A photonic phase-shifter qubit realises a fixed single-qubit gate in the…

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · The quantum circuit model and quantum algorithms · The quantum circuit model and quantum algorithms

Quantum Information and Technology, Ch The quantum Fourier transform and phase estimation

Problem

A photonic phase-shifter qubit realises a fixed single-qubit gate

U=\mathrm{diag}(1,e^{2\pi i/5})

in the computational basis. The state

|u\rangle=|1\rangle

is an eigenstate,

U|u\rangle=e^{2\pi i\varphi}|u\rangle

with

\varphi=1/5

, which has no finite binary expansion. Phase estimation is run with

t=4

control qubits, built exactly as in Section 7.5, with no noise. ((a)) Give

2^t\varphi

and the nearest integer reading

b

. ((b)) Using the Dirichlet-kernel formula of Eq. (7.14), compute

P(b)

for the nearest reading and

P(b+1)

for the next reading beyond it, and compare their sum with the guaranteed single-reading bound

4/\pi^2

and with the bound

8/\pi^2\approx0.811

(quoted here without proof) for the combined probability of the two nearest readings together. ((c)) An eigenphase is now to be resolved to

m=4

bits with failure probability at most

\epsilon=0.05

. Using the register-size rule

t=m+\lceil\log_2(2+1/2\epsilon)\rceil

, find the enlarged number of control qubits

t

needed, and state which of its

2^{t}

readings count as a success.

Browse the problem library