Circuit-noise-resilient virtual distillation

Noisy intermediate-scale quantum (NISQ) devices suffer from environmental and hardware-induced noise that limits algorithmic accuracy. Quantum error mitigation (QEM) techniques aim to reduce noise effects without full error correction. Virtual distillation (VD) is a prominent QEM method that, for an N-qubit noisy state ρ and observable O, uses multiple copies of ρ to suppress errors exponentially with the order n, estimating Tr[ρ^n O]/Tr[ρ^n]. Despite its advantages, VD faces two key challenges when targeting high-accuracy estimates: (1) coherent mismatch between the ideal and noisy states, which can be mitigated with techniques such as quantum subspace expansion or polynomial fitting over multiple VD orders; and (2) noise in the VD circuit itself, which uses Hadamard-test-like circuits with controlled derangement operators and controlled-O, introducing multi-qubit gate errors that can exceed the original state-preparation error. Existing approaches either treat the VD circuit as part of the target circuit and apply additional QEM or avoid the VD circuit via shadow estimation at the cost of exponential measurement overhead. This work targets the second challenge and proposes circuit-noise-resilient virtual distillation (CNR-VD), which integrates a calibration step using easily prepared states to compensate VD-circuit noise and recover ideal-VD results.

The paper situates its contribution within QEM and VD literature. VD has been developed as a state-noise-agnostic method that achieves exponential error suppression for observable estimation. Prior work has addressed coherent mismatch via quantum subspace expansion, polynomial fitting across VD orders, and shadow estimation variants (e.g., shadow distillation) to approximate nonlinear VD functionals. Noise in the VD circuit has been mitigated using general QEM such as zero-noise extrapolation or avoided by shadow techniques, though with significant measurement overhead. Measurement error mitigation (e.g., TPN and related methods) effectively cleans single-qubit readout, relevant because VD measurements target an ancilla. Randomized compiling (RC) tailors noise to stochastic Pauli channels and underpins several mitigation strategies by bringing hardware noise closer to Pauli models. Against this backdrop, the authors propose a calibration-based approach specific to VD-circuit noise, compatible with Pauli and certain general Markovian noise models, and extendable to Hadamard test circuits.

• Noisy VD model: Represent the noisy state as ρ = (1 − ε)|ψ⟩⟨ψ| + ε Σ_λ λ |ψ_λ⟩⟨ψ_λ|. The n-th order VD aims to estimate ⟨O⟩_VD = Tr[ρ^n O]/Tr[ρ^n], which suppresses errors as O(ε^n). Practically, a Hadamard-test-like VD circuit implements controlled derangement D_n and controlled-O to obtain Tr[ρ^n O] and Tr[ρ^n].
• Circuit noise characterization: Focus on stochastic Pauli gate noise on m-qubit gates with error rate p_m: E_m(ρ) = (1 − p_m)ρ + p_m Σ_s δ_m^s P^s ρ P^s. Assume clean ancilla readout due to effective measurement error mitigation. Analyze second-order VD, where the noisy estimator 2 p̃_O(ρ,O) − 1 can be written as a sum over contributions indexed by j, the number of unintended SWAPs induced by Pauli errors on CSWAP control, and combinations t of their positions. This yields a decomposition with p-independent coefficients a_j(·) and state-dependent terms A_j(t).
• CNR-VD estimator (one-weight Pauli observables): For single-qubit Pauli observables O (and for I), A_j(t) = Tr[ρ^2 O] for j ∈ {0,1} and t = 0, reducing the noisy relation to 2 p̃_O(ρ,O) − 1 = a(P_1,P_2,P_3,O) Tr[ρ^2 O], where a(·) is independent of gate error rates p_i given temporal stability. Calibrate a(·) using an easily prepared state s that is an eigenstate of O with eigenvalue +1, so Tr[s O] = 1. Obtain a(·) estimates via the same noisy VD circuit applied to s, with a small bias from imperfect s preparation. Similarly calibrate with O = I to estimate Tr[ρ^2]. Define the noisy VD ratio estimator Õ_VD(ρ) = (2 p̃_O(ρ,O) − 1)/(2 p̃_O(ρ,I) − 1) and construct the CNR-VD estimator: Õ_CNR-VD(ρ) = (Õ_VD(ρ)/Õ_VD(s)) Õ_VD(s), which cancels VD-circuit noise factors under the calibration model.
• Extension to higher-weight observables: Apply CNR-VD to higher-weight Pauli observables under moderate noise by neglecting terms with j > 1 in the expansion, incurring an approximation error dependent on Pauli error contributions O(P_1,P_2,P_3). If the native noise deviates from Pauli stochastic noise, use randomized compiling to tailor it towards Pauli-like behavior.
• General Markovian noise and effective noise model: For an n-order VD circuit with l noisy layers U_l = Π_j U_j and layerwise Markovian noise ε_j, commute all noise to the end to define an effective channel ε_eff with Kraus operators K_k. The noisy VD equals an ideal VD followed by ε_eff. Under either condition: (1) each Kraus operator factors as K_k = K_k^R ⊗ K_k^P with unitary factors, or (2) ε_eff = ε_eff^R ⊗ ε_eff^P with ε_eff^R single-qubit unital and ε_eff^P CPTP, the CNR-VD estimator reduces ε_eff-induced error when s-preparation error is negligible relative to ε_eff. Depolarizing noise satisfies Condition 2, implying high tolerance to low-rate Pauli-like noise.
• Extension beyond VD to Hadamard test: Generalize D and O to an arbitrary Hermitian operator U to form a Hadamard-test circuit. Choose calibration state s with Tr[s U] = 1 and apply the same estimator construction to mitigate controlled-U circuit noise when estimating Tr[U].
• Randomized compiling for CNR-VD: Employ RC to approach the stochastic Pauli noise assumption. For second-order VD, apply twirling separately to the controlled-O chain (Clifford) using random multi-qubit Pauli operators and to the CSWAP chain by selecting a subset of 3-qubit Pauli gates preserved under CSWAP conjugation. Compile twirling layers to minimize overhead: only three Pauli layers are needed, with some gates merged into state-preparation layers and only the first-qubit Pauli required in the final layer due to single-qubit measurement. RC suppresses off-diagonal elements in the Pauli transfer matrix, reducing non-Pauli noise components.

• Accuracy improvement: CNR-VD significantly improves estimation accuracy over noisy VD, achieving up to an order-of-magnitude (tenfold) error reduction in simulations for low-weight observables and moderate VD-circuit noise levels.
• Positive mitigation at higher noise: CNR-VD provides positive error mitigation even in regimes where standard noisy VD fails, effectively raising the noise threshold for beneficial mitigation by up to one order of magnitude, as evidenced in noise-threshold analyses (Fig. 3b) and failure-rate maps (Fig. 3c).
• Robustness across settings: For 4-qubit random states with fixed state-preparation error ε = 0.10 and VD-circuit noise levels from 0.1 to 10.0, CNR-VD closely recovers ideal-VD outcomes at low observable weight or moderate noise (Fig. 3a). Results are averaged over 50 random-state experiments with error bars showing standard deviation.
• Comparative performance: Against Shadow Distillation (SD) and noisy VD with zero-noise extrapolation (ZNE-VD), CNR-VD maintains superior accuracy across qubit counts N = 2 to 8 and measurement budgets M from 10^2 to 10^7. With M = 2×10^4 and N ∈ {2,8}, CNR-VD outperforms noisy VD by more than an order of magnitude and consistently surpasses SD and ZNE-VD (Fig. 4a). For a 4-qubit case, as shots increase from 10^4 to 10^6, CNR-VD, ZNE-VD, and SD improve markedly, while noisy VD and unmitigated estimation saturate (Fig. 4b).
• Higher-order VD: Supplementary results show CNR-VD can recover ideal behavior for third-order VD under moderate noise.
• Compilation considerations: Post-compilation analyses (Supp. Note 5) indicate CNR-VD retains performance superiority among VD-based algorithms despite some variability introduced by compilation.

The study addresses the central challenge that noise within the VD circuit can negate the benefits of VD-based error mitigation. By incorporating a calibration procedure with easily prepared states and constructing a ratio estimator that cancels VD-circuit noise factors, CNR-VD restores the reliability of VD estimates. Simulations confirm that CNR-VD recovers ideal-VD behavior for relevant regimes, expands the range of noise levels where mitigation is effective, and maintains accuracy advantages over alternative VD-based approaches such as ZNE-VD and SD. The theoretical framing via effective noise channels explains why CNR-VD is resilient under broad classes of Markovian noise, particularly depolarizing and Pauli-like noise, and clarifies the role of randomized compiling in aligning hardware noise with the estimator’s assumptions. The extension to Hadamard-test circuits suggests that the calibration-based strategy is broadly applicable to circuits where controlled operations introduce additional noise, enhancing practical utility in near-term quantum algorithms.

The paper introduces Circuit-Noise-Resilient Virtual Distillation (CNR-VD), a calibration-augmented estimator that mitigates errors arising from noise in VD circuits. Through analytical modeling and numerical simulations, CNR-VD demonstrates substantial accuracy gains—often exceeding an order of magnitude over noisy VD—raises noise thresholds for effective mitigation, and outperforms other VD-based strategies (ZNE-VD, SD) across varied system sizes and shot budgets. The approach generalizes to Hadamard-test circuits, indicating broader applicability beyond VD. Future work includes exploring advanced CNR-VD techniques for more complex noise environments, implementing and validating the method on real quantum hardware, and extending calibration strategies to additional classes of observables and circuit architectures.

• Noise model assumptions: The core derivations assume stochastic Pauli gate noise; when hardware noise deviates from this model, randomized compiling is required to approximate Pauli-like behavior.
• Calibration stability: The method relies on temporal stability of gate error rates to reuse calibrated parameters; drift can degrade performance and necessitate re-calibration.
• State-preparation of calibration states: Imperfect preparation of the easy state s introduces estimation error; theoretical guarantees require this error to be negligible compared to VD-circuit noise.
• Higher-weight observables: For multi-qubit observables, accuracy relies on neglecting higher-order terms (j > 1) in the noise expansion, introducing approximation error under stronger noise.
• Noise regime: While CNR-VD extends the effective noise threshold, beyond certain high-noise regimes its advantage diminishes, as indicated by failure-rate boundaries.
• Measurement overhead: Although more efficient than shadow-only approaches, CNR-VD still requires calibration runs in addition to target-state measurements, impacting total shot budgets.