Richard Cleve DC 3524 cleve@cs.uwaterloo Course web site at:

1. Introduction to Quantum Information ProcessingCS 467 / CS 667Phys 467 / Phys 767C&O 481 / C&O 681 Richard Cleve DC 3524 cleve@cs.uwaterloo.ca Course web site at: http://www.cs.uwaterloo.ca/~cleve Lecture 6 (2005)

2. Contents • Phase estimation problem • Algorithm for the phase estimation problem • Order-finding via phase estimation

3. Phase estimation problem • Algorithm for the phase estimation problem • Order-finding via phase estimation

4. a a U b Uab a1 a1 : : am am b1 U Ua1 amb : bn Generalized controlled-U gates Example: 11010101 1101U11010101

5. mqubits Input: black-box for and a copy of  U nqubits Phase estimation problem U is a unitary operation on n qubits  is an eigenvector of U, with eigenvalue e2i (0≤  <1) Output: (m-bit approximation)

6. Phase estimation problem • Algorithm for the phase estimation problem • Order-finding via phase estimation

7.  Algorithm for phase estimation (I) 0 H Starts off as: 0 H 0 H  U 00 … 0 ab aUab  (0 + 1)(0 + 1) …(0 + 1) =(000 + 001 + 010 +011 + …+111) =(0 + 1 + 2 +3 + …+2m 1) (0+e2i1+(e2i)22+(e2i)33+…+(e2i)2m12m1)

8. Quantum Fourier transform where  = e2i/N (for n qubits, N= 2n)

9. H 4 H 8 4 H 16 8 4 H 32 16 8 4 H H m Computing the QFT (I) Quantum circuit for F32: reverse order Gates: For F2n costs O(n2) gates

10. Computing the QFT (II) One way on seeing why this circuit works is to first note that F2na1a2…an = (0+e2i(0.an)1)…(0+e2i(0.a2…an)1)(0+e2i(0.a1a2…an)1) It can then be checked that the circuit produces these states (with qubits in reverse order) for all computational basis states a1a2…an Exercise: (a) prove the above equation from the definition of the QFT; (b) confirm that the circuit produces these states

11. Algorithm for phase estimation (II) Note: this is exactly the state generated by our preliminary algorithm for phase estimation when = 0.a1a2…am Therefore, if = 0.a1a2…am then applying F† to this state yields a1a2…am (from which can be deduced exactly) What if is not of this nice form? Example:= ⅓ = 0.0101010101010101…

12. Phase estimation problem • Algorithm for the phase estimation problem • Order-finding via phase estimation

13. Order-finding algorithm (I) Let M be an m-bit integer Def: ZM*= {x  {1,2,…, M1} : gcd(x,M) = 1} ZM*is a group under multiplication moduloM Example: Z21*= {1,2,4,5,8,10,11,13,16,17,19,20} For a ZM*, consider all powers of a : a0, a1, a2, a3, … Ex: Fora = 10, the sequence is: 1, 10, 16, 13, 4, 19, 1, 10, 16, … Def: ordM (a) is the minimum r> 0 such that ar=1(mod M) Ex: ord21(10) = 6 Order-finding problem: given a and M, find ordM (a)

14. Order-finding algorithm (II) Define:U (an operation on m qubits) as: Uy =ay mod M Define: Then

15. 2nqubits U nqubits Order-finding algorithm (III) corresponds to the mapping: xy xaxy mod M Moreover, this mapping can be implemented with roughly O(n2) gates The phase estimation algorithm yields a2n-bit estimate of1/r From this, a good estimate of r can be calculated … Problem: how do we construct state1 to begin with?

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