Quantum Algorithms Preliminaria Artur Ekert

Computation INPUT OUTPUT Physics Inside (and outside) THIS IS WHAT OUR LECTURES WILL BE ABOUT

Classical deterministic computation Initial configuration (input) Final configuration (output) Configuration = complete specification of the state of the computer and data Physically allowed operations, computational steps Intermediate configurations

Classical deterministic computation Computational steps – moves from one configuration to another – are performed by elementary operations on bits

Boolean Networks NOT OR AND OR

Basic operations = logic gates AND 1 0 OR Wire, identity NOT 1 1 Logical AND 0 0 Logical OR Output 0 apart from the (1,1) input Output 1 apart from the (0,0) input Fan out X X X

Classical probabilistic computation Input Possible outputs

Quantum computation Constructive or destructive interference: enhance correct outputs suppress wrong outputs GOOD SIDE: extra computational power BAD SIDE: sensitive to decoherence

Quantum computation Initial configuration of the three qubits

Bits and Qubits BIT QUBIT

Quantum Boolean Networks H HH

Quantum operations H HH

Single qubit gates H Hadamard Continuous set of phase gates Discrete set of phase gates

Single qubit interference H H

Any single qubit interference H H INPUT OUTPUT in the matrix form

Any unitary operation on a qubit H H INPUT OUTPUT in the matrix form – the most general SU(2) operation on a single qubit

Possible implementations © ENS Paris

Two and more qubits Notation

Operations on two qubits Controlled-NOT Controlled-U U U

Quantum interferometry revisited H H H H U REMEMBER THIS TRICK !

Phases in a new way H H U

Entangled states H entangled separable

Bell & GHZ states H H

Useful decomposition of any U in SU(2) For any U in SU(2) A A -1 BB -1 Rotation by  around some axis a Rotation by  around some axis b Recall that  x represents rotation by  around axis x Rotation by twice the angle between axis a and b around the axis perpendicular to a and b

Building controlled-U operations A A -1 BB -1 U = A, A -1, B and B -1 are single qubit operations and can be constructed from the Hadamard and phase gates. Controlled-U can be constructed from single qubit operations and the controlled-NOT gates. Hence any controlled-U gate can be constructed from the Hadamard, the controlled-NOT and phase gates.

Toffoli Gate = H H

Controlled-controlled NOT Computes logical AND Quantum adder

Quantum Networks H H Quantum adder H H H H Quantum Hadamard transform

Quantum Hadamard Transform H H H H

H H H H H H H H

Is also known as the quantum Fourier transform on group group = the set with operation (addition mod 2) group = the set with operation (addition mod 2 bit by bit) example for n=15

Quantum Fourier Transform Quantum Fourier transform on group

Recall H Hadamard Discrete set of phase gates

Quantum Fourier Transform H H H H H H F1F1 F2F2 F3F3

H H H H Uniform family of networks n Hadamard gates and n(n-1)/2 phase shifts, the size of the network = n(n+1)/2

Quantum Fourier Transform H H H H

H H H F3F3 H H H Fy3Fy3

Quantum function evaluation f Boolean function

Quantum function evaluation can be viewed as m Boolean functions f m-1 f m-2 f0f0 …………………………………………

Quantum function evaluation Group X Group (Y, ) bit by bit addition – group modular addition – group