Lifting: Quantum State Discrimination Framework for Mid-Circuit Measurements

Chi Zhang; Neel Vora; Yilun Xu; Gang Huang; Fan Chen; Lei Jiang

quant-phOct 13, 2026

Lifting: Quantum State Discrimination Framework for Mid-Circuit Measurements

Authors: Chi Zhang, Neel Vora, Yilun Xu, Gang Huang, Fan Chen, Lei Jiang

Organizations: Indiana University Bloomington, Bloomington, Indiana, USA · Lawrence Berkeley National Laboratory, Berkeley, California, USA

Abstract

Quantum computing promises new capabilities beyond classical computation. Mid-circuit measurements (MCMs) enable error correction and outcome-dependent control, but their reliability is limited by noise on superconducting hardware. These operations depend on accurate quantum state discrimination (QSD), which assigns measured signals to state labels. Neural network-based (NN-based) QSD improves this assignment, yet its error remains bounded below by the Bayes risk of the available readout distributions. A complementary challenge is state deterioration as successive measurements accumulate and circuit execution advances, making static calibration less representative of the workload. We present Lifting, a QSD framework that addresses both challenges through repetition encoding, neural decoding, and in situ calibration. Encoding supplies joint in-phase and quadrature (I-Q) readout signals from the target and auxiliary qubits, while contextual calibration trains the decoder for the conditions encountered during execution. Our theoretical analysis shows that retaining additional channels from a fixed joint distribution cannot increase Bayes risk and establishes sufficient conditions for contraction of an overlap-based upper bound. Experiments across three IBM processors evaluate the framework on repeated stabilizer measurement, parity-check, and teleportation workloads.

Figures & tables

Panel of Figure 1 — Figure 1: State deterioration reflected in readout-distribution changes in the five-cycle QR-Stabilizer workload. (a) Core circuit with a measured data qubit and a phase-kickback ancilla. Lifting proxy wires are omitted. (b) SPAM references for preparation labels $0$ (blue) and $1$ (red) on qubit 51 of ibm_fez . 10,000 shots per class are sampled from the reference acquisitions for display. (c) Target-readout I–Q distributions ordered from cycles $c_{1}$ to $c_{5}$ . (d) Equal-prior kernel density estimation (KDE) Bayes-risk estimates comparing the SPAM- $0$ reference with each ideal- $1$ target cycle, SPAM-trained NN-based QSD error rates, and transferred digital-error estimates. The curve labeled ibm_fez transfers the accuracy offset between NN-based QSD and digital readout from a separate acquisition. It is not native digital readout of the same target shots.

Figure 2: Lifting workflow. (a) Context-matched calibration executions prepare reference labels and place encoded readout at the workload’s measurement locations. (b) Local repetition encoding couples the labeled MCM qubit to

d-1

auxiliary ancillas initialized to

\ket{0}

. Concurrent readout supplies their joint I–Q readout signals. (c) The

d

I–Q pairs are concatenated into a

2d

-component feature vector. Reference labels supervise neural training, and inference assigns a logical outcome to each target observation. Optional confidence selection retains a subset of observations before reporting retained-shot accuracy.

Figure 3: Error rates across five MCM cycles: the KDE Bayes-risk estimate, SPAM-trained NN-based QSD, the transferred ibm_fez digital estimate from Fig. 1 (d), and single-channel NN-based QSD with cycle-matched in situ calibration. The risk curve compares constructed reference–target distributions. The decoder curves evaluate target shots. Contextual calibration changes the training references, while this comparison retains the single-channel observation space.

Figure 4: Qualitative geometry of marginal and conditional overlap. (a) The preceding

d-1

readout channels have marginal Bhattacharyya overlap

\rho_{d-1}=\mathrm{BC}(f_{0}^{(d-1)},f_{1}^{(d-1)})

, which upper-bounds balanced-prior Bayes risk after multiplication by

1/2

. (b) Conditional overlap in the added I–Q channel is bounded by

\gamma_{\max,d}

. Shaded regions schematically illustrate the factorization.

Figure 5: Target-only and joint I–Q readout discrimination for $d=2$ on ibm_fez , using 10,000 shots per preparation label. (a,b) Target and auxiliary (proxy) I–Q scatter, with blue and red denoting reference labels $0$ and $1$ , respectively. (c,d) Signed quadratic-discriminant scores $s_{1}-s_{0}$ from the two-dimensional target observations and four-dimensional joint I–Q readout signals. Annotated risks are equal-prior KDE-based estimates of distributional overlap, rather than held-out classifier error rates.

Figure 6: IBM Fez assignment accuracy versus encoding distance

d

, comparing hardware measurements (emp.) with the fitted empirical model in Eq. ( 18 ). The curve labeled theo. uses

\kappa=0.45

and

\alpha=9.03

, fitted for the displayed distance sweep.

Panel of Figure 7 — Figure 7: Assignment accuracy versus distance on (a) ibm_fez , (b) ibm_kingston , and (c) ibm_marrakesh . The baseline averages native assignment accuracy over the selected physical qubits. Spatial repetition (rep- $d$ ) uses independent hard assignments followed by voting. The temporal comparator (tem- $d$ ) jointly decodes $d$ sequential I–Q observations of one qubit. Lifting (lif- $d$ ) jointly decodes concurrent I–Q observations from $d$ repetition-encoded qubits. The spatial schemes increase physical-qubit count with $d$ , whereas the temporal scheme increases measurement time. Both neural schemes use $2d$ scalar inputs.

Panel of Figure 8 — Figure 8: Distance-two assignment accuracy for physical-qubit pairs on (a) ibm_fez , (b) ibm_kingston , and (c) ibm_marrakesh . Bracketed labels identify the selected qubit pairs. The base0 and base1 bars report native assignment accuracy of the first and second qubits, respectively. The rep-2 bars use hard repetition decoding. Lifting , labeled lif-2, uses neural decoding of joint I–Q readout signals from the same two-qubit encoding. Pair selection spans heterogeneous native readout accuracies.

Figure 9: Per-cycle QR-Stabilizer discrimination accuracy on ibm_fez , in measurement order

c_{1}

–

c_{5}

. NN-based QSD uses target shots. Lifting with in situ calibration uses held-out calibration shots (Appendix B.2 ).

Figure 10: Circuit success rates for (a) QR-Stabilizer, (b) QEC-5, and (c) teleportation. Success requires a correct complete QR-Stabilizer trace, final QEC Pauli-frame bit, or recovered computational-basis input, respectively, as defined in Appendix B.1 . Green bars report success among retained shots. Decoder configurations and filtering rules are given in Appendix B.2 .

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 11 — Figure 11: MCM workloads. (a) Repeated stabilizer measurement with reset and reuse. (b) Repeated parity extraction with an outcome-conditioned $X$ correction. (c) Teleportation with outcome-conditioned $X$ and $Z$ corrections. Double lines carry classical measurement outcomes. Lifting auxiliary wires are omitted. The experiments account for the corrections in (b,c) through the Pauli-frame interpretation described in the text.

Figure 12: Physical connectivity used by the MCM workloads on ibm_fez , ibm_kingston , and ibm_marrakesh . Nodes identify physical qubits, and edges denote native couplings. The workload-specific qubit assignments are given in the text.

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Quantum Cloud Computing and Distributed Systems (qCLOUDS) Lab, School of Computing and Information Systems, The University of Melbourne, Melbourne, VIC, 3010, Australia

Late Fusion Quantum Machine Learning Hybrid Quantum-Classical ML Quantum Neural Networks

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Circuit cutting lets a large quantum neural network (QNN) run as independent subcircuits on small devices, but rebuilding its outputs by reconstruction carries a classical sampling overhead exponential in the number of cuts - the dominant runtime cost in prior work. We ask whether, for machine-learning tasks, this step is necessary, and replace it with late fusion: each subcircuit is trained and measured independently, and a small classical head combines their outputs - a linear-cost, decision-level combination borrowed from multimodal learning. To characterize the trade-off we introduce a quantumness dial

Q

, a tunable reconstruction budget interpolating from pure fusion to full reconstruction, and a cut-entanglement diagnostic that indicates how much reconstruction a task needs (Spearman

ρ=0.59

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runs). Across synthetic and standard datasets, independently trained late fusion matches full reconstruction accuracy within

0.04

at every point of the controlled sweep and on every classical benchmark, at exponentially lower cost; it is also markedly more robust to shot and device noise. Controlled entangled-data experiments locate the boundary where fusion must fail. We do not claim advantage over classical machine learning - consistent with recent benchmarking, quantum offers no accuracy edge on these datasets. Late fusion is thus an efficient, noise-robust, self-characterizing alternative to reconstruction for circuit-cutting QML.

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