Quantum Algorithms I

1. Quantum Algorithms I Andrew Chi-Chih Yao Tsinghua University & Chinese U. of Hong Kong

2. Quantum Computing: Confluence of deep ideas Chemistry Physics ComputerScience Mathematics Also, Material Science, Engineering, . . .

3. Quantum Information Processing Research Underway on Many Fronts: • Build quantum computers, quantum sources • Develop quantum information theory, error correction methods • Develop quantum algorithms, comm. protocols (cryptography, teleportation, etc.)

4. Quantum Preliminaries: In the simplest case • A quantum state is a unit vector u in C2 • A measurement is an orthogonal base {e1, e2}. u = u1+u2 will be measured as u1with prob |u1|2 as u2with prob |u2|2 • Features: If u is not known, then -- The result of a measurement cannot be predicted. -- A measurement can disturb the system. • In any physical process: -- U has to be a unitary operator (ie, a “rotation”)

6. Quantum Preliminaries: More generally • An n-qbitquantum state is a unit vector u in the Hilbert space • A measurement is a family of orthogonal linear subspaces decomposing H • A state will, upon meaurement M, become uiwith prob |ui|2 • Notation: Quantum states u often are written as |x> • Any physical process: where U is unitary

7. Outline of Talk I. What is a quantum computer ? II. Fast quantum algorithms -- Simon’s Problem III. Quantum cryptography -- Key distribution IV. Outlook

8. X=011001 f(x)=110101 I. What is a quantum computer ? classical computer Q Q’s state transition q0 q1 qT after time T . . . 2mpossible states eachqi specified withmbits Can be implemented with m flip-flops 1 0 0 0 1 1

9. Classical Computer (continued) • Turing, Church (1930’s): formulate the models • von Neumann (1950’s): circuit-based architecture • Example: parity function x = 01110110, p(x) = 1 iff x has even number of 1’s 0 1 1 1 0 1 1 0 moves from left to right, keeping track of current parity. q0

10. Quantum computer m spin-1/2 particles represented by a Hilbert space of dimension 2m, with a natural base |x>, x in {0,1}m after T steps x=011001 perform a measurement on vt to get f(x)=1100101 v0 v1 vt . . . determined by x

11. Quantum Computer (continued) • Benioff (1980) version of quantum Turing machine • Feymann (1985) version of quantum circuits • Deutsch (1985, 89) QTM, circuits, universal QTM • Bernstein, Vazirani (1993) efficient universal QTM for restricted classes • Solovay, Yao (1993) efficient universal QTM, equivalence of QTM & circuits

12. m For our purpose: an array of m spin-1/2 particles are processed left-to-right in each step. control unit with fixed number of bits

13. Implementation of Quantum Computer: Quantum circuits can be built using Hadamard gates, pi/8-gates, and CNOT.

14. A Typical Quantum Circuit

15. II. Fast Quantum Algorithms - Simon’s Problem A black-box 2 to 1 mapping: there exists a secret with f(x) = f(x+s) Problem: determine s Note: classical algorithms must make an exponential number of queries f(x)=?

17. Each hole x illuminates points (y, f(x)) on the screen with amplitudes • Write For any (y,z), the only amplitude contributions come from and , and is equal to 0 if since • Thus, all bright spots (y,z) satisfy . Finding n such columns y’s is sufficient to determine s

19. Hadmard’s Transform • For 1 bit • For n bits

20. Simon’s Algorithm

21. Now, Thus, Make a measurement on v, one sees only those |y>|z> satisfying . Repeat n times to solve for s as in the optical case. to Bob.

22. A very important result: Shor (1994) developed an efficient quantum algorithm for factoring large integers. His method uses an approach similar to Simon’s algorithm.t to Bob.

23. III. Quantum Cryptography -- key distribution Conjugate Coding(S. Wiesner 1970) • Two bases for a 2-state quantum system • A bit b can be stored as |b>p , where p is randomlychosen from{+, x} |1>+ |0>+ |1>x |0>x

24. Quantum Bank Note Method: An n-bit serial number b1b2 …bn is stored on the bank note by conjugate coding each bi as a quantum object Bi. The bank records what bases are used. When cashed, the bank measures Bi by using the recorded bases to see if there is any discrepancy. Advantages: • cannot be copied • hard to counterfeit • tampering destroys the bank note But it’s not practical ...

25. Key Distribution Problem Alice and Bob want a confidential communication. Is it possible to generate a key K by public discussions ? In the classical case … impossible information-theoretically; but feasible if assuming Eve cannot factor large integers. x y Bob Alice K K Eve

26. Observation : • It suffices to generate KAfor Alice,KBfor Bob that differ in at most 10% of the bits. It is possible to then use error correcting code to obtain a common K for both.

27. Quantum Key distribution (Bennett &Brassard 84) • Alice makes a random quantum bank note, and sends it to Bob. • Alice reveals all the n bases used on the bank note. • Then Bob can measure the Bi using these bases to get the original bits bi.

28. Quantum Key distribution (Bennett &Brassard 84) • Alice makes a random quantum bank note, and sends it to Bob. Bob measures randomly n/2 of the Bi’s, each with a random base; then he checks with Alice for any inconsistency. • Alice reveals all the n bases used on the bank note. • Then Bob can measure the Bi using these bases to get the original bits bi.

29. Heuristic Proof of Security: • After the test Alice and Bob have n/2-bit keys KA and KB • If Eve tampered with less than 5% of the original Bi’s, then KA and KB differ in at most 10% of their bits. This then gives a common key K. • If Eve tampered with more than 5% of the original Bi’s, then the test step would have caught it.

30. Remarks: Rigorous proof of security for quantum key distribution is often hard. Some strong results were first given by D. Mayers (1995). Also see Chau & Lo, Preskill & Shor.ee

31. IV. Outlook • Need more instances of fast quantum algorithms -- Graph isomorphism problem ? • Need more applications in quantum cryptography. Coin flipping with bounded bias can be done (Aharonov, Ta-Shma, Vazirani, Yao ‘00) -- What else can be accomplished ? • Beginning of a new interdisciplinary science between Physics and Computer Science.