Quantum Algorithms Oracles Artur Ekert Query Scenario BLACK
Quantum Algorithms Oracles Artur Ekert
Query Scenario BLACK BOX, ORACLE An ORACLE is very precious, you are charged some fixed amount of money each time you use it 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. f 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
Asymptotic notation for comparisons
Three Query Scenarios We analyse three scenarios in which we gain if we use quantum rather then classical oracles. • Deutsch’s problem (1985) – quantum oracles outperform classical oracles • Grover’s search (1996) – quadratic separation • Simon’s problem (1994) – exponential separation
Deutsch’s Problem Given is f constant or balanced ? David Deutsch four possible oracles CONSTANT BALANCED f
Deutsch’s Problem Classical 2 queries + 1 auxiliary operation f f Quantum H H H f CONSTANT BALANCED 1 query + 3 auxiliary operations
Deutsch’s Problem revisited INPUT: PROMISE: OUTPUT: CLASSICAL COMPLEXITY: either constant or balanced determine whether constant or balanced H H H H H f queries 00000 CONSTANT any other output BALANCED
Deutsch’s Problem revisited H H f
Fair comparison? classical deterministic: classical probabilistic with error prob. : quantum : 1 FAIR COMPARISON 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
Bernstein-Vazirani Problem INPUT: PROMISE: OUTPUT: is of the form binary string H H H H H f
Bernstein-Vazirani Problem H H f
Search Problem INPUT: PROMISE: OUTPUT: binary string Classical Complexity: Searching large and unsorted database containing 2 n 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 2 n items before you succeed with probability P=1 • you need to check 2 n-1 items before you succeed with probability P=0. 5
Grover’s algorithm INPUT: PROMISE: OUTPUT: Quantum Complexity: binary string ITERATION 1 ITERATION 2 H H H H H f f 0 … … …
Grover’s algorithm ITERATION f H H H H f 0
Grover’s algorithm ITERATION f H H H H f 0
Grover’s algorithm reflection about hyperplane orthogonal to f H H H H f 0 reflection about hyperplane orthogonal to
Grover’s algorithm H reflection about hyperplane orthogonal to H H H H reflection about hyperplane orthogonal to
Grover’s algorithm two reflections about the planes at angle rotate the vector by
Grover’s algorithm H H H H
Grover’s algorithm ITERATION 1 ITERATION 2 H H H H H f f 0 … … …
Grover’s algorithm After r iteration the state is rotated by from the hyperplane for large n We iterate until
Query complexity classical probabilistic: quantum : 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.
Simon’s Problem INPUT: PROMISE: OUTPUT: Classical Complexity: period Example: 000 001 010 011 100 101 110 111 010 100 110 111 010 s=110 is the period (in the group)
Simon’s algorithm n qubits H H
Simon’s algorithm n qubits H H n qubits 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
Classical Complexity Analysis Classical approach: Randomly choose: Evaluate: Search for collisions: Average number of collisions: Probability of at least one collision: Number of queries in a classical probabilistic approach : CLASSICAL
Quantum Complexity Analysis 1
Quantum Complexity Analysis 2
Summary H H Deutsch (1985), Deutsch and Jozsa (92): The first indication that quantum computers can perform better f H H H f 0 f Grover: Polynomial separation H f f 0 classical quantum Simon: Exponential separation H H f classical quantum
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