Randomized Benchmarking with Leakage Errors

TL;DR

Leakage errors can significantly bias RB fidelity estimates by moving population out of the computational subspace. The authors develop three leakage-aware RB methods—Comp. SPAM, Avg. MB, and LPS—and derive survival-probability expressions across four error regimes, validating them with 2Q simulations. They demonstrate how leakage affects fidelity reporting and show that leakage explains part of discrepancies in Quantinuum 2Q data, with Avg. MB and LPS providing robust, leakage-aware fidelity estimates. The work offers practical tools for accurately benchmarking near-term quantum devices and emphasizes the need to account for leakage as systems scale and approach error-corrected regimes.

Abstract

Leakage errors are unwanted transfer of population outside of a defined computational subspace and they occur in almost every platform for quantum computing. While prevalent, leakage is often overlooked when measuring and reporting the fidelity of quantum gates with standard methods. In fact, when leakage is substantial it can cause a large overestimation of fidelity from the typical method used to measure fidelity, randomized benchmarking. We provide several methods for properly estimating fidelity in the presence of leakage errors that are applicable in different error regimes with carefully chosen sequence lengths. Then, we numerically demonstrate the methods for two-qubit randomized benchmarking, which often have the largest errors. Finally, we reanalyze previously shared data from Quantinuum systems with some of the methods provided.

Figures (8)

• Figure 1: Clifford 2Q RB with leakage returns different fidelity estimates with measurement operators that contain different amounts of projection onto leakage subspaces. (a) An example process with a fixed infidelity $10^{-3}$ that is dominated by leakage as described in Sec. \ref{['sec:when_leakage_ignored']}. Clifford 2Q RB survival probability plot for three different measurement settings, from a measurement operator with no leakage population, some leakage population to all leakage population. (b) Infidelity comparisons between three different methods for the example process with a changing leakage magnitude but a fixed total infidelity of $10^{-3}$. The quantity $\lambda$ represents the magnitude of computational errors and $\tau$ represents the magnitude of leakage errors. (c) Energy level diagram of two qubits each with a leakage state $\ket{l}$.
• Figure 2: The leakage gadget uses an ancilla qubit "$a$” to detect if a target qubit "$q$” has leaked. The ancilla is initially prepared in $|1\rangle$. If "$q$” has leaked, the 2Q gates have no effect, and "$a$” is measured as $|1\rangle$. If instead "$a$” has leaked, then "$a$” will also be measured as $|1\rangle$ since the leakage is measured in a bright state. If neither "$q$” nor "$a$” has leaked, then "$a$” is measured as $|0\rangle$.
• Figure 3: Heat-map plots of the relative difference between the estimated values from each method compared to the values of the input error model using methods tailored to short sequences (e.g. $\tau \ell \ll 1, \lambda \ell \ll 1$). The x-axis of each subplot shows the injected value of $\lambda_s$ (the magnitude of the computational error) and the y-axis shows the injected value of $\tau_s$ (the magnitude of the leakage error). (a) Example decay curve for $\lambda_s=\tau_s=10^{-3}$ for the Comp. SPAM method, (b) Infidelity $1-F$ for the Comp. SPAM method, (c) Example decay curve for $\lambda_s=\tau_s=10^{-3}$ for the Avg. MB method (d) infidelity $1-F$ for the Avg. MB method, (e) $1-r$ for the Avg. MB method, (f) $\tau$ for the Avg. MB method, (g) Example decay curve for $\lambda_s=\tau_s=10^{-3}$ for the LPS method (h) infidelity $1-F$ for the LPS method, (i) $1-r$ for the LPS method, and (j) $\tau$ for the LPS method.
• Figure 4: Heat-map plots of the relative difference between the estimated values from each method compared to the values of the input error model using methods tailored to dominant computational errors (e.g. $\tau \ell \ll 1$ and $\tau < \lambda$). The x-axis of each subplot shows the injected value of $\lambda_s$ (the magnitude of the computational error) and the y-axis shows the injected value of $\tau_s$ (the magnitude of the leakage error). (a) Example decay curve for $\lambda_s=10^{-2},\tau_s=10^{-3}$ for the Comp. SPAM method, (b) Infidelity $1-F$ for the Comp. SPAM method, (b) $1-r$ for the Comp. SPAM method, (b) $\tau$ for the Comp. SPAM method, (c) Example decay curve for $\lambda_s=10^{-2},\tau_s=10^{-3}$ for the Avg. MB method (d) infidelity $1-F$ for the Avg. MB method, (e) $1-r$ for the Avg. MB method, (f) $\tau$ for the Avg. MB method, (g) Example decay curve for $\lambda_s=10^{-2},\tau_s=10^{-3}$ for the LPS method (h) infidelity $1-F$ for the LPS method, (i) $1-r$ for the LPS method, and (j) $\tau$ for the LPS method.
• Figure 5: Heat-map plots of the relative difference between the estimated values from each method compared to the values of the input error model using methods tailored to error processes without seepage (e.g. $\bar{\Lambda}_{CL} = 0$). The x-axis of each subplot shows the injected value of $\lambda_s$ (the magnitude of the computational error) and the y-axis shows the injected value of $\tau_s$ (the magnitude of the leakage error). (a) Example decay curve for $\lambda_s=\tau_s=10^{-3}$ for the Comp. SPAM method, (b) Infidelity $1-F$ for the Comp. SPAM method, (b) $1-r$ for the Comp. SPAM method, (b) $\tau$ for the Comp. SPAM method, (c) Example decay curve for $\lambda_s=\tau_s=10^{-3}$ for the Avg. MB method (d) infidelity $1-F$ for the Avg. MB method, (e) $1-r$ for the Avg. MB method, (f) $\tau$ for the Avg. MB method, (g) Example decay curve for $\lambda_s=\tau_s=10^{-3}$ for the LPS method (h) infidelity $1-F$ for the LPS method, (i) $1-r$ for the LPS method, and (j) $\tau$ for the LPS method.