Fig. 1: The procedures of channel spectrum benchmarking.
From: Benchmarking universal quantum gates via channel spectrum
The benchmarking circuits are composed of three parts: the first part Us prepares the initial state \(\left|\psi \right\rangle={c}_{a}\left|{\phi }_{a}\right\rangle+{c}_{b}\left|{\phi }_{b}\right\rangle\), which is a superposition of two eigenstates of target gate U; then the target gate U is repeated L times, where L is an integer in \([0,\, {L}_{\max }]\); finally, the operator \(O=\left|\psi \right\rangle \left\langle \psi \right|\) is measured. The choice of coefficients ca, cb in the initial state is flexible as long as they are comparable and admit an efficient preparation of the initial state. Throughout this work, we choose \({c}_{a}={c}_{b}=\frac{1}{\sqrt{2}}\). For each initial state, we estimate several noisy eigenvalues from the time series data \({\left\langle O\right\rangle }_{L}\) at different depth L using the matrix pencil method.
