A single-qubit gate suffers a Pauli error channel (Chapter 11) with probabilities ,

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Physical realizations and quantum technologies · Physical realizations and quantum technologies

Quantum Information and Technology, Ch What a qubit must do: requirements, tomography and benchmarking

Problem

A single-qubit gate suffers a Pauli error channel (Chapter 11) with probabilities

p_x = 0.0010

p_y = 0.0006

p_z = 0.0014

per application, arising from imperfect control. Averaging this error over the single-qubit Clifford group (using the Clifford group of Chapter 13, as randomised benchmarking does) turns it into a depolarising channel

\mathcal{D}_\lambda

with

\lambda = \tfrac43\left(p_x+p_y+p_z\right),

whose Bloch shrink factor

p = 1-\lambda

is what randomised benchmarking would report as its decay parameter, giving an error per Clifford

r = (1-p)(d-1)/d

with

d=2

. The gate takes

t_{\rm gate} = 30\,\mathrm{ns}

and is applied to a qubit with

T_1 = T_\phi = 120\,\mu\mathrm{s}

, for which the coherence-limited error is

r_{\rm coh} = 2t_{\rm gate}/(3T)

with

T = T_1 = T_\phi

. ((a)) Find the decay parameter

p

and the error per Clifford

r

that randomised benchmarking would report for this gate. ((b)) Find

r_{\rm coh}

and the fraction of

r

that is coherence-limited. ((c)) Decide whether this gate is limited by coherence or by control, and justify the decision.

Browse the problem library