Quantum Algorithms II Andrew C. Yao Tsinghua University & Chinese U. of Hong Kong

Outline of Talk I. Introduction II. Element Distinctness & Sorting III. Quantum Algorithm -- element distinctness IV. Lower Bounds -- sorting problems V. Conclusions

Quantum Mechanics More generally u A quantum state is a unit vector u in C N u A measurement is a family of orthogonal subspaces ( V 1, V 2, …,V k ): u = u 1 +u 2 +…+u k will be measured as u i with prob |u i | 2 u A computation step is a unitary operator (a ‘rotation’) u A Quantum Algorithm specifies > an initial quantum state > a sequence of unitary operators > a final measurement.

Classical algorithm Quantum Algorithm

I. Introduction u Shor (‘94): Fast factorization of integers u Grover (‘96): Searching n-item list in time Exactly one is 1, all other are 0 Problem: Find j

I. Introduction u Hilbert space with base vectors -- Start with -- Use a unitary operator to compute (where ) u Grover’s Theorem:

I. Introduction u Initially, u After t steps, u Take measurement in base the probability of seeing |j> is

Random Walk Hitting j by classical random walks takes n steps

Quantum Random Walk Hitting j by quantum random walks takes steps

I. Introduction Optimality Theorem (Bernstein et al ‘ 97 ): Any quantum algorithm must use steps to locate j Question: What other problems can be speeded up with quantum algorithms? This talk: Progress on sorting-like problems -- Element Distinctness Problem -- Sorting Problem

I. Introduction u Element Distinctness: Given decide whether there exist such that u Sorting: Given distinct determine such that u For conventional computers, essentially same problem (solvable in time ). u For quantum computers, very different...

II. Element Distinctness & Sorting u Theorem 1. Element Distinctness can be solved in quantum steps. * Buhrman et al ‘00 * Ambainis ‘03 u Theorem 2. Any quantum algorithm for Element Distinctness must use time. * Aaronson & Shi ‘04

II. Element Distinctness & Sorting u Sorting can be done in time classically. u Unlike Element Distinctness, it cannot be speeded up by quantum algorithms. Theorem 3. Any quantum algorithm for Sorting must use time. * Hoyer, Neerbek and Shi ’02 u Will illustrate Element Distinctness upper bound & Sorting lower bound.

III. Element Distinctness Quantum Algorithm for Element Distinctness u First, Grover’s list-searching implies: Computing takes only evals. u Def: Element x in S is called a repeater if x = some x’ Question: Any repeater in S ? F1F2FjFn

III. Element Distinctness Quantum Algorithm for Element Distinctness 1. Choose k, divide S into groups of size k 2. Design Repeater i : Output 1 iff some x in S i is a repeater

III. Element Distinctness Repeater 1 : Any repeater in ? 1. Sort ; if there are equal elements, halt and return Use Grover to decide if some outside = some in ; if yes, return 1 * Phase 1 takes time * Phase 2 takes quantum time to compute

III. Element Distinctness Quantum algorithm for Element Distinctness: Use Grover to evaluate

IV. Sorting for Partial Order P P: partial order on n objects u Given input consistent with P Determine such that u The standard sorting problem is the special case P = empty

IV. Sorting Problem for Partial Order P u e(P): # of linear extensions consistent with P Information bound: Upper bound: [Fredman ‘75] [Kahn & Saks ‘91] u Quantum Complexity Theorem 4 (Yao ‘04) constants

Proof Outline A. Quantum Partial Order Problem B. P=empty: Review of Proof C. Reducing A to a Combinatorial Problem D. Graph Entropy Chain Polytope

A. Quantum Partial Order Problem u = Linear extensions consistent with P Thus e(P) = | | u Quantum algorithm S for partial order P computes the function u Note is a “partial function”

Quantum Decision Tree u Initial state u Unitary Operators u Final measurement M With prob >, M applied to gives the correct u

B. Review of Case u An entropy type argument (extending Ambainis) u At any time, the current state has an entropy (uncertainty). Initially, it has L= n log n bits of entropy. After the sorting, it should have 0 entropy. If each operation can reduce that entropy by at most, then the number of operations must be at least

B. Review of Case [Hoyer, Neerbek, Shi ‘02] u For input x, after step j, the quantum state is u Initially all inputs x have states u In the end, every pair x, y have nearly orthogonal states u Define weight function w(x, y) for inputs x, y u Find lower bound to and upper bound to for any algorithm u

B. Review of Case u u Let Lemma 1 Lemma 2

Extension to u Use the natural extension of their approach u Need to show a lower bound L to

C. Reduced to a Combinatorial Problem (*) u This is a purely combinatorial problem -- recall u (*) can be proved by using combinatorial and information-theoretic tools developed by Korner (‘73), Stanley (‘86), Kahn & Kim (‘95).

D. Graph Entropy, Chain Polytope, etc. Proof outline for u The natural entropy for a partial order P would be simply log e(P). u Kahn and Kim (‘95) defined H(P), an alternative entropy (based on Korner’s graph entropy), and showed H(P) = (log e(P)) u Our proof uses Stanley’s chain polytope theory to show that

V. Conclusions u Quantum Complexity -- many open questions u Integer factorization has classical polynomial time algorithm? u Graph isomorphism has quantum polynomial time algorithm? u Rich source of problems: eg quantum complexity for computing graph properties.