Relations between the single-pass and multi-pass qubit probabilities

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. Instead, it is often determined by sequentially repeating the same gate multiple times, which leads to the accumulation 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 $P_N \approx 1 - N \epsilon$. %A more accurate calculation, which takes into account the exchange of probabilities between the qubit states adds correction terms to $p^N$ but in the limit of a tiny error the estimate $P_N \approx 1 - N \epsilon$ after $N$ passes remains in place. 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 particular, the relations suggest that in most cases of interest the $N$-pass transition probability degrades as $P_N = 1-N^2\epsilon$, i.e. dramatically faster than the classical probability estimate. Therefore, the actual single-pass fidelities in various experiments, calculated from $N$-pass fidelities, might have been far greater than the reported values.