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Digital zero noise extrapolation for quantum error mitigation

Tudor Giurgica-Tiron, Yousef Hindy, Ryan LaRose, Andrea Mari, William J. Zeng

arXiv:2005.10921v2quant-ph

TL;DR

Scalable logical qubits remain unavailable, motivating practical error mitigation for noisy quantum computations. This paper develops digital noise scaling and adaptive extrapolation methods for zero-noise extrapolation, finding that polynomial and exponential fits outperform Richardson extrapolation while adaptive extrapolation generally outperforms its non-adaptive counterpart.

  • Problem

    Because scalable logical qubits have not yet been demonstrated, quantum programmers need practical error-mitigation techniques for noisy intermediate-scale quantum computations.

  • Method

    The paper develops gate-level unitary folding, parameter noise scaling, and non-adaptive and adaptive extrapolation methods that treat zero-noise extrapolation as statistical inference.

  • Results

    Order-2 polynomial and exponential fits outperform Richardson extrapolation, while adaptive exponential extrapolation outperforms the non-adaptive approach at almost all sample levels.

  • Takeaways & Limitations

    Zero-noise extrapolation can be benchmarked on larger variational QAOA circuits than previously considered, extending evaluation beyond randomized benchmarking and VQE.

  • Takeaways & Limitations

    Unitary folding’s exact exponential noise scaling is established for odd positive-integer scale factors and depolarizing gate noise.

Abstract

from arXiv · show

Zero-noise extrapolation (ZNE) is an increasingly popular technique for mitigating errors in noisy quantum computations without using additional quantum resources. We review the fundamentals of ZNE and propose several improvements to noise scaling and extrapolation, the two key components in the technique. We introduce unitary folding and parameterized noise scaling. These are digital noise scaling frameworks, i.e. one can apply them using only gate-level access common to most quantum instruction sets. We also study different extrapolation methods, including a new adaptive protocol that uses a statistical inference framework. Benchmarks of our techniques show error reductions of 18X to 24X over non-mitigated circuits and demonstrate ZNE effectiveness at larger qubit numbers than have been tested previously. In addition to presenting new results, this work is a self-contained introduction to the practical use of ZNE by quantum programmers.

I. INTRODUCTION · II. NOISE SCALING METHODS · A. Unitary Folding

The paper motivates zero-noise extrapolation as a practical alternative while scalable fault tolerance remains unavailable, and formalizes it as measuring observables at increased noise levels before estimating the noiseless value. It improves both noise scaling and extrapolation, including gate-level unitary folding that increases physical operations without changing the ideal computation.

  • I. INTRODUCTION: Scalable logical qubits have not yet been demonstrated, motivating practical error-mitigation techniques for today’s noisy quantum hardware.The paper contrasts these techniques with the long-term potential of fault tolerance and quantum error correction.
  • I. INTRODUCTION: Zero-noise extrapolation alters a quantum program to run at different effective processor-noise levels, then extrapolates the result to the noiseless value.The noise level is parameterized by a dimensionless scale factor λ.
  • I. INTRODUCTION: ZNE measures E(λ) at m values with λ ≥1 and infers E(0), where E(1) is the hardware-noise observable and E(0) is the unmeasurable noiseless target.λ = 0 denotes removed noise, while λ = 1 matches the physical hardware’s true noise level.
  • I. INTRODUCTION: The work introduces improvements to both ZNE components: digital noise scaling methods and extrapolation methods that treat estimation as an inference problem.The introduction identifies unitary folding, calibration-noise scaling, non-adaptive extrapolation, and adaptive extrapolation as contributions.
  • II. NOISE SCALING METHODS: Time-scaling increases effective noise by executing the same unitary evolution over longer durations, but it requires low-level physical control and repeated pulse recalibration.These requirements can be unavailable on some hardware and make calibration resource intensive.
  • II. NOISE SCALING METHODS: Gate-level noise scaling instead increases the total number of gates or circuit depth, requiring only gate-level system access rather than longer gate durations.The approach reverses the usual compiler objective of reducing circuit depth.
  • A. Unitary Folding: Unitary folding replaces a unitary circuit or gate U with an operation containing U†U factors, preserving the ideal computation because U†U equals the identity.The framework includes both circuit folding and gate folding, with U representing the full circuit or selected local gates.
  • A. Unitary Folding: 1 + 2n is the physical-operation scaling factor for unitary folding, which increases noise on real quantum hardware while leaving the ideal circuit logically unchanged.The paper reports that this effect is visible in the quantum-computing experiment shown in Figure 6.

1) Circuit folding · 2) Gate (or Layer) folding · 3) Advantages and limitations of unitary folding

Unitary folding provides digital, hardware-agnostic noise scaling through global circuit, gate, or layer folding, with partial folds enabling finer scale resolution. Its physical validity depends on the noise model: it exactly yields exponential depolarizing-noise scaling under specified conditions, but does not reliably amplify coherent or SPAM errors.

  • 1) Circuit folding: Global circuit folding applies the replacement rule to the entire circuit and scales effective depth by odd integers.A circuit of depth d can also receive a final partial fold over its last s layers.
  • 1) Circuit folding: For any real λ, folding selects the nearest integer k to d(λ−1)/2, divides k by d, and uses the quotient and remainder for integer and partial folds.Physically, folding drives the qubit Hamiltonian forwards and backwards while preserving the ideal unitary dynamics.
  • 2) Gate (or Layer) folding: Gate or layer folding applies the replacement rule in place to individual gates or layers, scaling their initial number d by the odd integer 1+2n.Each Lj may represent a one- or two-qubit gate or a layer of several gates.
  • 2) Gate (or Layer) folding: Folding gates or layers uniformly samples the input circuit and converges smoothly to λ = 1+2n, where every input gate is folded exactly n times.This rationale motivates distributing folds across the circuit rather than appending them only globally.
  • 3) Advantages and limitations of unitary folding: Unitary folding is digital and does not require knowledge of the underlying noise model, but it may fail to amplify systematic and coherent errors.Applying a gate inverse can undo such errors instead of increasing them.
  • 3) Advantages and limitations of unitary folding: Unitary folding does not scale SPAM noise, but is expected to scale incoherent noise associated with gate applications or computation duration.For odd positive integer λ = 1 + 2n under global depolarizing noise, both circuit and gate folding exactly implement pj → pλ.
  • 3) Advantages and limitations of unitary folding: The depolarizing-noise equivalence holds only for odd scaling factors λ, while gate-independent depolarizing noise makes all folding methods equivalent for all λ.For gate-dependent global depolarizing noise, p = Πjpj and folding induces exponential scaling of the global operation.

4) Numerical Results

Numerical simulations show that unitary-folding ZNE works across randomized benchmarking, random six-qubit circuits, and a variational MAXCUT algorithm. The benchmarks compare mitigation methods and extend evaluation beyond prior single- and two-qubit studies.

  • Benchmark scope: Density-matrix simulations benchmarked unitary folding on random circuits and a variational algorithm at six and more qubits.These results extend prior work focused on single- and two-qubit cases.
  • Randomized benchmarking: 99.0% randomized benchmarking decay was achieved with mitigation, versus 97.9% without mitigation, under 1% depolarizing noise.The simulation used circuit folding and order-2 polynomial extrapolation at λ = 1, 1.5, 2.0.
  • Random circuits: Six-qubit random-circuit experiments quantified ZNE improvement as the unmitigated error divided by the mitigated error.The study defines Ru/Rm using absolute errors relative to the zero-noise expectation value.
  • Method comparison: Table II compares different combinations of folding and extrapolation techniques on randomized benchmarking circuits.The table is referenced as providing the technique-combination comparison for the numerical results.
  • Variational algorithm: Unitary-folding ZNE was evaluated on random MAXCUT instances using the quantum approximation optimization algorithm’s percentage closer to optimal.The variational-algorithm study used exact density-matrix simulation.

B. Parameter Noise Scaling · 1) Parameter Noise Scaling Theory

Parameter noise scaling applies ZNE to stochastic calibration errors by injecting classical noise into gate-control parameters rather than modifying circuit structure. Its theory provides a variance-based scaling rule and shows that mitigation can match unitary folding without adding gates.

  • B. Parameter Noise Scaling: Parameter noise scaling targets stochastic calibration errors in parametric quantum gates, including pulse-area miscalibration from control fluctuations or uncertain hardware parameters.The model also applies to variational quantum circuits.
  • B. Parameter Noise Scaling: The method injects classical noise into control parameters to increase native hardware noise without changing the quantum circuit structure.This directly enables ZNE for the modeled noise source.
  • 1) Parameter Noise Scaling Theory: A parameterized gate is modeled using classical controls θ = (θ1, θ2, . . . θl) and Hermitian operators H1, H2, . . . Hl.These controls may represent processor tuning parameters or user-programmable variational-gate parameters.
  • 1) Parameter Noise Scaling Theory: Additional zero-mean Gaussian control noise with variance (λ −1)σ2_j scales the overall parameter-noise variance by λ ≥1.The required noise variances can be estimated from repeated-gate tomography.
  • 1) Parameter Noise Scaling Theory: The scaling procedure does not require knowledge of the Hermitian operators H_j or the corresponding error channel’s Kraus operators.This makes the method applicable without explicitly identifying the detailed noise structure.
  • 1) Parameter Noise Scaling Theory: For single-parameter rotation-like gates, the calibration-noise channel has two Kraus operators and mixes identity with the unitary H using Q = 1 2(1 −e−2σ2).The identity occurs with probability 1 −Q and H with probability Q.
  • 1) Parameter Noise Scaling Theory: Parameter noise mitigation performs as well as unitary folding while adding no new gates to the circuit.The authors therefore suggest it may be less sensitive to other noise sources such as decoherence.

III. NON-ADAPTIVE EXTRAPOLATION METHODS: ZERO NOISE EXTRAPOLATION AS STATISTICAL INFERENCE

This section formulates zero-noise extrapolation as a statistical estimation problem: infer E(0) from noisy estimates measured at scale factors λj ≥1. Under a model for E(λ), non-adaptive ZNE becomes regression using polynomial or poly-exponential ansätze, including Richardson extrapolation as a special case.

  • Statistical formulation: With finite samples, each expectation estimate is statistical and can be modeled as a random prediction with variance σ2 = σ2_0/N.The single-shot variance is σ2_0, and N denotes the number of measurement samples.
  • Statistical formulation: ZNE seeks an estimator ˆE(0) for E(0) whose bias and variance are both reasonably small, commonly evaluated by mean squared error.The target is the zero-noise expectation at λ = 0, inferred from measurements at nonzero noise scales.
  • Model assumptions: Because E(λ) is otherwise unconstrained, extrapolation is impossible; physical models such as linear, polynomial, or exponential dependence make inference possible.For depolarizing noise, an exponential ansatz is suggested.
  • Non-adaptive models: The main non-adaptive models are polynomial and poly-exponential ansätze, with Richardson’s extrapolation appearing as a particular case.These models generate a broad family of specific extrapolation algorithms.

A. Polynomial extrapolation

Polynomial extrapolation models the noise dependence with a degree-d polynomial, justified as a Taylor approximation in the weak-noise regime. It requires at least d + 1 data points and can use many points while keeping d small to reduce over-fitting.

  • A. Polynomial extrapolation: Polynomial extrapolation uses a degree-d polynomial model with d + 1 unknown real parameters.The model is physically justified as a Taylor series approximation in the weak noise regime.
  • A. Polynomial extrapolation: The problem is well defined only when the number of data points m is at least d + 1.This matches the number of free parameters in the model.
  • A. Polynomial extrapolation: Keeping extrapolation order d small while using many data points helps avoid over-fitting statistical fluctuations.Increasing d too much forces the model to follow random fluctuations in the data.

B. Linear extrapolation · C. Richardson extrapolation

Linear extrapolation provides a simple polynomial-extrapolation method with an analytic least-squares intercept estimator. Richardson extrapolation maximizes polynomial order for the available data points, achieving asymptotic error O(m) but incurring exponentially growing statistical uncertainty as m increases.

  • B. Linear extrapolation: Linear extrapolation is the simplest method and a particular case of polynomial extrapolation.
  • B. Linear extrapolation: Its zero-noise estimator has a simple analytic form corresponding to the ordinary least-squares estimator of the intercept.
  • B. Linear extrapolation: The linear estimator is unbiased relative to Elinear(0), while equal statistical uncertainties determine its variance.The passage specifies equal statistical uncertainty σ2 for each yj.
  • C. Richardson extrapolation: Richardson extrapolation sets d = m − 1, maximizing the polynomial order for the number of data points.
  • C. Richardson extrapolation: Its fitted polynomial is poly (λ) = c0 + c1λ + . . . cm−1λm−1.
  • C. Richardson extrapolation: O(m) is the asymptotic error relative to the true expectation value because the polynomial perfectly interpolates the m data points as N →∞.This interpolation result assumes distinct elements of λ.
  • C. Richardson extrapolation: Exponential statistical uncertainty limits Richardson extrapolation as the number of data points increases.The O(m) error applies only asymptotically; for finite N, the variance grows exponentially with m, including under equally spaced noise scales λk = k λ1.

D. Poly-Exponential extrapolation · E. Exponential extrapolation · F. Benchmark comparisons of ZNE methods

The paper extends exponential extrapolation to a poly-exponential model that can capture deviations from ideal exponential behavior, then benchmarks ZNE methods across simulated and IBMQ London circuits. Polynomial and exponential fits outperform Richardson in the reported London comparison, while Richardson can become unstable and nonphysical when using more data points.

  • D. Poly-Exponential extrapolation: The degree-d poly-exponential ansatz models the extrapolation as a ± e^z(λ), with z(λ) a degree-d polynomial in the noise scale.The model uses d + 2 parameters: a, z_0, z_1, …, z_d.
  • D. Poly-Exponential extrapolation: When the asymptotic value a is known, measurements are transformed logarithmically, polynomially extrapolated to λ = 0, and converted back to estimate E(0).A small regularizing constant ϵ > 0 is used, and weights can account for nonlinear statistical-error propagation.
  • E. Exponential extrapolation: Exponential extrapolation is a special case of the poly-exponential method, using an equivalent parameterization of the same ansatz.The coefficients a, b, c correspond to an alternative parameterization of a, z_0, z_1.
  • E. Exponential extrapolation: Increasing the poly-exponential order, such as to d = 2, can capture small deviations from ideal exponential behavior and potentially improve zero-noise estimates.This generalizes the exponential model within the paper’s extrapolation framework.
  • F. Benchmark comparisons of ZNE methods: ZNE improves on unmitigated noise in all benchmark cases except Richardson extrapolation, but performance varies substantially and no method strictly dominates.The comparisons are reported in Table II across different scaling and extrapolation methods.
  • F. Benchmark comparisons of ZNE methods: 50 circuits were evaluated on IBMQ London, where order-2 polynomial and exponential fits outperform Richardson extrapolation for randomized benchmarking circuits with ideal expectation value 1.Random gate folding was used to scale noise across the two-qubit circuits.
  • F. Benchmark comparisons of ZNE methods: Richardson extrapolation becomes unstable with more than the first three data points, producing nonphysical results and failing to provide stable mitigation on the measured data.The instability is attributed to the behavior described earlier in Section III-C.

IV. ADAPTIVE ZERO NOISE EXTRAPOLATION · A. Exponential extrapolation with two scale factors

The adaptive protocol reduces computational overhead by choosing scale factors and sample counts adaptively. With a known asymptotic value, exponential extrapolation uses two measured scale factors to estimate the zero-noise limit, whose error depends on both scale factors and sample allocations.

  • IV. ADAPTIVE ZERO NOISE EXTRAPOLATION: Adaptive extrapolation chooses scale factors and sample counts to reduce computational overhead.The procedure is described in Algorithm 2.
  • IV. ADAPTIVE ZERO NOISE EXTRAPOLATION: Adaptive scale factors need not increase monotonically, and λnext may repeat a previous λj to improve E(λj) estimation with additional samples Nnext.Any λnext above or equal to 1 is allowed.
  • IV. ADAPTIVE ZERO NOISE EXTRAPOLATION: The adaptive example uses the exponential ansatz Eexp(λ) = a+be−cλ.This ansatz was introduced previously as Eq. (35).
  • IV. ADAPTIVE ZERO NOISE EXTRAPOLATION: When a is known, at least two scale factors are required to fit b and c, with the method generalizable to an arbitrary number chosen adaptively.The two-scale-factor case is developed before the generalized procedure.
  • A. Exponential extrapolation with two scale factors: The two-scale-factor setup uses λ1 and λ2, typically with λ1 = 1, to estimate E(λ1) and E(λ2) statistically.The corresponding measurement variances are σ1^2 = σ0^2/N1 and σ2^2 = σ0^2/N2 when the single-shot variance is λ-independent.
  • A. Exponential extrapolation with two scale factors: Using measured results y1 and y2, the unique exponential interpolant estimates b and c when a is known.The measurement results follow the statistical distribution specified by Eq. (18).
  • A. Exponential extrapolation with two scale factors: The zero-noise estimator is ˆEexp(0) = a + ˆb, so its error arises only from statistical noise in ˆb and depends on λ1, λ2, N1, and N2.Measurement errors in y1 and y2 propagate to ˆb, with the leading-order analysis based on σ1^2 and σ2^2.

1) Error minimization

The section minimizes zero-noise extrapolation error by optimizing the allocation of circuit evaluations and the second noise-scale factor. Because these optimizations require prior knowledge of c, it motivates an adaptive exponential extrapolation algorithm.

  • Error minimization: Given a total evaluation budget Nmax = N1 + N2 and fixed λ1, λ2, the error is minimized by optimizing N1 and N2.The resulting minimized error is presented in the corresponding equation.
  • Error minimization: With λ1 usually fixed to 1, optimizing λ2 minimizes the extrapolation error through a numerically solved condition.The solution uses the numerical constant α ≃1.27846 to determine the optimal λ2 for fixed λ1.
  • Error minimization: Because the analytical optimizations require prior knowledge about c, the section introduces an adaptive exponential extrapolation algorithm.The algorithm uses a maximum total sample count Nmax, a fixed batch size Nbatch, and a minimum scale factor λ1, typically 1.
  • Error minimization: The adaptive procedure begins with c ←−1, sets α ←−1.27846, iteratively evaluates expectations at λ1 and λ2, and refits c until Nused reaches Nmax.It returns the extrapolated expectation value Eexp(0; a, b, c).

B. An adaptive exponential extrapolation algorithm · V. CONCLUSION

The paper presents adaptive exponential extrapolation within a digital zero-noise extrapolation framework and concludes that ZNE is an inference problem with broad opportunities for benchmarking and extension. It also applies ZNE to larger QAOA/MAXCUT circuits and considers specialization to different noise models.

  • B. An adaptive exponential extrapolation algorithm: Adaptive exponential extrapolation uses the ansatz Eexp(λ) = a + be−cλ, with a known constant, and generally outperforms the non-adaptive approach across sample levels.The comparison is reported in Figure 7 and Algorithm 3.
  • V. CONCLUSION: The work makes ZNE digital through unitary folding, enabling error mitigation with instruction-set-level access, and develops improved non-adaptive and adaptive extrapolation methods.The conclusion frames ZNE as an inference problem with many avenues for further optimization.
  • V. CONCLUSION: ZNE is benchmarked on MAXCUT problems solved with QAOA, extending evaluation to larger variational quantum circuits than previously considered.Earlier benchmarks used randomized benchmarking circuits or VQE.
  • V. CONCLUSION: The study specializes ZNE to different noise models, using calibration noise as an example, and identifies multidimensional extrapolation as relevant for more sophisticated combined models.The example combined calibration noise with amplitude damping.
  • B. An adaptive exponential extrapolation algorithm: With a fixed sample budget, the adaptive method generally produces more accurate extrapolations using fewer samples, although its advantage is not particularly large in this example.The non-adaptive scale factors were already close to their optimal values.
  • V. CONCLUSION: Viewing ZNE as an inference problem is presented as a first step, with possible extensions using priors, constraints, and historical data from similar executions.These additions could incorporate observable, noise, or circuit structure into inference.
  • V. CONCLUSION: Error mitigation is expected to remain a critical toolkit for NISQ-era quantum programmers, making continued improvement and benchmarking important.The conclusion emphasizes the continuing role of these techniques.
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