# Relations between the single-pass and multi-pass qubit gate errors

## TL;DR

This paper analyzes how repeating quantum gates multiple times affects error measurement, revealing interference effects that can cause errors to amplify faster than classical predictions, with implications for quantum gate fidelity assessment.

## Contribution

It provides a rigorous theoretical framework linking single-pass and multi-pass qubit gate errors using SU(2) symmetry and Chebyshev polynomials, including practical recipes for error determination.

## Key findings

- Multi-pass transition probability can degrade as 1 - N^2 ε, faster than classical estimates.
- Explicit analytic relations between single-pass and N-pass propagators are derived.
- Recipes are proposed for accurate error estimation in both Clifford and non-Clifford gates.

## Abstract

In quantum computation the target fidelity of the qubit gates is very high, with the admissible error being in the range from $10^{-3}$ to $10^{-4}$ and even less, depending on the protocol. The direct experimental determination of such an extremely small error is very challenging by standard quantum-process tomography. Instead, the method of randomized benchmarking, which uses a random sequence of Clifford gates, has become a standard tool for determination of the average gate error as the decay constant in the exponentially decaying fidelity. In this paper, the task for determining a tiny error is addressed by sequentially repeating the \emph{same} gate multiple times, which leads to the coherent amplification of the error, until it reaches large enough values to be measured reliably. If the transition probability is $p=1-\epsilon$ with $\epsilon \ll 1$ in the single process, then classical intuition dictates that the probability after $N$ passes should be $P_N \approx 1 - N \epsilon$. However, this classical expectation is misleading because it neglects interference effects. This paper presents a rigorous theoretical analysis based on the SU(2) symmetry of the qubit propagator, resulting in explicit analytic relations that link the $N$-pass propagator to the single-pass one in terms of Chebyshev polynomials. In particular, the relations suggest that in some special cases the $N$-pass transition probability degrades as $P_N = 1-N^2\epsilon$, i.e. dramatically faster than the classical probability estimate. In the general case, however, the relation between the single-pass and $N$-pass propagators is much more involved. Recipes are proposed for unambiguous determination of the gate errors in the general case, and for both Clifford and non-Clifford gates.