Characterization of Unlearnable Noise with Mid-Circuit-Measurement-Based Cycle Benchmarking

Technical Summary: Characterization of Unlearnable Noise with Mid-Circuit-Measurement-Based Cycle Benchmarking

Problem Statement
Quantum noise characterization is a critical bottleneck for error mitigation and the validation of quantum error correction (QEC) protocols. Standard Cycle Benchmarking (CB) effectively isolates State Preparation and Measurement (SPAM) errors and characterizes Pauli channels by amplifying gate errors through repeated cycles. However, CB faces a fundamental limitation when applied to general $n$-qubit Clifford gates: the induced Pauli transfer graphs often contain coupled error parameters. Specifically, Clifford gates can induce irreducible cycles of Pauli propagation where distinct Pauli fidelities are coupled (e.g., $\lambda_{P_\alpha}\lambda_{P_\beta}$), rendering individual fidelities "unlearnable" due to gauge degrees of freedom. Standard CB cannot resolve these coupled parameters, and the number of unlearnable degrees of freedom grows exponentially with the number of qubits. Furthermore, standard CB assumes Markovian noise dynamics, a condition that may not hold in real hardware, yet standard protocols lack the tools to diagnose non-Markovian deviations.

Methodology
The authors propose Generalized Cycle Benchmarking (GCB), a framework that integrates Mid-Circuit Measurements (MCMs) as active tomographic primitives to resolve unlearnable noise. The methodology relies on three core theoretical and algorithmic pillars:

1. Pauli Weight Mismatch and Deferred Feed-Forward: The authors define the Pauli weight mismatch $\delta(P_\alpha, P_\beta)$ as the bitwise difference between the support of input and output Pauli operators under a Clifford gate $G$. In standard CB, if $\delta \neq 0$, the fidelity is unlearnable. The GCB protocol inserts MCMs at the qubit locations specified by $\delta(P_\alpha, P_\beta)$. By utilizing the Deferred Feed-Forward Principle, the protocol demonstrates that classical feed-forward operations (conditional bit-flips based on MCM outcomes) can be implemented entirely in classical post-processing. This allows the MCMs to effectively "reset" the Pauli operator back to its initial state, breaking the Clifford-induced cycle and decoupling the noise parameters without requiring real-time quantum control.

2. Uniform Stochastic Instruments (USI) and Learnability Conditions: The framework models noisy MCMs as Uniform Stochastic Instruments.

   • Exact Learning (Theorem 3): If the MCMs provide high-fidelity projections onto the $|0\rangle$ and $|1\rangle$ states (specifically, no quantum errors occur post-projection), all Pauli fidelities of the MCM and the interleaved Clifford gate are exactly learnable. Crucially, classical readout assignment errors do not affect learnability as they do not couple quantum noise parameters.
   • Approximate Learning (Theorem 4): In scenarios where only $|0\rangle$ preparation is reliable (common in superconducting devices), the authors show that unlearnable pairs can still be approximated. By post-selecting on the $|0\rangle$ outcome and utilizing a geometric-arithmetic mean decomposition, the protocol can isolate individual fidelities from the coupled product, provided the noise difference is sufficiently pronounced.

3. Binomial Analysis for Non-Markovianity: The protocol introduces a diagnostic tool based on the statistics of "flip variables" (changes in consecutive MCM readouts). Under the assumption of separable Markovian noise, these flip statistics follow a clean binomial distribution. Deviations from this distribution serve as a direct, quantitative signature of non-Markovian dynamics (e.g., memory effects or leakage), which are invisible to standard CB that collapses all information into a single final measurement.

Key Contributions

• Resolution of Unlearnable Noise: The paper provides a systematic method to break Pauli cycles in general $n$-qubit Clifford gates. The cost of breaking a cycle is quantified by the Hamming weight of the Pauli weight mismatch, specifying the minimal number of MCMs required.
• SPAM Robustness with MCMs: The framework achieves partial SPAM immunity for unlearnable noise components, a capability previously restricted to simple Pauli channels.
• Unified Noise Characterization: The authors demonstrate that MCMs with classical feed-forward can be treated as Clifford gates within the consistent Pauli noise learning framework (Gate Set Tomography), allowing for self-consistent characterization even when projection quality is imperfect.
• Non-Markovian Diagnostics: The work establishes a method to detect and quantify non-Markovian noise (specifically leakage-induced memory and readout asymmetries) using the statistical properties of repeated MCMs.

Experimental and Numerical Results
The authors validate the framework through numerical simulations and experiments on IBM superconducting quantum processors (ibm-aachen and ibm-pittsburgh).

• Numerical Simulations: Simulations on CNOT gates and 3-qubit CNOT ladders demonstrate the feasibility of extracting unlearnable noise pairs (e.g., $\{ \lambda_{0z}, \lambda_{zz} \}$) and show that the protocol scales effectively, particularly when leveraging local noise models to reduce MCM overhead.
• Experimental Validation:
  • Unlearnable Noise: On both devices, the GCB protocol successfully extracted unlearnable Pauli fidelities for CNOT gates. The results were benchmarked against physical CPTP (Completely Positive Trace-Preserving) bounds and conventional tomography. The GCB results were found to be within CPTP bounds and often closer to the physical limits than conventional tomography, demonstrating superior precision in disambiguating coupled noise.
  • Non-Markovian Signatures: Binomial analysis of MCM readouts revealed two distinct non-Markovian signatures:
    1. A "zig-zag" statistical bias favoring even flip counts, attributed to classical readout assignment errors.
    2. An exponential tail in the flip distribution at large flip counts, attributed to leakage-induced memory (non-Markovian correlations). The authors modeled this tail to extract a decay rate $\lambda_d$, quantifying the memory duration.

Significance and Claims
The paper claims that Mid-Circuit Measurements are not merely passive diagnostic tools but active resources for noise characterization. The significance of this work lies in:

1. Overcoming Fundamental Limits: It resolves the "unlearnable" degrees of freedom in Clifford gate noise that have limited the scalability of noise characterization protocols.
2. Hardware Agnosticism: While demonstrated on superconducting devices, the framework is hardware-agnostic and applicable to any platform supporting MCMs and classical feed-forward.
3. Diagnostic Capability: It provides a new method to detect and characterize non-Markovian dynamics, defining the operational regime where the Pauli noise model is valid.
4. Practical Utility: The authors demonstrate that state-of-the-art superconducting MCMs possess sufficient projection quality to enable physically consistent noise learning with partial SPAM immunity, paving the way for characterizing larger Clifford gates and advancing error mitigation strategies.

The paper concludes that this framework extends the reach of Cycle Benchmarking beyond unitary Clifford gate sets to the broader class of dynamic circuits available in modern quantum processors.