Lifting: A New Framework for Quantum State Discrimination in Mid-Circuit Measurements

When Quantum Computing Meets Noise: The Reliability Challenge of Mid-Circuit Measurements

One of the core promises of quantum computing is to push beyond the boundaries of classical computation. But to truly deliver on that promise, a fundamental problem must be solved: noise.

This is especially true for mid-circuit measurements (MCMs), which underpin quantum error correction and measurement-based conditional control—the cornerstones of fault-tolerant quantum computing. Yet on superconducting hardware, MCM reliability has long been constrained by noise. These operations rely on precise quantum state discrimination (QSD), which maps measurement signals to quantum state labels.

Neural-network-based QSD has already demonstrated superior signal classification performance compared to traditional methods, but its error remains bounded by the Bayesian risk lower bound of the available readout distribution—a theoretical ceiling that no better classifier alone can breach. At the same time, as the number of measurements increases and circuit execution progresses, the quantum states themselves degrade, making static calibration increasingly unable to represent the true conditions in actual workloads.

These two challenges compound to form the core bottleneck in current MCM reliability.

Quantum state discrimination (QSD) is a fundamental task in quantum information processing: given a signal produced by measuring a quantum system (e.g., the I-Q voltage trajectory of a superconducting qubit), determine which quantum state label it corresponds to. In the ideal case, different quantum states map to non-overlapping regions in signal space, and discrimination is trivial. But in real hardware, amplifier noise, quantum decoherence, and state preparation errors cause the readout distributions of different states to overlap, turning discrimination into a statistical inference problem. The goal of QSD is to guess the true state label with minimum error probability given an observed signal. This minimum error probability is closely related to Bayesian risk: given known prior probabilities and conditional distributions, Bayesian risk provides the error rate lower bound that no classifier can breach. Therefore, the performance ceiling of QSD is not determined by the classifier algorithm, but by the signal-to-noise ratio and distribution overlap of the readout signal itself.

The Bayesian risk lower bound is a classical concept in statistical decision theory. In quantum state discrimination, if the prior probabilities of each quantum state and the conditional probability distributions of the readout signal are known, then the average error rate of the Bayes-optimal classifier is the Bayesian risk. For two-state discrimination, it can be expressed analytically as half the overlap area of the two distributions; for multi-state discrimination, it involves more complex optimization. The key point is: no matter how powerful the neural network or how advanced the signal processing technique, as long as the readout distribution remains fixed, the classifier's error rate cannot fall below this lower bound. This fact means that to truly improve MCM reliability, optimizing the classifier alone is insufficient—more information usable for discrimination must be extracted from the information source itself, namely the joint distribution of readout signals. This is precisely the theoretical starting point for the "repetition encoding" mechanism in the Lifting framework.

Lifting's Core Approach: Encoding, Decoding, and In-Situ Calibration

To address these problems, the research team proposed the Lifting framework, which improves quantum state discrimination accuracy through three synergistic mechanisms.

Repetition Encoding: Extracting More Information from a Fixed Distribution

The first mechanism of Lifting is repetition encoding. It collects joint in-phase and quadrature (I-Q) readout signals from target qubits and ancillary qubits. The key insight is: rather than using only the signal from a single measurement, it retains more channels from a fixed joint distribution—and theoretical analysis proves that doing so does not increase Bayesian risk.

This conclusion provides a rigorous mathematical foundation for the framework, meaning that the introduction of additional information channels is safe and will not lead to degraded discrimination performance.

Neural Network Decoding and Contextual Calibration

The second mechanism of Lifting is neural decoding. Building on the rich signals provided by encoding, a neural network decoder maps these signals to quantum state labels.

But a statically trained decoder faces a practical problem: quantum states degrade during circuit execution, and the data distribution at training time may not match the conditions encountered during actual execution. Lifting's solution is in-situ calibration—the decoder performs contextual calibration based on the actual conditions encountered during execution, keeping the discrimination model closely aligned with the true state of the current workload.

Repetition encoding is a classical concept in quantum error correction: encoding the same logical information repeatedly onto multiple physical qubits so that information can still be recovered when some qubits fail. But repetition encoding in Lifting has a different meaning—it does not redundantly encode the quantum information itself, but rather expands the channels of the readout signal. Specifically, in the joint I-Q readout of target qubits and ancillary qubits, beyond the typically used single-measurement signal, multiple samples, time-resolved trajectories, or correlation signals between ancillary and target qubits can be used as additional information channels. These additional channels come from the same fixed joint distribution, and therefore do not change the statistical structure of the problem. Theoretical analysis shows that adding these channels does not increase Bayesian risk, because the optimal classifier is free to ignore useless channels. Conversely, if these channels contain additional information correlated with the quantum state, the optimal classifier can exploit them to reduce the actual error rate.

Theoretical Guarantees: Why More Channels Cannot Make Results Worse

One of Lifting's compelling theoretical contributions is proving that retaining additional channels from a fixed joint distribution cannot increase Bayesian risk.

Bayesian risk is the performance lower bound of QSD, and traditional methods can only approach it asymptotically. Lifting does not attempt to circumvent this lower bound, but instead reduces the practically achievable error by expanding information channels.

Furthermore, the research establishes sufficient conditions for overlap-based upper bound contraction. This means that under certain conditions, the upper bound on discrimination error can be narrowed, providing stronger guarantees for practical performance. This analysis bridges engineering design and statistical learning theory, avoiding the blindness of purely empirical hyperparameter tuning.

Experimental Validation: Real Workloads on Three IBM Processors

Lifting was evaluated on three IBM quantum processors, covering three typical workload categories:

• Repeated stabilizer measurements: fundamental operations in quantum error correction
• Parity check measurements: a critical component of error detection
• Quantum teleportation: a complete protocol involving conditional control and measurement

What these workloads share is their dependence on MCM accuracy. Experiments were conducted across three different processors, providing a degree of cross-hardware consistency validation.

Significance and Limitations

Lifting's value lies in expanding QSD from an isolated classification problem into a dynamic discrimination framework oriented toward the entire circuit execution process. The encoding layer mines the information potential of the joint distribution, the decoding layer uses neural networks to fit complex mappings, and the calibration layer addresses distribution drift caused by state degradation—three layers corresponding to three different sources of error.

It should be noted that the framework's experimental validation is limited to IBM's superconducting platform, and its applicability to other quantum hardware architectures (such as ion traps and photonic quantum computing) remains to be verified. Additionally, in-situ calibration requires extra execution overhead, and how its cost-benefit ratio varies across circuits of different scales warrants further study.

For researchers focused on quantum error correction and fault-tolerant quantum computing, Lifting offers a viable path to improving MCM reliability at existing noise levels—it does not require hardware-level breakthroughs, but instead mines information from existing readout signals through signal processing and statistical learning.

Superconducting quantum processors are among the most mature hardware platforms in quantum computing today. They use Josephson junctions as qubits, operating at extremely low temperatures of approximately 10 millikelvin in dilution refrigerators. Readout of superconducting qubits is typically achieved through dispersive coupling: the state of a qubit slightly shifts the resonant frequency of a microwave cavity coupled to it, and this frequency shift is then heterodyne-demodulated into a voltage signal in the in-phase and quadrature (I-Q) plane. This readout method is fast, but the signal is noisy, fidelity is limited, and the readout process itself destroys the quantum state. IBM's quantum processors (such as the Eagle and Heron series) use fixed-frequency or tunable-frequency transmon qubit architectures, with readout fidelities typically between 98% and 99.5%. Mid-circuit measurement requires readout in the middle of executing a quantum circuit, which demands not only fast readout but also the ability to reset qubits after measurement and have them continue participating in subsequent gate operations—placing higher demands on both hardware and discrimination algorithms.