Quantum Algorithms Preliminaria Artur Ekert Computation INPUT OUTPUT

Quantum Algorithms Preliminaria Artur Ekert

Computation INPUT OUTPUT 1 1 0 0 1 0 1 Physics Inside (and outside) THIS IS WHAT OUR LECTURES WILL BE ABOUT

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

Classical deterministic computation 000 001 110 Computational steps – moves from one configuration to another – are performed by elementary operations on bits

Boolean Networks 0 OR 0 0 AND OR NOT 1 1 0

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

Classical probabilistic computation Input Possible outputs

Quantum computation nstructive 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 H H

Quantum operations H H H

Single qubit gates Hadamard H 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 INPUT H 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 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) Rotation by twice the angle between axis a and b around the axis perpendicular to a and b A A-1 B B-1 Recall that x represents rotation by around axis x Rotation by around some axis a Rotation by around some axis b

Building controlled-U operations = A A-1 B B-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 Quantum Hadamard transform

Quantum Hadamard Transform H H

Quantum Hadamard Transform H H H H

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

Quantum Fourier Transform Quantum Fourier transform on group

Recall Hadamard H Discrete set of phase gates

Quantum Fourier Transform F 1 H H F 2 H H F 3 H H

Quantum Fourier Transform 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

Quantum Fourier Transform H F 3 H H H Fy 3

Quantum function evaluation Boolean function f

Quantum function evaluation can be viewed as m Boolean functions … … … f 0 … … fm-1 fm-2

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