Mid-Circuit Measurement in Quantum Circuits

• Mid-circuit measurement is a quantum process inserted within circuits that yields both a classical measurement result and a post-measurement quantum state.
• Advanced methodologies like QILGST enable detailed characterization using error metrics such as half diamond distance, readout fidelity, and output state fidelity.
• Experimental results on superconducting qubits reveal timing-dependent non-Markovian errors, highlighting the need for cavity reset in fault-tolerant quantum computing.

Mid-circuit measurement refers to quantum measurement operations inserted into the internal layers of a quantum circuit, as opposed to traditional measurements performed only at circuit termination. With both classical (measurement result) and quantum (post-measurement state) outputs, mid-circuit measurements are essential for quantum error correction, real-time feedback, state initialization, and adaptive protocols. Unlike terminating measurements, they introduce unique error channels, demand precise characterization, and require specialized hardware and control for integration with subsequent quantum operations.

1. Theoretical Model: Quantum Instruments and Tomography

Mid-circuit measurements in quantum circuits are formally modeled as quantum instruments—a collection of completely positive (CP) maps $\{Q_0, Q_1, ..., Q_{m-1}\}$ such that $\sum_i Q_i$ is trace-preserving. Each $Q_i$ corresponds to outcome $i$: it produces both a classical bit (the measurement result) and a quantum state (the post-measurement state). This generalizes POVMs by specifying the state update as well as the outcome probability.

Quantum Instrument Linear Gate Set Tomography (QILGST) extends standard gate set tomography by enabling self-consistent reconstruction of these quantum instruments. In QILGST, circuits are constructed that insert the mid-circuit measurement into informationally complete fiducial sequences, and the instrument is estimated (via linear inversion or maximum likelihood methods) as a stacked process matrix. For a qubit ($d=2$) with two outcomes, the superoperator matrix is $4\times8$. Performance and error modes can then be extracted not only for the classical measurement but also for the full quantum channel conditioned on the outcome (Rudinger et al., 2021).

2. Error Metrics and Characterization

QILGST enables error diagnosis and quantification via several key metrics:

• Half Diamond Distance: $$\epsilon_\diamond = \frac{1}{2}\|\hat{Q} - Q_\text{target}\|_\diamond$$. This is a worst-case error rate incorporating both classical outcome errors and quantum state preparation imperfections. In experimental application with sufficient post-measurement delay, $\epsilon_\diamond = 8.1 \pm 1.4\%$.
• Readout Fidelity: $F = \frac{1}{2}(P_{0|0} + P_{1|1})$, quantifying classical discrimination between basis states. Example value: $97.0 \pm 0.3\%$.
• Output State Fidelity: Fidelity between conditioned output quantum state and ideal target for each outcome; e.g., $96.7\pm0.6\%$ for '0', $93.7\pm0.7\%$ for '1'.

An explicit form for a two-outcome $\sigma_z$ dispersive measurement:

$$Q_{\mathrm{target},k}[\rho] = \mathrm{Tr}\left[\frac{1}{2}(I + (-1)^k z) \rho\right] (I + (-1)^k z),\quad k=0,1.$$

This mapping both projects and repopulates the eigenstate corresponding to the measurement outcome (Rudinger et al., 2021).

3. Non-Markovianity: Error Sources and Timing Dependence

The fidelity of mid-circuit measurements is strongly dependent on the quantum hardware’s ability to fully reset all degrees of freedom coupled to the measured qubit. In 3D transmon circuits with dispersive readout, residual cavity photons after the measurement pulse induce a Stark shift on the measured qubit. For short inter-operation delays ($t_d$ below $\sim 1\,\mu s$), this causes non-Markovian errors in the gates applied immediately after measurement.

This regime is evidenced by high log-likelihood model violation (e.g., $N_\sigma\approx400$ standard deviations above expectation) and total variation distances (TVD) up to $37\%$. When the photon population is given time to decay ($t_d\gtrsim 1\,\mu s$ given a $\sim 240$ ns cavity lifetime), these effects recede (TVD $<6\%$), and performance saturates: $\epsilon_\diamond = 8.1\%$, $F\approx97\%$, output state fidelities in the mid-90%s. Stark-shift models capture these effects as exponentially decaying phase errors appearing as $$G_k(\alpha, r, i, m) = \exp\left(\log \hat{G}_k + \alpha_i(t_d) e^{-mr_i} \mathcal{Z}\right)$$ (Rudinger et al., 2021).

4. Implications for Quantum Error Correction and Fault Tolerance

For fault-tolerant quantum computing, mid-circuit measurements are indispensable—they allow repeated measurement of error syndromes (stabilizers) on ancilla qubits without disturbing logical data qubits. Detailed characterization of these measurements is critical because several distinct error channels must be diagnosed:

• Swapping or misidentification of the syndrome bit
• Errors in the conditional quantum state
• Residual non-Markovian coupling with hardware

QILGST provides a comprehensive, circuit-level method for assessing total error and individual error modes in the context most relevant for real-world QEC: operating circuits containing both logic and measurement. This is superior to detector tomography or application of readout fidelity alone, which miss quantum-state-altering errors. The half diamond distance metric in particular quantifies whether fault-tolerance thresholds are met for measurement-based protocols (Rudinger et al., 2021).

5. Experimental Protocol and Results

The methods were validated on a five-qubit superconducting transmon device employing dispersive readout. Measurements were performed by injecting $\sim 1\,\mu s$ pulses into a coupled cavity and analyzing the output with a Josephson traveling-wave parametric amplifier (JTWPA). Around 100–128 GST circuits (36 with mid-circuit measurement) were used, with 1024 shots per circuit.

Key findings (for $t_d > 1\,\mu s$):

Metric	Value
Half diamond error ($\epsilon_\diamond$)	$8.1 \pm 1.4\%$
Readout fidelity ($F$)	$97.0 \pm 0.3\%$
Output state fidelity ('0')	$96.7 \pm 0.6\%$
Output state fidelity ('1')	$93.7 \pm 0.7\%$

Shorter delays led to significant error inflation due to photon-induced frequency shifts, while operation beyond $t_d\sim 1\,\mu s$ matched the expected error budget. These results confirm the necessity of cavity reset or active cooling in fault-tolerant operation (Rudinger et al., 2021).

6. Methodological Impact and Outlook

QILGST enables the systematic, self-consistent characterization of mid-circuit measurements as full quantum operations, diagnosing both classical and quantum error modes. This technique is expected to be integral to validating and calibrating measurement primitives in future multi-qubit, error-corrected devices. As mid-circuit measurements become instrumental in real-time feedback and adaptive algorithms, QILGST and related methodologies will be key to understanding error propagation, designing error mitigation, and informing architecture-level choices for scalable quantum computing (Rudinger et al., 2021).