Global Randomized Error Cancellation (GREC)

• GREC is an error mitigation technique for parametric quantum circuits that computes ideal outputs using a global linear combination of randomized noisy measurements.
• It employs a constrained least-squares regression on data from a classically accessible parameter subset to determine correction coefficients applied across the entire range.
• GREC extends to adiabatic evolutions and shows improved performance over local methods like ZNE by effectively reducing errors in reconstructed quantum observables.

Searching arXiv for GREC and related work. arXiv search query: "Global Randomized Error Cancellation quantum error mitigation" Global Randomized Error Cancellation (GREC) is an error-mitigation scheme for parametric quantum circuits in which, for some parameter region, the circuit’s ideal output can be computed classically. Its defining construction is a global linear combination of noisy expectation values obtained from additionally randomized versions of the circuit, with real coefficients learned by regression against exact classical data in a classically accessible subset of parameter space; the same coefficients are then used throughout the target range, including regions where classical simulation is not feasible (Sazonov et al., 2021). In later work, the same principle was extended from static parametric circuits to adiabatic evolution, where the coefficients are learned separately for each time point in order to account for error accumulation along the evolution (Kaikov et al., 9 Jul 2025).

1. Parametric setting and the meaning of “global”

GREC is formulated for a circuit family $U(\lambda)$ parametrized by a real scalar $\lambda \in \Lambda \subset \mathbb{R}$. The circuit acts on the initial state

$\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$

and the ideal expectation value of an observable $A$ is

$$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$

The parameter $\lambda$ is typically a physical parameter such as a coupling or field strength; in the Ising example, it is the transverse field (Sazonov et al., 2021).

The structural assumption is the existence of an “easy” subset $\Lambda_{\rm e}\subset \Lambda$ such that, for every $\lambda\in\Lambda_{\rm e}$, the ideal expectation value $\langle A(\lambda)\rangle^{\rm ideal}$ is efficiently computable on a classical computer, while for all $\lambda\in\Lambda$, the noisy quantum expectation value is experimentally accessible. The classically accessible region is used to train the mitigation coefficients, and the same coefficients are then applied to all $\lambda \in \Lambda \subset \mathbb{R}$0. Examples explicitly mentioned for this paradigm include lattice QCD at imaginary chemical potential versus real chemical potential, and the Hubbard model at imaginary chemical potential versus real chemical potential away from half-filling (Sazonov et al., 2021).

The “global” adjective refers to the fact that the coefficients are global in the parameter $\lambda \in \Lambda \subset \mathbb{R}$1: they are determined once using data from an easy parameter region and then used for all $\lambda \in \Lambda \subset \mathbb{R}$2 in the target range. This is not a pointwise correction applied independently at each parameter value. In the original formulation, GREC therefore learns what the paper describes as a global functional correction from noisy randomized outputs to the ideal output (Sazonov et al., 2021).

2. Estimator, regression problem, and baseline form

For a set of $\lambda \in \Lambda \subset \mathbb{R}$3 randomized circuits indexed by $\lambda \in \Lambda \subset \mathbb{R}$4, GREC constructs the mitigated estimator

$\lambda \in \Lambda \subset \mathbb{R}$5

where $\lambda \in \Lambda \subset \mathbb{R}$6 is the noisy expectation value measured from randomized circuit $\lambda \in \Lambda \subset \mathbb{R}$7, and $\lambda \in \Lambda \subset \mathbb{R}$8 together with $\lambda \in \Lambda \subset \mathbb{R}$9 are real coefficients chosen by regression fit to exact classical data in $\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$0 (Sazonov et al., 2021).

The coefficients are obtained by solving the constrained least-squares problem

$\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$1

With the optimal coefficients, the mitigated curve is

$\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$2

The constraint $\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$3 is imposed primarily to make the stability analysis more transparent (Sazonov et al., 2021).

The simplest baseline variant uses no extra random circuits and reduces the correction to a two-parameter affine fit,

$\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$4

This baseline is trained by linear regression on the overlap region where classical and noisy quantum data are both available. In the original numerical study, it yields reasonable error reduction but leaves residual biases that differ between subranges, underscoring that the noise error is not simply a global affine function of the noisy output (Sazonov et al., 2021).

3. Randomized circuit family and operational workflow

The randomized part of GREC is implemented at the circuit level. Starting from the original circuit $\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$5, one constructs $\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$6 randomized versions by inserting $\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$7 auxiliary parametric gates $\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$8 into the circuit and drawing their parameters independently according to

$\rho_0 = |0\rangle\langle 0|^{\otimes N_{\rm q}},$9

Setting $A$0 recovers the original non-randomized circuit. The randomization is therefore in the gate content or gate parameters of the circuit; the noise model itself is not artificially modified, in contrast to zero-noise extrapolation with explicit noise scaling (Sazonov et al., 2021).

In the Ising implementation, each distinct one-qubit gate $A$1 is “equipped” by appending a general one-qubit unitary $A$2,

$A$3

which results in about $A$4 auxiliary gates for the full four-qubit circuit. For each randomized circuit $A$5, the experiment measures the curve

$A$6

and these curves are then linearly combined by the fitted coefficients (Sazonov et al., 2021).

The workflow can be summarized as follows. The inputs are the parametric circuit $A$7, the easy subset $A$8 and classical data $A$9, the number of randomized circuits $$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$0, and the number of auxiliary gates $$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$1. One then generates the randomized circuits, measures the noisy curves for each randomized circuit, solves the constrained least-squares problem, and finally produces the mitigated curve by the global linear combination (Sazonov et al., 2021).

The operational role of the randomized circuits is not noise symmetrization in the sense of randomized compiling or Pauli twirling. Instead, the randomization generates a diverse set of error profiles $$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$2, and the regression attempts to combine these profiles so that the total residual error is small. This makes GREC structurally close to generic linear-error-mitigation formulas, while shifting the source of the coefficients from calibrated noise decompositions to classical training data (Sazonov et al., 2021).

4. Analytical formulation, extrapolation picture, and resource profile

For each randomized circuit $$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$3, the noisy output is written as

$$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$4

Using $$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$5, the mitigated estimator becomes

$$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$6

If the learned coefficients make the residual term

$$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$7

small for all $$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$8, then the mitigated estimator approximates the ideal expectation value (Sazonov et al., 2021).

The stability analysis treats GREC as an analytic continuation or extrapolation problem in $$\langle A(\lambda)\rangle^{\rm ideal} = \operatorname{tr}\!\big[A\,\hat\rho(\lambda)\big],\qquad \hat\rho(\lambda)=\mathcal U(\lambda)\rho_0\mathcal U^\dagger(\lambda).$$9. The error functions $\lambda$0 are assumed analytic in $\lambda$1 inside a Bernstein ellipse $\lambda$2 with foci at $\lambda$3 and parameter $\lambda$4, and uniformly bounded there by $\lambda$5. Under these assumptions, each error function admits a Chebyshev expansion

$\lambda$6

with truncation remainder bounded by

$\lambda$7

Summing over randomized circuits with $\lambda$8 yields the total truncation error

$\lambda$9

The corresponding extrapolation bound improves as the total error decreases, but worsens near the edge of the extrapolation domain, where the analytic continuation becomes unstable (Sazonov et al., 2021).

This analytic-function perspective is central to the way GREC differs from calibration-heavy cancellation methods. The method does not require knowledge of a Kraus decomposition or a gate-by-gate noise model, and it is agnostic to whether the noise is gate- or time-dependent. Its practical performance, however, relies on the error curves being sufficiently smooth in $\Lambda_{\rm e}\subset \Lambda$0 (Sazonov et al., 2021).

The resource usage is correspondingly simple. If $\Lambda_{\rm e}\subset \Lambda$1 is the number of randomized circuits and $\Lambda_{\rm e}\subset \Lambda$2 is the number of parameter points, then the total shot cost is $\Lambda_{\rm e}\subset \Lambda$3. In the original experiments, $\Lambda_{\rm e}\subset \Lambda$4, $\Lambda_{\rm e}\subset \Lambda$5, and $\Lambda_{\rm e}\subset \Lambda$6. Once the $\Lambda_{\rm e}\subset \Lambda$7 circuits have been measured, the mitigation step is purely classical. Unlike probabilistic error cancellation in its standard quasi-probability form, GREC does not introduce quasi-probability sampling over a huge combinatorial set of trajectories, and there is no intrinsic multiplicative variance factor of the kind associated with the $\Lambda_{\rm e}\subset \Lambda$8 norm of quasi-probabilities (Sazonov et al., 2021).

5. Demonstration on the four-spin antiferromagnetic Ising model

The original demonstration considers the antiferromagnetic Ising chain with a modified periodic-boundary term,

$\Lambda_{\rm e}\subset \Lambda$9

with $\lambda\in\Lambda_{\rm e}$0 spins and transverse field $\lambda\in\Lambda_{\rm e}$1. The Hamiltonian is diagonalized by a unitary $\lambda\in\Lambda_{\rm e}$2 composed of the Jordan–Wigner transformation, Fourier transform, and Bogoliubov transformation. For each $\lambda\in\Lambda_{\rm e}$3, the circuit prepares the ground state and measures the average magnetization

$\lambda\in\Lambda_{\rm e}$4

The exact magnetization used as reference is

$\lambda\in\Lambda_{\rm e}$5

and the study focuses on $\lambda\in\Lambda_{\rm e}$6 (Sazonov et al., 2021).

The parameter range is $\lambda\in\Lambda_{\rm e}$7. Two easy ranges are defined: $\lambda\in\Lambda_{\rm e}$8 with training and validation splits

$\lambda\in\Lambda_{\rm e}$9

$\langle A(\lambda)\rangle^{\rm ideal}$0

In classical simulations, 10 points are used in each training and validation subrange. On the real device, due to time constraints, 5 points per subrange are used. The noise sources are classical simulations of the IBM backend ibmq_manila with its noise model and actual runs on the superconducting device ibmq_manila (Sazonov et al., 2021).

The randomization settings in this example are $\langle A(\lambda)\rangle^{\rm ideal}$1, $\langle A(\lambda)\rangle^{\rm ideal}$2, and $\langle A(\lambda)\rangle^{\rm ideal}$3. Each distinct one-qubit gate in the original circuit is followed by a general one-qubit $\langle A(\lambda)\rangle^{\rm ideal}$4 with three parameters, and the parameters are sampled uniformly in $\langle A(\lambda)\rangle^{\rm ideal}$5. The full GREC mitigation generates nine randomized magnetization curves, fits $\langle A(\lambda)\rangle^{\rm ideal}$6 on either $\langle A(\lambda)\rangle^{\rm ideal}$7 or $\langle A(\lambda)\rangle^{\rm ideal}$8, and then evaluates the mitigated curve on both training and validation subranges (Sazonov et al., 2021).

The reported behavior is consistent across simulation and hardware. In simulation, the mitigated curves are much closer to the exact analytical line than the raw noisy data, and the improvement holds both within the training range and in the validation range, indicating successful extrapolation in $\langle A(\lambda)\rangle^{\rm ideal}$9. On ibmq_manila, GREC significantly reduces the discrepancy between measured magnetization and the exact analytic curve relative to unmitigated data, although the performance is noisier than in simulation. In the supplementary comparison against linear zero-noise extrapolation implemented with the Mitiq library, using nine noise-scaling factors in $\lambda\in\Lambda$0 to match GREC’s circuit count, GREC slightly outperforms linear ZNE in reconstructing the magnetization curve (Sazonov et al., 2021).

6. Extensions, related methods, and limitations

GREC is conceptually related to both probabilistic error cancellation (PEC) and zero-noise extrapolation (ZNE), but its mechanism is different. PEC expresses the ideal quantity as a linear combination of noisy expectation values derived from a detailed decomposition of each ideal gate into implementable noisy basis operations. GREC also uses a linear combination of noisy outputs and allows real, possibly negative coefficients, but it does not rely on calibrated gate-level decompositions; instead, it learns the coefficients from classical training data over the circuit parameter $\lambda\in\Lambda$1. ZNE is likewise agnostic to the detailed microscopic noise model, but it scales noise in a controlled way and usually extrapolates locally, mitigating each $\lambda\in\Lambda$2 point using only data taken at that same $\lambda\in\Lambda$3. GREC is global in $\lambda\in\Lambda$4, uses parametric structure and classical data, and does not require an explicit noise-scaling parameter (Sazonov et al., 2021).

Method	Core resource	Distinctive feature
PEC	Detailed decomposition of each ideal gate into noisy basis operations	Coefficients come from the decomposition
ZNE	Circuits with scaled effective noise strength	Extrapolates each parameter point locally
GREC	Classically accessible parameter regime plus randomized circuits	Same coefficients are used globally in $\lambda\in\Lambda$5

A common misconception is to treat any “global” cancellation effect as GREC. A distinct line of work on analog quantum simulation studies stochastic error cancellation under the assumption that the actual Hamiltonian differs from the target Hamiltonian by many independent, unbiased local perturbations. In that setting, the error in general observables scales as

$\lambda\in\Lambda$6

and the fidelity obeys

$\lambda\in\Lambda$7

with high probability. This is conceptually related to global cancellation of many error contributions, but it is not the regression-based parametric-circuit procedure of GREC (Cai et al., 2023).

The 2025 extension to adiabatic evolution adapts GREC to time evolution of eigenstates in the lattice Schwinger model. There, the original constant-coefficient assumption is relaxed because the error of the time evolution on a noisy device accumulates with each additional time step. The coefficients are therefore learned separately for each time point and each energy level. The method partitions the parameter domain into a learning region $\lambda\in\Lambda$8 and a prediction region $\lambda\in\Lambda$9, learns from adiabatic “training lines” in $\lambda \in \Lambda \subset \mathbb{R}$00, and then applies the fitted coefficients to a target line traversing $\lambda \in \Lambda \subset \mathbb{R}$01. In a six-site Schwinger-model simulation with a custom mild noise model, the reported prediction-region errors are $\lambda \in \Lambda \subset \mathbb{R}$02, $\lambda \in \Lambda \subset \mathbb{R}$03, and $\lambda \in \Lambda \subset \mathbb{R}$04 for $\lambda \in \Lambda \subset \mathbb{R}$05, and $\lambda \in \Lambda \subset \mathbb{R}$06, $\lambda \in \Lambda \subset \mathbb{R}$07, and $\lambda \in \Lambda \subset \mathbb{R}$08 for $\lambda \in \Lambda \subset \mathbb{R}$09. In that setting, adiabatic GREC is reported to transfer between different parameter regimes and, in particular, between different phases of the model, while also being more cost-efficient than ZNE in terms of the total number of gates used for the simulations (Kaikov et al., 9 Jul 2025).

More broadly, recent work on generalized physical implementability formalizes quasi-probability cancellation with respect to an arbitrary convex free set of implementable channels, rather than the full set of CPTP maps. In that framework, direct simulation of the inverse of a global error channel can be both more efficient and more accurate than separate layer-by-layer simulation when the implementing operations are themselves noisy. This is not a definition of GREC, but it provides a resource-theoretic language for analyzing global cancellation strategies in realistic noisy settings (Jin et al., 2024).

The main structural limitation of GREC remains the requirement of a classically accessible parameter regime. The method is expected to be effective when the error functions are reasonably smooth and analytic in $\lambda \in \Lambda \subset \mathbb{R}$10, when the easy region is nontrivial and sufficiently close to the target region for analytic extrapolation to remain stable, and when classical data in $\lambda \in \Lambda \subset \mathbb{R}$11 is available with sufficient precision. It is less effective when one must extrapolate very far from the training region, when the dependence of noise on the parameter is highly non-smooth or non-analytic, or when no parameter regime is classically tractable. The original paper explicitly identifies lattice QCD at finite chemical potential, the Hubbard model at finite density, and simpler lattice field theories such as $\lambda \in \Lambda \subset \mathbb{R}$12 and Schwinger models as promising directions for future applications (Sazonov et al., 2021).