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Exact quantum algorithms

Exact quantum algorithms. Andris Ambainis University of Latvia. Types of quantum algorithms. Bounded-error: correct answer with probability at least 2/3. Exact: correct answer with certainty (probability 1). Grover's search. 0. 1. 0. 0. x 1. x 2. x 3. x N. Is there i : x i =1 ?

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Exact quantum algorithms

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  1. Exact quantum algorithms Andris Ambainis University of Latvia

  2. Types of quantum algorithms • Bounded-error: correct answer with probability at least 2/3. • Exact: correct answer with certainty (probability 1).

  3. Grover's search ... 0 1 0 0 x1 x2 x3 xN Is there i:xi=1? Classically, N queries required. Quantum: O(N) queries [Grover, 96]. Quantum, exact: N queries.

  4. Model

  5. Query model • Function f(x1, ..., xN), xi{0,1}. • xi given by a black box: i xi Complexity = number of queries

  6. Queries in the quantum world • Basis states: |1,1, |1, 2, …, |N, M. • Query: • |i, j  |i, j, if xi=0; • |i, j  -|i, j, if xi=1;

  7. Query 1,1|1, 1+1,2|1, 2- 2,1|2, 1+3,1|3,1 Example x1 x2 x3 1,1|1, 1+1,2|1, 2+2,1|2, 1+3,1|3,1 0 1 0

  8. Quantum query model • Fixed starting state. • U0, U1, …, UT – independent of x1, …, xN. • Q – queries. • Measuring final state gives the result. U0 Q Q U1 … UT

  9. Known exact algorithms

  10. Deutsch’s problem 0 1 • Determine x1x2, with query access to xi. • [Cleve et al., 1998]: 1 quantum query, always the correct answer. x1 x2

  11. Dutsch-Jozsa ... • Distinguish whether: • x1 = x2 = ... = xN or • xi=0 (xi=1) for exactly ½ of i{1, 2, ..., N}. • Deterministic: N/2+1 queries. • Quantum: 1 query. 0 1 0 0 x1 x2 x3 xN

  12. Grover's search ... 0 1 0 0 x1 x2 x3 xN Is there i:xi=1? Promise: there is 0 or 1 i: xi=1. Classically: N queries. Quantum, exact: O(N) queries.

  13. Exact algorithms for total functions?

  14. Deutsch’s problem 0 1 • Determine x1x2, with query access to xi. • [Cleve et al., 1998]: 1 quantum query, always the correct answer. x1 x2 x1x2...xN can be computed with N/2 queries

  15. Montanaro et al., 2011. • EXACT24(x1, x2, x3, x4)=1 if there are exactly 2 i:xi=1. • Classical: 4 queries. • Quantum: 2 queries, exact. Is there a total function f(x1, ..., xN) for which QE(f) < D(f)/2? quantum exact deterministic

  16. Our results

  17. Superlinear separation • Theorem There is f(x1, ..., xN) such that • D(f)=N; • QE(f)=O(N0.86...). What should f be?

  18. Polynomial degree lower bound • deg(f) – degree of f(x1, ..., xN) as a multilinear polynomial. • [Nisan, Szegedy, 92, Beals et al., 98]

  19. Basis function D(f)=3, deg(f)=2

  20. NE NE NE NE x1 x2 x3 x4 x5 x6 x7 x8 x9 Iterated NE d levels  D(f)=3d, deg(f)=2d

  21. NE NE NE NE x1 x2 x3 x4 x5 x6 x7 x8 x9 Our result • Theorem For d levels, QE(f)=O(2.593...d).

  22. Step 1 • Algorithm for NE(x1, x2, x3). • Starting state: • Result:

  23. Step 2 • p-algorithm: • |start  |start if f=0; • |start  p|start + | with ||start, if f=1. p=0  exact quantum algorithm

  24. f f NE f f Step 3 • p-algorithm: • |start  |start if f=0; • |start  p|start + | with ||start, if f=1. • NE(x1, x2, x3) – 2 queries, p = -7/9 p-algo, k queries p’-algo, 2k queries

  25. NE NE NE NE x8 x7 x6 x5 x9 x3 x2 x1 x4 Step 3: result • d levels, 3d variables; Bad p! • p-algorithm with 2d queries.

  26. f f Step 4 • Amplification 2k queries, smaller p p-algo, k queries Form of amplitude amplification [Brassard et al., 2000]

  27. Final algorithm 1 level, 3 variables, 2 queries Iterate 2 levels, 9 variables, 4 queries Iterate 3 levels, 27 variables, 8 queries Amplify 3 levels, 27 variables, 16 queries ...

  28. Final result • 211 queries for each 8 levels. • N=38 variables, 211 queries. • N=38k variables, 211k queries. QE(f)=N0.86...

  29. Other exact quantum algorithms

  30. EXACT ... • Determine whether xi=1 for exactly k of N variables. • Montanaro et al., 2011: • Algorithm: 2 out of 4, 2 queries; • Computer optimization: 3 out of 6, 3 queries; • Conjecture: N/2 out of N, N/2 queries. 0 1 0 0 x1 x2 x3 xN

  31. A, Iraids, Smotrovs • Exact algorithms for determining: • if xi=1 for exactly N/2 i, N/2 queries; • if xi=1 for exactly k i, max(k, N-k) queries; • Provably optimal. • Natural computational problems; • Simple algorithms.

  32. Algorithm: summary 1 query 1 query ... ...

  33. Threshold functions ... • Is it true that xi=1 for k of N variables? • Exact algorithm, max(k, N-k+1) queries. • Easiest: k=N/2, N/2+1 queries. • Hardest: k=0 or k=N, N queries. 0 1 0 0 x1 x2 x3 xN

  34. Summary • A function that requires N queries classically, O(N0.86...) queries for exact quantum algorithms. • First separation by more than a factor of 2. • Several other exact quantum algorithms. Advantages for exact quantum algorithms are more common that I thought

  35. Open problems • d-level NE function (with 3d variables): • O(2.593...d) query exact algorithm; • Lower bound: (2.11...d). • Other iterated functions? • Other symmetric functions? • More exact algorithms?

  36. Open problems • Lower bound methods for exact quantum algorithms? Currently known: • Bounded-error quantum lower bounds; • QE(f)  deg(f)/2; For NEd, both of them fail.

  37. More information • A. Ambainis. Superlinear advantage for exact quantum algorithms, arxiv:1211.0721. • A. Ambainis, J. Iraids, J. Smotrovs. Exact quantum query complexity of EXACT and THRESHOLD, arxiv:1302.1235.

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