Detailed, interpretable characterization of mid-circuit measurement on a transmon qubit

TL;DR

This work addresses the interpretability gap in mid-circuit measurement (MCM) characterization by extending the error generator formalism to quantum instruments. The authors present a perturbative, gauge-inspired framework that maps two-qubit error processes into a compact, physically meaningful set of MCM error strengths (FOMGI), enabling detailed decomposition into amplitude damping, readout error, and imperfect collapse. Through gate set tomography on a transmon device across readout amplitudes, they reveal dominant $T_1$-related errors, leakage at high drive, and non-Markovian AC Stark-shift effects, and they demonstrate how reduced models—notably CPTP+Stark—capture the essential physics with significantly fewer parameters. The findings offer a platform-agnostic diagnostic tool and scalable modeling approach to improve MCM fidelity, with implications for debugging and simulating large syndrome-extraction circuits in quantum error correction. These insights advance practical MCM calibration and pave the way for more efficient GST-based diagnostics across architectures.

Abstract

Mid-circuit measurements (MCMs) are critical components of the quantum error correction protocols expected to enable utility-scale quantum computing. MCMs can be modeled by quantum instruments (a type of quantum operation or process), which can be characterized self-consistently using gate set tomography. However, experimentally estimated quantum instruments are often hard to interpret or relate to device physics. We address this challenge by adapting the error generator formalism -- previously used to interpret noisy quantum gates by decomposing their error processes into physically meaningful sums of "elementary errors" -- to MCMs. We deploy our new analysis on a transmon qubit device to tease out and quantify error mechanisms including amplitude damping, readout error, and imperfect collapse. We examine in detail how the magnitudes of these errors vary with the readout pulse amplitude, recover the key features of dispersive readout predicted by theory, and show that these features can be modeled parsimoniously using a reduced model with just a few parameters.

Figures (12)

• Figure 1: Interpreting mid-circuit measurement (MCM) errors in experimental hardware. (a) A noisy MCM implemented on a superconducting qubit device can be characterized using gate set tomography, which constructs an estimated quantum instrument $\hat{\mathcal{Q}}$. (b) We propose a new framework for interpreting $\hat{\mathcal{Q}}$. First, we write $\hat{\mathcal{Q}}=\bar{\mathcal{Q}}+\Delta \mathcal{Q}$, where $\bar{\mathcal{Q}}$ is the target (ideal) quantum instrument and $\Delta \mathcal{Q}$ is the deviation from $\bar{\mathcal{Q}}$ (error). (c) We leverage a circuit gadget representation of an MCM. Here, we treat MCM errors (red splat, left) as being produced by an error process $e^L$ (orange splat, right) acting on a joint physical and virtual qubit system. (d) Assuming a linearized regime, we use the elementary error generator representation blume-kohout2022a of $L$ to establish an MCM error decomposition into physically interpretable error strengths. Notably, the error decomposition is sparse for most theory-derived MCM noise models as well as the experimental data shown here.
• Figure 2: Graphical interpretation of Eq. \ref{['eq:QI_circuit_probs']}. (a) A quantum circuit containing an MCM can be interpreted as two different quantum processes, conditional on the measurement outcome: (b) $c=0$ and (c) $c=1$.
• Figure 3: A single-qubit MCM (left) can be written as a circuit gadget (right) on two qubits with no mid-circuit measurement. (a) The noiseless case $\bar{\mathcal{Q}}_c[\rho]$: a virtual qubit is initialized ideally in $|0\rangle$ followed by a CNOT gate with the physical qubit as control and the virtual qubit as target. An ideal terminating measurement on the virtual qubit leaves the physical qubit in an output state conditional on the measurement outcome $c$. (b) The noisy case $\mathcal{Q}_c[\rho]$ (MCM with red splat): a post-gate error process generated by $L$ (orange splat) corrupts the CNOT. The dashed line indicates the noiseless initial state $\rho'$ of the physical and virtual auxiliary qubits and is given in Eq. \ref{['jointstate']}.
• Figure 4: Two-qubit error processes are mapped to single-qubit instrument errors [Eq. \ref{['eq:QIfromError']}]. (a) The deviation $\Delta \mathcal{Q} = \mathcal{Q}-\bar{\mathcal{Q}}$ between the ideal QI $\bar{\mathcal{Q}}$ produced by an identity "error" process and the noisy QIs $\mathcal{Q}$ generated by elementary error generators (EEGs) $S_{IX}$ with weight $s_{ix}$, $S_{IY}$ with weight $s_{iy}$, and $H_{zy}$ with weight $h_{zy}$. The two-qubit PTM has Pauli labels $II$,$IX$, $\dots$$ZZ$. Note that $\Delta Q^{(s_{ix})} = \Delta \mathcal{Q}^{(s_{iy})}\ne \Delta \mathcal{Q}^{(h_{zy})}$. (b) $H_{ZY}$ produces both first-order effects (inner diagonal sub-block) and second-order effects (anti-diagonal corners), with strengths $\mathcal{O}(h_{zy})$ and $\mathcal{O}(h_{zy}^2)$ respectively. Further, the second-order effects produce the same deviation as the first-order effects of $S_{IX}$ and $S_{IY}$ in (a).
• Figure 5: MCM error mechanisms as a function of the readout pulse's amplitude $V$. (a) A mid-circuit measurement is implemented through dispersive readout. This setup enables non-destructive state measurement of individual qubits by leveraging the interaction between the qubit (green) and a readout resonator (blue). The readout amplitude is systematically swept to control the drive strength, allowing for the investigation of measurement errors as a function of amplitude. (b) We plot single-shot IQ data from an experiment used to calibrate the MCM for four representative values of $V$ as point clouds (bottom) and log-scaled histograms of counts projected on the I axis (top). (c)-(d) To better understand the error mechanisms present in the MCM, we extract error strengths from the GST estimates as a function of $V$ using the error decomposition. Shaded regions show $2\sigma$ error bars, which in some cases are smaller than the plotted line width. (c) Errors that only appear in the low-amplitude regime, where the signal-to-noise ratio is low: weakness (blue, orange, and pink lines) and pure readout error/measurement bias (yellow and blue lines). Panel (d) Errors that can be attributed to $T_1$ effects, occurring either before or after the MCM. The sum of all $T_1$ effects (dashed green line) can be compared with the $T_1$ effects predicted by idling for the same period as the measurement (dashed black line). In the absence of added $T_1$ effects from the MCM, these two quantities should be equal. "X"-shaped markers show data where leakage events have been removed via post-selection; see Section \ref{['ssec:leakage']}.