Quantum Algorithms Towards quantum codebreaking Artur Ekert More

Quantum Algorithms Towards quantum codebreaking Artur Ekert

More general oracles Quantum oracles do not have to be of this form e. g. generalized controlled-U operation n qubits m qubits

Phase estimation problem n qubits m qubits

Phase estimation algorithm Suppose p is an n-bit number: Recall Quantum Fourier Transform:

Phase estimation algorithm STEP 1: n qubits H m qubits Recall Quantum Fourier Transform:

Phase estimation algorithm STEP 2: Apply the reverse of the Quantum Fourier Transform n qubits H F ny m qubits But what if p’ has more than n bits in its binary representation ?

1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000 Probability Phase estimation algorithm

Phase estimation - solution n qubits m qubits H F ny

Order-finding problem PRELIMINARY DEFINITIONS: This is a group under multiplication mod N For example

Order-finding problem PRELIMINARY DEFINITIONS: For example (period 6)

Order-finding problem Order finding and factoring have the same complexity. Any efficient algorithm for one is convertible into an efficient algorithm for the other.

Solving order-finding via phase estimation n qubits m qubits Suppose we are given an oracle that multiplies y by the powers of a

Solving order-finding via phase estimation H Fn y Estimate of p 1 with prob. | |2 Estimate of p 2 with prob. | |2

Solving order-finding via phase estimation

Shor’s Factoring Algorithm 2 n qubits H F 2 ny n qubits Quantum factorization of an n bit integer N

Wacky ideas for the future • Particle statistics in interferometers, additional selection rules ? • Beyond sequential models – quantum annealing? • Holonomic, geometric, and topological quantum computation? • Discover (rather than invent) quantum computation in Nature?