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Quantum algorithms oracles 12249412

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An optimal quantum algorithm for the oracle identification problem

In the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set C of size M and our task is to identify x. We present a quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm considerably simplifies and improves the previous best algorithm due to Ambainis et al. Our algorithm also has applications in quantum learning theory, where it improves the complexity of exact learning with membership queries, resolving a conjecture of Hunziker et al.
The algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered γ2-norm semidefinite program, which characterizes quantum query complexity. Our composition theorem is quite general and allows us to compose quantum algorithms with input-dependent query complexities without incurring a logarithmic overhead for error reduction. As an application of the composition theorem, we remove all log factors from the best known quantum algorithm for Boolean matrix multiplication.

https://arxiv.org/abs/1311.7685

Published in: Science
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Quantum algorithms oracles 12249412

  1. 1. Quantum Algorithms Oracles Artur Ekert
  2. 2. Query Scenario f { } { }: 0,1 0,1 n f ® ( )x y x y f x® Å An ORACLE is very precious, you are charged some fixed amount of money each time you use it BLACK BOX, ORACLE Typical scenario: Given an ORACLE that computes f your goal is to determine some properties of f making as few queries to the ORACLE as possible, i.e. you want to minimize your expenditure. You are not allowed to look inside the ORACLE, but you can embed it into any Boolean network composed of any logic gates of your choice, we assume you are not charged for extra logic gates
  3. 3. Asymptotic notation for comparisons
  4. 4. Three Query Scenarios • Deutsch’s problem (1985) – quantum oracles outperform classical oracles • Grover’s search (1996) – quadratic separation • Simon’s problem (1994) – exponential separation We analyse three scenarios in which we gain if we use quantum rather then classical oracles.
  5. 5. Deutsch’s Problem f0 David Deutsch { } { }: 0,1 0,1f ®Given ( )1f x ( )2f x ( )3f x ( )4f xx 0 1 0 0 0 0 1 1 1 1 CONSTANT BALANCED x x ( )f x ?is f constant or balanced (0) (1) 0 or 1f fÅ = four possible oracles
  6. 6. Deutsch’s Problem H H f1 0 1- f (0) (1)f fÅf0 0 1 H 0 0 1 CONSTANT BALANCED 2 queries + 1 auxiliary operation 1 query + 3 auxiliary operations Quantum Classical
  7. 7. Deutsch’s Problem revisited H H f 0 1- 0 00000 CONSTANT any other output BALANCED H H H H H H H H 0 0 0 0 0 1- 0 0 0 0 0 INPUT: either constant or balancedPROMISE: OUTPUT: { } { }: 0,1 0,1 n f ® determine whether constant or balanced CLASSICAL COMPLEXITY: 1 2 1n- + queries
  8. 8. Deutsch’s Problem revisited 0 nÄ 0 1- HH f 0 1- ( )0 1 x x -å ( ) ( ) ( )( ) ( ) ( ) 1 1 0 1 x f x x x f x f x x - Å = - - å å ( ) ( )( ) 1 0 1 f x x y y x y Åæ ö - -ç ÷ è ø å å ! ( ) 22 ( ) 00...0 {0,1} 1 1 2 n f x n x P Î æ ö = -ç ÷ è ø å
  9. 9. Fair comparison? classical deterministic: quantum : 1 1 2 1n- + classical probabilistic with error prob. :e 2 1 logO e æ öæ ö ç ÷ç ÷ è øè ø Query in k places, if the queries had at least one 0 and one 1 then the function is balanced, otherwise assume it is constant. Probability that it is balanced when declared constant is 1 2 1 1 log 1 2 k ke e - æ ö æ ö = Þ = +ç ÷ ç ÷ è ø è ø FAIR COMPARISON
  10. 10. Bernstein-Vazirani Problem H H f 0 1- 0 H H H H H H H H 0 0 0 0 0 1- 4a 3a 2a 1a 0a INPUT: is of the formPROMISE: OUTPUT: { } { }: 0,1 0,1 n f ® binary string f ( )f x a x= ! a
  11. 11. Bernstein-Vazirani Problem 0 nÄ 0 1- HH f 0 1- ( )0 1 x x -å ( ) ( ) ( ) ( ) ( ) 1 1 0 1 x a x x x f x f x x - Å = - - å å ! ( )( ) ( )1 0 1 a y x y x y Åæ ö - -ç ÷ è ø å å ! a
  12. 12. Search Problem INPUT: PROMISE: OUTPUT: { } { }: 0,1 0,1 n f ® binary string ( ) 1 ( ) 0f a f x a= ¹ = a Searching large and unsorted database containing 2n items • Example of a sorted database: • a phone book if you are given a name and looking for a telephone number • n lookups suffice • Example of an unsorted database: • a phone book if you are given a number and looking for a name • you need to check 2n items before you succeed with probability P=1 • you need to check 2n-1 items before you succeed with probability P=0.5 Classical Complexity:
  13. 13. Grover’s algorithm H H f 0 1- 0 H H H H H H 0 0 0 0 1- 3a 2a 1a 0a INPUT: PROMISE: OUTPUT: { } { }: 0,1 0,1 n f ® binary string ( ) 1 ( ) 0f a f x a= ¹ = a f0 H H H H H f H H H f0 H H H H ITERATION 1 ITERATION 2 … … … … … … … … … … … Quantum Complexity:
  14. 14. Grover’s algorithm H f H H H 0 1- f0 H H H H ITERATION 0 1- 0 0 ( ) 0 for 0 (0) 1 f x x f = ¹ = ( ) 0 for ( ) 1 f x x a f a = ¹ =
  15. 15. Grover’s algorithm H f H H H 0 1- f0 H H H H ITERATION 0 1- ( ) ( ) ( )( ) 0 1 1 0 1 f x x x- ® - - ( ) ( ) ( )0 ( ) 0 1 1 0 1 f x x x- ® - -
  16. 16. Grover’s algorithm H f H H H 0 1- f0 H H H H 0 1- ( ) ( ) ( )( ) 0 1 1 0 1 f x x x- ® - - ( ) ( ) ( )0 ( ) 0 1 1 0 1 f x x x- ® - - ,x x a a® ® - , 0 0x x® ® - 1 2U a a= - 0 1 2 0 0U = - reflection about hyperplane orthogonal to a reflection about hyperplane orthogonal to 0
  17. 17. Grover’s algorithm H H H H H H H H 1 2U a a= - 0 1 2 0 0HU H H H= - reflection about hyperplane orthogonal to a reflection about hyperplane orthogonal to 0H a ^ ( )0H ^
  18. 18. Grover’s algorithm 1 2U a a= - 0 1 2 0 0HU H H H= - a ^ ( )0H ^ q 2q two reflections about the planes at angle rotate the vector by q 2q 1 0 sin 2n a H q= =
  19. 19. Grover’s algorithm 2q H H H H H H H H a a ^ 1 0 sin 2n a H q= =
  20. 20. Grover’s algorithm H H f 0 1- 0 H H H H H H 0 0 0 0 1- 3a 2a 1a 0a f0 H H H H H f H H H f0 H H H H ITERATION 1 ITERATION 2 … … … … … … … … … … … q q 2q q 2q 0H 2q a
  21. 21. Grover’s algorithm a 1 1 0 sin 2 2n n a H q q= = Þ ! q 2q 2q After r iteration the state is rotated by 2rq q+ a ^ from the hyperplane a ^ for large n ( )2 1 2 r p q+ ! We iterate until 2 4 n r p !
  22. 22. Query complexity quantum : classical probabilistic: Quadratic speedup compared to classical search algorithms Cryptanalysis: Attack on classical cryptographic schemes such as DES (the Data Encryption Standard) essentially requires a search among 256=7 £ 1016 possible keys. If these can be checked at a rate of, say, one million keys per second, a classical computer would need over a thousand years to discover the correct key while a quantum computer using Grover's algorithm would do it in less than four minutes.
  23. 23. Simon’s Problem INPUT: PROMISE: OUTPUT: { } { }: 0,1 0,1 n n f ® period ( ) ( ) , {0,1}n f x f x s x s= Å Î s Example: s=110 is the period (in the group) ( )f xx 000 001 010 011 100 101 110 111 111 010 100 110 100 110 111 010 ( ) 3 2Z ( ) ( )110f x f xÅ = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 000 000 110 110 111 001 001 110 111 010 010 010 110 100 100 011 011 110 101 110 f f f f f f f f f f f f = Å = = = Å = = = Å = = = Å = = Classical Complexity:
  24. 24. Simon’s algorithm 0 { } { }: 0,1 0,1 n n f ® 0 such that 0 y s y =! f n qubits n qubits H H ( )0 0 0 x x x x f x® ®å å ( ) ( )0 0 0measurement x x s f x® + Å ( ) ( )( ) ( ) ( )( )0 0 0 1 1 1 1 1 x y x s y x y s y y y y y y Å é ù- + - = - + - ® ë ûå å ! ! ! ! 0( )f x ( ) ( ) 1 0 if 1 1 if 0 2n P y y s P y y s- = = = = ! !
  25. 25. Simon’s algorithm 0 { } { }: 0,1 0,1 n n f ® 0 such that 0 y s y =! f n qubits n qubits H H 0( )f x 1 2 3 1 0 0 0 ... 0n y s y s y s y s- = = = = ! ! ! ! Solve the system of linear equations Probability of failure of generating linearly independent vectors y is less than 0.75 Needs roughly n queries. Quantum complexity
  26. 26. Classical Complexity Analysis Classical approach: 1 2 3, , ,... kx x x x ( ) ( ) ( ) ( )1 2 3, , ,... kf x f x f x f x ( ) ( )if theni j i jf x f x x x s= Å = Randomly choose: Evaluate: Search for collisions: ( ) 2 1 2 3 1 1 2 1 2 3 ... 212 n n k p p p k k æ ö ç ÷ -è ø < - + + =Average number of collisions: ( )2n k O= Number of queries in a classical probabilistic approach : CLASSICAL Probability of at least one collision: 2 2n k <
  27. 27. Quantum Complexity Analysis 1
  28. 28. Quantum Complexity Analysis 2
  29. 29. Summary HH f Deutsch (1985), Deutsch and Jozsa (92): The first indication that quantum computers can perform better H f Grover: Polynomial separation Simon: Exponential separation H H f H f0 f0 HH f classical quantum classical quantum

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