Randomized Benchmarking Protocol for Dynamic Circuits

TL;DR

This work addresses the benchmarking gap for dynamic quantum circuits that include mid-circuit measurements and feedforward by extending randomized benchmarking to interleave dynamic blocks with the data-qubit RB. The authors formalize a protocol using blocks $\mathcal{F}$ that ideally act as the identity on data qubits but propagate measurement readout errors into the data-qubit error rate, derive leading-order expressions for these errors, and validate them with simulations and IBM Eagle experiments. Key findings show that readout assignment errors can dominate and that dynamical decoupling substantially suppresses coherent and measurement-induced errors, enabling a fast diagnostic of dynamic-circuit faults. The protocol provides a practical, scalable tool for mapping dynamic-circuit crosstalk and informing fault-tolerance benchmarks and calibration in current devices.

Abstract

Dynamic circuit operations -- measurements with feedforward -- are important components for future quantum computing efforts, but lag behind gates in the availability of characterization methods. Here we introduce a series of dynamic circuit benchmarking routines based on interleaving dynamic circuit operation blocks $F$ in one-qubit randomized benchmarking sequences of data qubits. $F$ spans between the set of data qubits and a measurement qubit and may include feedforward operations based on the measurement. We identify six candidate operation blocks, such as preparing the measured qubit in $|0\rangle$ and performing a $Z$-Pauli on the data qubit conditioned on a measurement of `1'. Importantly, these blocks provide a methodology to accumulate readout assignment errors in a long circuit sequence. We also show the importance of dynamic-decoupling in reducing ZZ crosstalk and measurement-induced phase errors during dynamic circuit blocks. When measured on an IBM Eagle device with appropriate dynamical decoupling, the results are consistent with measurement assignment error and the decoherence of the data qubit as the leading error sources.

Figures (8)

• Figure 1: (a) Example of the protocol with $l=3$ sequence length. Sequences of one-qubit Clifford gates are applied on data qubits simultaneously, interleaved with dynamic circuit blocks $\mathcal{F}$ which expand the space to include a measurement qubit. Each $\mathcal{F}$ is ideally equivalent to the identity operation. (b) The specific dynamic circuit blocks $\mathcal{F}$ that we use in this work, as described in the main text.
• Figure 2: (a) Simulation results of Z_c0 for varying values of $T_1$ for the data qubit ($T_1=T_2$) as a function of readout assignment error $\epsilon_R$. We fit to the first-order expression as described in the main text. (b) Simulation results of H_CNOT for varying values of $\epsilon_R$, as a function of the CNOT depolarizing error. (c) Simulation results of H_CNOT compared to Z_c0 at $T_1=T_2=250\,\mu{\rm s}$. The H_CNOT block has an added 2Q depolarization probability of 0.01 ($7.5\times10^{-3}$ gate error). Both fit to the first order expression, with 2/3 and 4/9 linear dependence on the assignment error for H_CNOT and Z_c0 respectively.
• Figure 3: Simulation results for 5 of our dynamic circuit blocks as a function of ZZ crosstalk between the measured and the data qubit. Noise parameters are taken to be $\epsilon_R=0.02$, $T_1=T_2=250\,\mu{\rm s}$, $\epsilon_{2Q}=0.01$ and $\Delta=10$ kHz.
• Figure 4: The error rate for the different protocols as measured on three sets of qubits. The highlighted qubits are the nearest-neighbor (connected) qubits and the rest have large distance separation on the device. The large difference between these groups can be ascribed to crosstalk. The ratio between Z_c0 and Z_c1 is due to asymmetric behavior of the measured qubits, see text for details. In the case of the set with measured qubit 77, the assignment error is high, $\approx0.1$. No dynamical decoupling is performed in this experiment.
• Figure 5: (a) Dynamical decoupling (DD) sequences that we consider (not to scale). (Top) Measurement DD (MDD), which involves X2 DD within the measurement time on the data qubit. (Bottom) FFDD, to suppress the ff delay time, where the full delay is sliced to two sequences of X2, one with the ff delay time and the second with the remaining delay $\tau_{M}-\tau_{FF}$ assuming $\tau_{M}>\tau_{FF}$. For the device used in this work, $\tau_M=1512~\text{ns}$ and $\tau_{FF}=1060~\text{ns}$. (b) Experimental results of the H_CNOT protocol with and without DD on qubits 8 and 9. (c) Error rate of 30 connected pairs, extracted as in (b), each pair measured separately. The FFDD outperforms the MDD in almost all pairs, although the biggest improvement is gained by doing any type of DD. Once DD is applied the data agrees well to a model adding the assignment error and the CNOT error as reported from the device, see text for details.