A two-qubit entangling gate takes

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

t_{\rm gate} = 200\,\mathrm{ns}

. Two-qubit randomised benchmarking is run using the two-qubit Clifford group (dimension

d = 4

), and fitting the survival probability to

F(m) = Ap^m + B

gives decay parameter

p = 0.990

. Both qubits involved in the gate have

T_1 = T_\phi = 50\,\mu\mathrm{s}

(relaxation time equal to pure dephasing time). For a single qubit sitting idle for a time

t

with

T_1 = T_\phi = T

, the decoherence-limited average gate infidelity is

r_{\rm coh,1} = \dfrac{2}{3}\dfrac{t}{T}

(as derived from the depolarising and dephasing channels of Chapter 11). Because the two qubits decohere independently during the gate, their process (Pauli) errors add. Converting each single-qubit contribution with

e_P=\tfrac32 r_{\rm coh,1}

(

d=2

), adding the two, and converting back with

r=\tfrac45 e_P

(

d=4

) gives the two-qubit coherence limit

r_{\rm coh} = \frac45\times2\times\frac32\,r_{\rm coh,1} = \frac{12}{5}r_{\rm coh,1} = \frac{8}{5}\frac{t_{\rm gate}}{T}.

((a)) Using the fact that averaging any error over the two-qubit Clifford group of Chapter 13 turns it into a depolarising channel, find the error per Clifford

r

implied by the fitted decay parameter. ((b)) Find the coherence-limited error

r_{\rm coh}

for this gate. ((c)) What fraction of the measured error

r

is accounted for by coherence, and is this gate better described as coherence-limited or control-limited?

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