The Impact of Markovian Errors on Random Circuits

Randomized benchmarking (RB) returns a single decay parameter or gate error rate, but when the noise is gate-dependent that parameter is notoriously difficult to interpret. We study RB and related random-circuit benchmarks under small gate-local Markovian noise by factorizing each noisy gate into its ideal action followed by the exponential of a small error generator and then perturbing the RB $\mathscr{L}$ matrix, which models the RB decay. For any low-error gate set that satisfies the requirements of RB (it is a sequence-asymptotic unitary 2-design), we show that Pauli-stochastic error rates contribute linearly to the RB decay parameter, while Hamiltonian error rates contribute through a gauge-invariant quadratic form. Pauli-correlation and active generators are absent from the linear term and enter only at higher order. For the single-qubit native gate set $\{X_{π/2},Y_{π/2}\}$ we derive this quadratic form explicitly in terms of three first-order gauge-invariant (FOGI) coordinates: the two intrinsic over-rotations and one relational axis-tilt error. The resulting formula explains the dependence of DRB on the sampling distribution and the compilation dependence of CRB. These results turn RB decay rates into interpretable functions of experimentally meaningful elementary error rates.

Source: arXiv:2610.31500v1 - http://arxiv.org/abs/2610.31500v1 PDF: https://arxiv.org/pdf/2610.31500v1 Original Link: http://arxiv.org/abs/2610.31500v1