Subspace Leakage Error Randomized Benchmarking of Mølmer-Sørensen Gates

TL;DR

This work introduces Subspace Leakage Error Randomized Benchmarking (SLERB), a method that repurposes single-qubit RB to benchmark two-qubit Mølmer-Sørensen gates in trapped ions by confining operations to a two-state subspace $\mathcal{S}_{RB}$ and tracking leakage into $\mathcal{S}_{leak}$. By varying the MS gate phase $\phi_{MS}$, SLERB constructs Clifford decompositions within $\mathcal{S}_{RB}$ and utilizes a transfer-matrix and group-theoretical (twirl) framework to extract intra-subspace errors $\epsilon_{RB}$ and leakage errors $\epsilon_{leak}$, enabling an estimate of the average two-qubit gate infidelity. The authors validate the approach with numerical simulations (population-transfer rates, full MS dynamics, random-unitary noise) and demonstrate an experimental implementation on laser-free MS gates in Ca$^{+}$ ions, achieving a two-qubit error of $\epsilon_{2Q}=2.6(2)\times 10^{-4}$ and confirming the symmetry of errors via asymptotic population behavior. SLERB offers a scalable, SPAM-insensitive diagnostic that isolates error origins (subspace-conserving vs. leakage) without requiring single-qubit rotations, with potential applicability to other platforms and more complex qubit interactions.

Abstract

We demonstrate a new technique that adapts single-qubit randomized benchmarking to two-qubit Mølmer-Sørensen gates. We use the controllable gate phase to generate Cliffords that act on a two-state subspace, enabling benchmarking of two-qubit gates without single-qubit operations. In addition to quantifying the gate infidelity, the protocol provides valuable information about the type of error by distinguishing between those that conserve the two-state subspace and those that result in leakage out of it. We demonstrate the protocol for calibrating and validating all-electronic maximally entangling gates in a trapped-ion quantum computer, achieving a two-qubit gate error of $2.6 (2)\times10^{-4}$.

Figures (9)

• Figure 1: Conversion of single-qubit to two-qubit sequences. SLERB sequences are generated from (a) sequences of single-qubit rotations ($R_\phi(\pi/2)$, $\phi=\phi_\textrm{1Q}$) (a), which are converted to (b) sequences of two-qubit MS gates, ($R_{\phi,\phi}(\pi/2)$, $\phi=\phi_\textrm{MS}=\phi_\textrm{1Q}/2$). Each pulse ideally rotates the states by an angle $\pi/2$ within their respective two-state subspaces, as shown in the Bloch spheres for (c) single and (d) two qubits. Each rotation is color-matched to its respective Clifford in (a) and (b), and corresponds to a specific axis of the Bloch sphere specified by $\phi_\textrm{1Q}$ or $\phi_\textrm{MS}$. (e) Overview of population transfers. SU2 errors keep the states within the $\{\ket{00},\ket{11}\}$ subspace, with a corresponding error rate $\epsilon_\textrm{RB}$, while leakage errors result in populations in the $\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})$ state with a corresponding error rate $\epsilon_\textrm{leak}$.
• Figure 2: Monte Carlo simulations (colored solid lines) of population transfer for a SLERB sequence of length $l=500$, averaging over $5\times10^{3}$ random circuits of single-qubit Cliffords on $\mathcal{S}_{\text{RB}}$, all of which decompose into Eq. (\ref{['eq:ideal_unitary']}). For this run, we multiplied each Clifford by error unitaries representing two noise unitaries: one in $\mathcal{K}_{\text{RB}}$ and one in $\mathcal{K}_{\text{leak}}$. Using the results of Sec. \ref{['sec:transfer_matrix']}, we calculate $\epsilon_{\text{RB}}$ and $\epsilon_{\text{leak}}$ analytically, then apply the resultant transfer matrix $T$ to the initial population vector $\vec{P}_{0}=(1,0,0)$ (black dashed lines). The result illustrates how the analytical expressions agree with numerical simulations.
• Figure 3: Numerical simulations of SLERB sequences for (a) amplitude and (b) detuning errors. For each sequence length, we sample 50 random sequences; the error bars correspond to the standard deviation for each population from the 50 circuits. The solid lines correspond to fits to the data following Eq. \ref{['eq_slerb_decay']}. (a) For amplitude errors, there is no leakage and the states stay within the $\ket{00}, \ket{11}$ subspace, with asymptotes of 1/2 for $P_\textrm{survival}$ and $P_\textrm{leak}$. (b) For detuning errors, there is residual spin-motional entanglement, which results in leakage. The asymptotes are instead $1/3$ for all three populations.
• Figure 4: Estimated vs true channel fidelity for 1000 random unitary error channels. Each point corresponds to a full simulation of a benchmarking experiment where a random unitary error channel was sampled, the numeric twirl of the resulting channel representation was calculated, and the signals were calculated and fit to extract an estimate of the fidelity. Transfer matrix points correspond to the fidelity estimator of Eq. \ref{['eq:slerb_avg_fidel']}, and group theory points correspond to Eq. \ref{['eq:group_theory_avg_fidel']}. The lines represent fits of the two estimated distributions to a linear model.
• Figure 5: SLERB implementation and results. (a) Pulse sequence for an individual MS gate with phase $\phi_\textrm{MS}$. Each MS gate consists of three frequencies: blue and red sideband tones at $\omega_0 \pm(\omega_m+\delta)$, and a carrier at $\omega_0$ for dynamical decoupling. We set the phases of all the tones to $\phi_\textrm{MS}$ in the first half of the gate, and to $\phi_\textrm{MS} + \pi$ in the second half of the gate. (b) SLERB results with up to 200 Cliffords. The populations $P_\textrm{survival}$ (purple circles), $P_\textrm{flip}$ (blue crosses), and $P_\textrm{leak}$ (green diamonds) indicate the populations in the target state, in the incorrect state within the $\{\ket{00},\ket{11}\}$ subspace, and the population that has leaked out of this subspace into the $\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})$ state, respectively. The measurements are performed with 50 randomizations and 50 shots each; error bars indicate the standard deviation of the populations across all the randomizations for each sequence length.