideal for number-theoretic problems (factoring).

22. Order finding • The order of aZN * modulo N is the smallest integer r>0 such that ar1 (mod N) • For example, order of 4 mod 7 is 3: 41 4, 42 =162, 43 =641 (mod 7). • Factoring reduces to order-finding.

23. Reduction • If ar1(mod N), then N divides ar-1. • If r even, ar-1=(ar/2-1)(ar/2+1). • If N is product of two or more primes, gcd(ar/2-1, N) is a nontrivial factor of N with probability at least 1/2.

24. Shor’s algorithm Repeat O(log n) times: • Generate random a{1, …, N-1}; • Check if (a, N)=1; • r = order(a); • If r even, check (ar/2-1, N).

25. Period finding • Function F:NN such that F(x)=F(x+r) for all x. • Find smallest r. |x> |x> F |0> |F(x)>

26. Simon’s problem • Function F:{0, 1}n {0, 1}n. • F(x+y)=F(x) for all x, + bitwise addition. • Find y. |x> |x> F |0> |F(x)>

27. Algorithm [Simon, 1994] H H |0> |y> F H H H H |f(x)> |0> Repeat n times and combine results y1,..., yn.

28. Hadamard transform

29. Hadamard on n qubits |0> H |0> H

30. Simon’s algorithm step-by-step H H |0> |y> F H H H H |F(x)> |0>

31. Measuring F(x) • Partial measurement. • We get some value y=F(x). • The state • collapses to part consistent with y=F(x).

32. Last step • We now have the state • How do we get z? • Measuring the first register would give only one of x and x+z.

33. Simon’s algorithm H H |0> |y> F H H H H |f(x)> |0>

34. Hadamard transform

35. Hadamard transform |x1> H |x2> H ... ... ... |xn> H

36. Hadamard transform Signs are the same iff zi yi= 0 mod 2.

37. Summary • Measuring the final state gives a vector y such that • n-1 such constraints uniquely determine z, with high probability.

38. Summary • Quantum parallelism: computing F for many values simultaneously. • Quantum interference: Hadamard transform.

39. Period finding • Function F:NN such that F(x)=F(x+r) for all x. • Find r. |x> |x> F |0> |F(x)>

40. Algorithm [Simon, 1994] H H |0> H H F H H |0> Repeat n times and combine results y1,..., yn.

41. Algorithm [Shor, 1994] QFT QFT |0> F |0> Find factor by continued fraction expansion.

42. Shor’s algorithm step-by-step QFT QFT |0> F |0>

43. Shor’s algorithm step by step • Measuring the second register leaves the first register in a state consisting of all x with the same F(x): |d>+|d+r>+…+|d+ir>

44. Quantum Fourier transform If M=2, this is Hadamard transform.

45. QFT detects periods • Assume r divides M. • Then, • If j relatively prime with r,

46. QFT detects periods • Assume r does not divide M. • Then, most of T| consists of |k> with

47. QFT detects periods r does not divide M r divides M 0 0 Can we find r?

48. Continued fraction expansion • Number theory algorithm. • Given k, M, finds j, r such that is smallest among all j and r  r0. • If M=(r2), correct w.h.p.

49. Summary of Shor’s factoring • Reduce factoring to period-finding. • Generate a quantum state with period r. • In the easy case, QFT transforms a state with period r into multiples of M/r. • General case: same but approximately. • Continued fraction algorithm finds the closest multiple of M/r.

50. Hidden subgroup • Function F:GS such that F(g)=F(hg) iff hH. • Find H. |x> |x> F |0> |F(x)>