Measuring error rates of mid-circuit measurements

Abstract

High-fidelity mid-circuit measurements, which read out the state of specific qubits in a multiqubit processor without destroying them or disrupting their neighbors, are a critical component for useful quantum computing. They enable fault-tolerant quantum error correction, dynamic circuits, and other paths to solving classically intractable problems. But there are few methods to assess their performance comprehensively. In this work, we address this gap by introducing the first randomized benchmarking protocol that measures the rate at which mid-circuit measurements induce errors in many-qubit circuits. Using this protocol, we detect and eliminate previously undetected measurement-induced crosstalk in a 20-qubit trapped-ion quantum computer. Then, we use the same protocol to measure the rate of measurement-induced crosstalk error on a 27-qubit IBM Q processor, and quantify how much of that error is eliminated by dynamical decoupling.

Fig. 1. Quantifying measurement-induced crosstalk in a trapped-ion quantum computer using QIRB.

Quantum instrument randomized benchmarking (QIRB) uses the mid-circuit-measurement-containing (MCM-containing) random circuits shown in (a, b) to measure the average error rate (r_Ω) of MCM-containing circuit layers. QIRB retains the simplicity and elegance of RB methods for measuring gate errors, as it estimates r_Ω as the decay rate of the average circuit success rates (${\overline{F}}_d$) versus circuit depth, as shown in the examples of (c). We used QIRB to study MCMs in Quantinuum’s H1-1 system, depicted in (g), which arranges 20 ¹⁷¹Yb⁺ ions into auxiliary ("A'') and gate ("G'') zones. Our initial experiments on H1-1 applied micromotion hiding only to unmeasured ions in gate zones during MCMs, as ions in auxiliary zones are distant from measured ions. We observed higher error rates in six-qubit QIRB than predicted [compare left with center bars in (d, e)]. We added micromotion hiding for all unmeasured ions, reran QIRB, and observed that the six-qubit error rate was approximately halved [right bars in (d, e)] and was now consistent with the predictions of Quantinuum’s emulator. Results from bright-state depumping experiments (f) independently confirmed the existence of long-range, MCM-induced crosstalk and its mitigation by extra micromotion hiding. These experiments quantify MCM-induced crosstalk errors by measuring the rate at which unmeasured ions, prepared in the $∣1⟩$ state, leak out of the computational space as gate-zone ions are measured. All error bars represent bootstrapped 1σ deviations from the sample mean.

Fig. 2. Quantifying the impact of dynamical decoupling on measurement-induced errors on ibmq_algiers.

a, b Success decay curves from performing 5-qubit (blue), 10-qubit (orange), and 15-qubit (green) QIRB on ibmq_algiers. Experiments were performed with (a) and without (b) dynamical decoupling (DD) on idling qubits during MCMs. c Even with DD, the observed QIRB error rate (r_Ω) and the MCM error rate (ε_MCM) are much larger than predicted by IBM's reported measurement error data, suggesting significant MCM-induced crosstalk errors that are not mitigated by DD. All error bars are 1σ bootstrapped error bars.

Fig. 3. A “dressed” QIRB circuit layer.

The bulk of a QIRB circuit consists of a sequence of “dressed” layers $\left\{{\tilde{L}}_i\right\}$, each composed of three sublayers: (i) l_i,1, (ii) l_i,2, and (iii) l_i,3. The middle sublayer, l_i,2 is an Ω-distributed circuit layer, while the other two sublayers are specially crafted. Ideally, the state of the processor before the dressed layer ${\tilde{L}}_i$ is stabilized by the Pauli s_i−1. The first sublayer, l_i,1 is designed to rotate the state of the processor into an eigenstate of $s^{\mathrm{pre}}$, whose support, $s_{\mathrm{meas}}^{\mathrm{pre}}$, on the measured qubits in l_i,2 is a tensor-product of Pauli-Z and I operators. Likewise, l_i,3 is designed to (ideally) rotate the post-measurement state of the processor into an eigenstate of s_i, whose support on the measured qubits in l_i,2 is a uniform randomly chosen m-qubit Pauli operator, $s_{\mathrm{meas}}^i$.

Fig. 4. Simulation of QIRB on few-qubit QPUs with a simple error model.

a, b RB curves from two two-qubit [(a)] and two [(b)] six-qubit QIRB-r simulations performed using $p_{\mathsf{cnot}}=.35$ and $p_{\mathsf{MCM}}=.01$ (blue) or $p_{\mathsf{MCM}}=.5$ (orange). The points show ${\overline{F}}_d$, and the violin plots show the distributions of $⟨s_{\tilde{C}}⟩$ over circuits of that depth. QIRB has succeeded in all four cases as evidenced by the exponential decay of ${\overline{F}}_d$. c A plot comparing r_Ω from eight two-, four-, and six-qubit simulations (blue, orange, and green, respectively) with varying $p_{\mathsf{MCM}}$ and fixed $p_{\mathsf{cnot}}=.35$ (the points) against our predictions for r_Ω (the lines) based on the data-generating model. The error bars are 1σ error bars generated by performing each simulation eight times. These results validate QIRB as the r_Ω observed in each simulation matches that predicted by our theory. d Table containing the extracted one-qubit gate, $\mathsf{cnot}$, and MCM error rates (resp. ${\epsilon }_{\mathrm{1Q}},{\epsilon }_{\mathrm{2Q}},and{\epsilon }_{\mathsf{MCM}}$). We see close agreement between the estimated error rates and those used to generate the model. See Supplementary Note 5 for additional details.