Quantum Algorithms Oracles Artur Ekert

Query Scenario f 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 Deutsch’s problem (1985) Grover’s search (1996) 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 f Given is f constant or balanced David Deutsch four possible oracles f CONSTANT BALANCED

Deutsch’s Problem f f H H H f Classical 2 queries + 1 auxiliary operation f f Quantum CONSTANT H H 1 query + 3 auxiliary operations BALANCED H f

Deutsch’s Problem revisited CLASSICAL COMPLEXITY: INPUT: queries PROMISE: either constant or balanced OUTPUT: determine whether constant or balanced H H 00000 CONSTANT H H H H any other output BALANCED H H H H f

Deutsch’s Problem revisited 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: is of the form OUTPUT: binary string H H H H H H H H H H f

Bernstein-Vazirani Problem H H f

Search Problem binary string INPUT: PROMISE: OUTPUT: Classical Complexity: PROMISE: OUTPUT: binary string 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

Grover’s algorithm binary string H H H H H H H H H H H H H H H H H H H INPUT: Quantum Complexity: PROMISE: OUTPUT: binary string ITERATION 1 ITERATION 2 … H H H H H H H H H H H H H H H H H H H H f f0 f f0

Grover’s algorithm H H H H H H H H f f0 ITERATION When we add a C-gate, which can stand for the interaction with an external system (environment, rest of the universe, measuring device), the interference is no longer perfect... (a|0>+b|1>)m0 -> a|0> m0 +b|1>m1 , m1<->U(m0) < m0 |m1><->0 (perfect measurement)

Grover’s algorithm H H H H H H H H f f0 ITERATION When we add a C-gate, which can stand for the interaction with an external system (environment, rest of the universe, measuring device), the interference is no longer perfect... (a|0>+b|1>)m0 -> a|0> m0 +b|1>m1 , m1<->U(m0) < m0 |m1><->0 (perfect measurement)

Grover’s algorithm H H H H H H H H f f0 reflection about hyperplane orthogonal to reflection about hyperplane orthogonal to H H H H f f0 When we add a C-gate, which can stand for the interaction with an external system (environment, rest of the universe, measuring device), the interference is no longer perfect... (a|0>+b|1>)m0 -> a|0> m0 +b|1>m1 , m1<->U(m0) < m0 |m1><->0 (perfect measurement)

Grover’s algorithm H H H H H H H H reflection about reflection about hyperplane orthogonal to reflection about hyperplane orthogonal to H H H H H H

Grover’s algorithm two reflections about the planes at angle rotate the vector by

Grover’s algorithm H H H H H H H H

Grover’s algorithm H H H H H H H H H H H H H H H H H H H H f f0 f f0 ITERATION 1 ITERATION 2 … H H H H H H H H H H H H H H H H H H H H f f0 f f0

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 period INPUT: PROMISE: OUTPUT: Example: Classical Complexity: PROMISE: OUTPUT: period Example: s=110 is the period (in the group) 000 001 010 011 100 101 110 111 111 010 100 110

Simon’s algorithm H H n qubits n qubits

Simon’s algorithm H H n qubits 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 f H H H H f0 f0 f f H H f Deutsch (1985), Deutsch and Jozsa (92): The first indication that quantum computers can perform better H f Grover: Polynomial separation H H H H f0 f0 f f classical quantum Simon: Exponential separation H H f classical quantum