A physical qubit has relaxation time

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Open quantum systems, decoherence and error correction · Open quantum systems, decoherence and error correction

Quantum Information and Technology, Ch Quantum error correction

Problem

A physical qubit has relaxation time T1=50 μsT_1 = 50\ \mu\mathrm{s}. During each code cycle of duration t=0.5 μst = 0.5\ \mu\mathrm{s} it undergoes amplitude damping with damping parameter γ=1−e−t/T1\gamma = 1 - e^{-t/T_1}, with Kraus operators (Chapter 11) K0=(1001−γ),K1=(0γ00).K_0 = \begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix}, \qquad K_1 = \begin{pmatrix}0&\sqrt{\gamma}\\0&0\end{pmatrix}. ((a)) Compute γ\gamma for this code cycle. ((b)) To first order in the small quantity γ\gamma, expand K0K_0 in the Pauli basis I,X,Y,ZI, X, Y, Z and give the coefficient of ZZ; expand K1K_1 and give its overall coefficient of (X+iY)(X+iY). ((c)) Explain, using the discretisation argument, why a code that corrects arbitrary single-qubit bit-flip, phase-flip and YY errors on this qubit also corrects this amplitude-damping error to the order worked in part (b).

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