Digital zero noise extrapolation for quantum error mitigation

I. INTRODUCTION

• In the NISQ era, programmers must mitigate errors because scalable logical qubits have yet to be demonstrated, while fault tolerance could reduce logical errors asymptotically.The introduction also situates ZNE among other techniques, including probabilistic error cancellation, and randomized compiling.Source: §I · p. 1
• ZNE benchmarks computations at increased noise levels and extrapolates measured expectation values to estimate the otherwise inaccessible noiseless value E(0).The natural hardware noise corresponds to λ = 1; noise scaling measures E(λ) at m values λ ≥ 1 before extrapolation.Source: §I · p. 1
• The paper improves both ZNE steps, introducing unitary folding and calibration-noise scaling, and studying non-adaptive and adaptive extrapolation methods.The adaptive approach treats extrapolation as statistical inference to improve performance and reduce resource overhead.Source: §I · p. 2

II. NOISE SCALING METHODS

• Time-scaling re-calibrates control pulses to execute the same unitary over longer durations and has suppressed errors in superconducting circuits, but requires low-level access and repeated calibration.The approach is described in and.Source: §II · p. 2
• Digital noise scaling increases circuit depth or gate count using only gate-level access, avoiding low-level pulse control and processor-specific physical knowledge.Unlike time-scaling, it does not require pulse recalibration for each duration and scaling level, which can be resource intensive.Source: §II · p. 2

A. Unitary Folding

• Unitary folding replaces a circuit or gate U with an ideal-equivalent construction; on hardware, its 1 + 2n physical operations increase noise.The operation uses U†U identity structure, so it has no logical effect in an ideal circuit; the noise increase is reported as visible in Figure 6.Source: A. Unitary Folding · p. 2
• Unitary folding can be applied to the full input circuit or to selected local gates, extending earlier noise-increasing CNOT-pair insertion approaches [20],.Source: A. Unitary Folding · p. 2

1) Circuit folding

• Circuit folding applies folds globally and can additionally fold the final s layers, adding 2s layers and enabling depth-scaling resolution of 2/d.For an arbitrary real λ, the procedure rounds d(λ−1)/2 to the closest integer and divides it by d to determine the full and partial folds.Source: 1) Circuit folding · p. 2

2) Gate (or Layer) folding

• Gate or layer folding applies folds in place to a subset of gates or layers, producing the same total count d(2n+1)+2s as circuit folding.Each L_j may represent a one- or two-qubit gate or a layer of gates.Source: 2) Gate (or Layer) folding · pp. 2–3
• Choosing folded indices from the left, right, or at random inserts noise at different circuit positions, so the optimal subset can depend on the circuit and noise model.The paper reports these three subset-selection prescriptions in Table I.Source: 2) Gate (or Layer) folding · p. 3
• The authors distinguish their random gate folding from [20], which focuses mainly on CNOTs and samples with replacement, whereas their method folds any gate or layer without replacement.They state that this samples the input circuit more uniformly and converges smoothly to odd λ=1+2n, where each gate is folded exactly n times.Source: 2) Gate (or Layer) folding · p. 3

3) Advantages and limitations of unitary folding

• Folding uses gate-level abstraction without requiring noise-model details, but may undo coherent errors and cannot scale SPAM; it is suited to incoherent gate- or time-dependent noise.The paper reports numerical and experimental demonstrations of unitary folding for ZNE.Source: 3) Advantages and limitations of… · p. 3
• For odd scale factors λ=1+2n, unitary folding exactly maps gate-dependent depolarizing parameters p_j to p_j^λ, yielding a globally depolarized output.This equivalence holds for both circuit folding and gate folding.Source: 3) Advantages and limitations of… · p. 3
• The exponential depolarizing formulas require depolarizing noise and odd λ; local depolarizing noise generally breaks the commutation simplification except in single-qubit circuits.With gate-independent depolarizing noise, all folding variants are equivalent and induce the stated exponential scaling for every λ.Source: 3) Advantages and limitations of… · p. 3

4) Numerical Results

• The benchmarks extend prior single- and two-qubit-focused work,,, by testing random circuits and a variational algorithm at six or more qubits.Source: 4) Numerical Results · pp. 3–4
• 99.0% randomized-benchmarking decay versus 97.9% without mitigation was obtained using circuit folding and order-2 polynomial extrapolation under 1% depolarizing noise.Some extrapolated expectation values exceeded the physical upper limit of 1; the passage attributes this to fit noise and notes that a Bayesian estimator could enforce physical results.Source: Figure 2 · p. 4, 4) Numerical Results · p. 4
• For randomly generated six-qubit circuits, ZNE improvement is quantified as the unmitigated absolute error divided by the mitigated absolute error.The passage defines the errors as Ru = |E(1) − E(0)| and Rm = |Em − E(0)|, respectively.Source: 4) Numerical Results · p. 4
• Unitary-folding ZNE was evaluated on random MAXCUT instances using the percentage by which the quantum approximation optimization algorithm [22] came closer to the optimum.Source: 4) Numerical Results · p. 4

B. Parameter Noise Scaling

• The method targets stochastic calibration errors in parametric gates, including pulse-area errors caused by control fluctuations or uncertainty in physical parameters.The model also applies to variational quantum circuits, where programmers access the gates' parameter dependence.Source: B. Parameter Noise Scaling · p. 4
• Instead of changing circuit structure, parameter noise scaling injects classical noise into control parameters to increase hardware noise for ZNE.The passage contrasts this with unitary folding.Source: B. Parameter Noise Scaling · p. 4

1) Parameter Noise Scaling Theory

• The theory models gates using classical control parameters, with finite-precision implementation producing stochastic calibration errors assumed to be Gaussian.The parameter-noise variances are assumed known and could be estimated by tomography of repeated gate applications.Source: 1) Parameter Noise Scaling Theory · pp. 4–5
• Adding zero-mean Gaussian control noise with variance (λ − 1)σ_j² scales the total parameter-noise variance by λ ≥ 1.This procedure does not require knowing the Hermitian operators H_j or the error-channel Kraus operators.Source: 1) Parameter Noise Scaling Theory · p. 5
• For a single-parameter rotation-like gate, the noise channel mixes identity with unitary H, with Q = 1/2(1 − e^−2σ²) giving the probability of H.For small σ², the same channel approximates the noise for gates that are not rotation-like, up to O[(σ²)²] corrections.Source: 1) Parameter Noise Scaling Theory · p. 5
• Parameter noise mitigation performs as well as unitary folding on random six-qubit circuits, without adding gates to the circuit.The authors say this may make it less sensitive to other noise sources such as decoherence.Source: Figure 5 · p. 6, 1) Parameter Noise Scaling Theory · p. 6

• With N finite samples, each measured expectation is statistically estimated, with sampling variance equal to the single-shot variance divided by N.Source: §III · p. 6
• Given measurements at m scale factors λ_j ≥ 1, non-adaptive ZNE seeks an estimator of E(0) with reasonably small bias and mean-squared error.Source: §III · p. 6
• Without regularity assumptions, E(0) cannot be inferred from values at λ ≥ 1; physical linear, polynomial, or exponential models make extrapolation possible.Source: §III · p. 6
• Choosing a generic model E_model(λ; Γ) turns non-adaptive ZNE into regression: fit the model to scale-factor data and evaluate it at λ = 0.Source: §III · p. 6
• The section focuses on polynomial and poly-exponential ansätze, including Richardson’s extrapolation as a special case.Source: §III · pp. 6–7

A. Polynomial extrapolation

• The method fits a degree-d polynomial with d + 1 unknown real parameters, corresponding to a Taylor-series approximation justified in the weak-noise regime.The model is well defined only when the number of data points m is at least d + 1.Source: A. Polynomial extrapolation · p. 7
• Unlike Richardson’s extrapolation, polynomial extrapolation can keep d small while using many data points, helping avoid overfitting and large generalization error.Increasing d reduces bias but can increase variance enough to raise total mean squared error.Source: A. Polynomial extrapolation · p. 7

B. Linear extrapolation

• Linear extrapolation is a particular case of polynomial extrapolation and is described as the simplest method.Source: B. Linear extrapolation · p. 7
• Linear extrapolation has an analytic ordinary least-squares solution for estimating the intercept parameter.Source: B. Linear extrapolation · p. 7
• The linear estimator is unbiased for the model’s zero-noise value Elinear(0); the text also gives its variance for equal statistical uncertainty across y_j.Source: B. Linear extrapolation · p. 7

C. Richardson extrapolation

• Richardson extrapolation uses polynomial degree m − 1, maximizing the order for the m available data points.It is a particular case of polynomial extrapolation.Source: C. Richardson extrapolation · p. 7
• Richardson’s polynomial perfectly interpolates the m data points, and its error relative to the true expectation value is O(m) in the infinite-sample limit.The explicit Lagrange estimator assumes all elements of λ are different.Source: C. Richardson extrapolation · p. 7
• Richardson’s O(m) error is asymptotic: at finite sample size, its variance term grows exponentially as m increases.Source: C. Richardson extrapolation · p. 7
• Richardson’s zero-noise estimate has statistical uncertainty that scales exponentially with the number of data points.At finite sample size, the variance term grows exponentially as the number of data points increases.Source: C. Richardson extrapolation · p. 7

D. Poly-Exponential extrapolation

• The degree-d poly-exponential ansatz is polyexp(λ) = a ± e^z(λ), where z(λ) is a degree-d polynomial and a, z0, …, zd are d + 2 parameters.Source: D. Poly-Exponential extrapolation · p. 7
• Assuming a finite asymptotic value, when a is known the transformation log(|y_k−a|+ε) enables polynomial extrapolation in log space, followed by conversion back to estimate E(0).This maps nonlinear regression to a polynomial fit linear in its parameters, which the authors describe as more stable.Source: D. Poly-Exponential extrapolation · pp. 7–8

E. Exponential extrapolation

• The exponential model, discussed in, is a special case of poly-exponential extrapolation; increasing the degree, such as to d = 2, can capture deviations and may improve accuracy.Source: E. Exponential extrapolation · p. 8

F. Benchmark comparisons of ZNE methods

• Across benchmarks, ZNE improves on unmitigated noise except with Richardson extrapolation; on IBMQ London, order-2 polynomial and exponential fits outperform Richardson.Fig. 6 compares extrapolations averaged over 50 two-qubit randomized benchmarking circuits.Source: F. Benchmark comparisons of ZNE methods · p. 8, Figure 6 · p. 8
• Table II reports exact-density-matrix simulations averaging 20 two-qubit randomized benchmarking circuits of mean depth 27, with depolarizing noise p = 1% or amplitude damping γ = 0.01.Non-adaptive methods used λ = {1, 1.5, 2, 2.5}; adaptive extrapolation ran for up to four scale factors.Source: Table II · p. 8
• Richardson extrapolation is unstable when more than the first three data points are used, producing nonphysical results and illustrating the limits of the vanilla method.Source: F. Benchmark comparisons of ZNE methods · p. 8

IV. ADAPTIVE ZERO NOISE EXTRAPOLATION

• Adaptive extrapolation chooses scale factors and sample counts to reduce computational overhead.Source: §IV · p. 9
• Unlike non-adaptive extrapolation, the adaptive procedure permits non-monotonically increasing scale factors and can repeat a factor to improve its expectation estimate.Source: §IV · p. 9
• For the exponential ansatz Eexp(λ) = a+be−cλ, assuming known a requires at least two scale factors to fit b and c, with generalization to adaptively chosen arbitrary numbers of factors.Source: §IV · p. 9

A. Exponential extrapolation with two scale factors

• With two scale factors, the expectation estimates have variances σ_1^2 = σ_0^2/N_1 and σ_2^2 = σ_0^2/N_2, assuming single-shot variance σ_0^2 is independent of λ.Source: A. Exponential extrapolation with two… · p. 9
• Interpolating the two measured points estimates b and c, and the zero-noise estimator is Ehat_exp(0) = a + bhat, whose error comes from bhat's statistical noise.Source: A. Exponential extrapolation with two… · p. 9
• Measurement errors in y1 and y2 propagate to the estimator of b, and the resulting error depends on both the scale factors and their sample counts.Source: A. Exponential extrapolation with two… · p. 9

1) Error minimization

• With fixed λ1 and λ2, the method minimizes the error by choosing how to divide the total evaluation budget Nmax = N1 + N2.The section also gives the corresponding minimized error, but its formula is not present in the supplied passage.Source: 1) Error minimization · p. 9
• With λ1 usually fixed to 1, optimizing λ2 yields a numerical condition with α ≃1.27846 for selecting the scale factor that minimizes extrapolation error.For a fixed λ1, the condition determines the optimal λ2; using the stated optimality equations requires prior knowledge about c.Source: 1) Error minimization · p. 9
• The adaptive algorithm iterates within Nmax, samples expectations at λ1 and λ2, refits c, and returns Eexp(0; a, b, c).It uses a known or estimated a, a fixed batch size Nbatch, and a minimum scale factor λ1 that is typically 1.Source: 1) Error minimization · p. 10

V. CONCLUSION

• Unitary folding makes zero-noise extrapolation digital with instruction-set-level access, and the paper demonstrates improved performance through non-adaptive and adaptive extrapolation methods.The paper frames ZNE generally as an inference problem with many avenues for further optimization.Source: §V · p. 10
• MAXCUT benchmarks solved with QAOA let the authors smoothly assess ZNE on larger variational quantum circuits than previously considered.The paper contrasts this benchmark scope with prior ZNE benchmarks on randomized benchmarking circuits or VQE.Source: §V · p. 10
• The paper specializes ZNE to different noise models using calibration noise and notes that combined calibration-noise and amplitude-damping models may motivate multidimensional noise extrapolation.Source: §V · p. 10

B. An adaptive exponential extrapolation algorithm

• Adaptive exponential extrapolation generally outperforms its non-adaptive counterpart at almost all sample levels, though its advantage is modest in the illustrated example.The algorithm uses the ansatz Eexp(λ) = a + be−cλ, where a is a known constant; the figure attributes the modest advantage to already-good non-adaptive scale factors.Source: Figure 7 · p. 10, B. An adaptive exponential… · p. 10

V. CONCLUSION

• Future inference extensions could incorporate priors or constraints from observables, noise, or circuit structure, and historical data from similar executions.The authors describe this work as a first step toward treating ZNE as an inference problem.Source: §V · p. 10
• Error mitigation is likely to remain a critical toolkit for NISQ-era quantum programmers, making continued improvement and benchmarking important.Source: §V · p. 10