Averaged Circuit Eigenvalue Sampling (ACES)

• ACES is a scalable noise metrology method that simultaneously estimates gate-resolved Pauli error rates in quantum circuits.
• It employs randomized compiling and Pauli eigenvalue sampling to reconstruct detailed noise models from Clifford circuit experiments.
• The protocol enables practical calibration for quantum error correction by characterizing errors in syndrome-extraction circuits to inform decoders.

Averaged Circuit Eigenvalue Sampling (ACES) is a method for scalable noise metrology of quantum circuits that stands for Averaged Circuit Eigenvalue Sampling. It was introduced by Flammia as a protocol that simultaneously estimates the individual error rates of all the gates in collections of quantum circuits, and can even account for space and time correlations between these gates (Flammia, 2021). In subsequent work, ACES was developed into a practical framework for characterizing the Pauli error channels of individual gates across a device, for reconstructing layer- and gate-resolved Pauli noise models of syndrome-extraction circuits, and for calibrating noise-aware decoders in quantum error correction (Pelaez et al., 2024, Hockings et al., 2024, Hockings et al., 28 Feb 2025, Lee et al., 15 Apr 2026).

1. Origins and conceptual scope

ACES was introduced as a scalable and flexible method for noise metrology in Clifford circuits with Pauli noise, combined with randomized compiling (Flammia, 2021). Its central objective is to characterize the performance of noisy quantum circuits by estimating the Pauli error probabilities of all gates in a circuit simultaneously rather than by sequentially characterizing one operation at a time. In the original formulation, ACES strictly generalizes randomized benchmarking, interleaved randomized benchmarking, simultaneous randomized benchmarking, and several other related techniques, while providing more detailed gate-resolved information under weaker assumptions (Flammia, 2021).

The protocol is motivated by the observation that decoders, compilers, and mitigation pipelines require hardware-relevant noise models rather than global averages. Later work made this motivation explicit for near-term devices and quantum error correction. In particular, ACES was presented as a rapid, scalable, and automated circuit-level characterization tool whose core motivation is to characterize errors gate-by-gate and layer-by-layer within the realistic context of an entire syndrome extraction sequence, to provide accurate, context-dependent Pauli error models for all operations including mid-circuit measurements, and to bridge the gap between hardware characterization and decoding (Lee et al., 15 Apr 2026).

The framework has been deployed in several regimes. One line of work implemented a full version of ACES on real NISQ devices and deployed it to Superstaq, with experiments on IBM Algiers and Osaka devices (Pelaez et al., 2024). Another line of work developed a scalable protocol suited to syndrome extraction circuits of topological codes, together with rigorous protocol optimization and a Julia implementation in the package QuantumACES (Hockings et al., 2024). In quantum error correction, ACES was used to calibrate prior noise models for minimum-weight perfect matching decoders and to improve error suppression in surface-code simulations (Hockings et al., 28 Feb 2025). It was also used experimentally in a heavy-hex superconducting-qubit array implementing a distance-5 dynamic compass code, where ACES supplied the noise characterization data used to construct a shot-independent base detector error model for decoding (Lee et al., 15 Apr 2026).

2. Mathematical formulation

ACES is built around Pauli channels. A Pauli channel on $n$ qubits is written as

$$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$

where $P_a$ are $n$-qubit Pauli operators and $p_a$ are associated error probabilities (Pelaez et al., 2024). For such a channel, Pauli operators are eigenvectors: $\mathcal{E}^P(P_b) = \lambda_b P_b,$ with eigenvalues

$\lambda_b = \sum_a (-1)^{\langle a, b\rangle} p_a,$

and the inverse relation

$$p_a = \frac{1}{2^n} \sum_b (-1)^{\langle a, b \rangle} \lambda_b$$

recovers the Pauli error probabilities from the eigenvalues (Pelaez et al., 2024). In the more general ACES formulation, this relation is also described as a Walsh-Hadamard or Fourier transform between Pauli eigenvalues and Pauli error rates (Flammia, 2021, Hockings et al., 2024).

For a sequence of noisy Clifford gates forming a circuit, the circuit eigenvalue factorizes into gate-wise eigenvalues. In one common notation,

$$\Lambda_{\mathcal{C}, a_1} = \prod_{k=1}^T \lambda_{k, a_k},$$

where the Pauli label is propagated through the Clifford circuit (Pelaez et al., 2024). The original ACES paper states the same idea as

$$\widetilde{C}^P(P_{a_1}) = \left(\prod_{k=1}^T \lambda_{k,a_k}\right) C(P_{a_1}),$$

so that

$$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$0

Taking logarithms yields a linear inverse problem: $$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$1 or equivalently

$$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$2

where $$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$3 is the design matrix, $$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$4 is the vector of gate eigenvalue parameters, and $$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$5 collects measured circuit eigenvalues after logarithmic transformation (Flammia, 2021, Pelaez et al., 2024, Hockings et al., 28 Feb 2025).

For syndrome-extraction circuits, a layer-based formulation is used. The circuit is partitioned into layers, each containing mutually commuting gates implemented simultaneously on disjoint qubit subsets, and ACES infers marginals of the corresponding layer-local Pauli channels onto the support of individual gates (Hockings et al., 2024). This is the sense in which ACES can capture averaged spatial correlations between gates applied simultaneously, while still returning gate-resolved quantities derived from layer-resolved data (Hockings et al., 2024).

3. Protocol structure and estimation procedure

The operational workflow of ACES begins with circuit construction. In one implementation, a collection of random Clifford circuits is built, each including a mirror section and a random section of additional depth, and a set of Pauli operators is selected for probing circuit eigenvalues (Pelaez et al., 2024). In another implementation targeted at syndrome extraction, the underlying Clifford circuit is partitioned into layers, and ACES experiments are constructed by rearranging these layers into tuples that generate varied composite Clifford circuits whose noise can be related, via a linear model, to the noise of individual gates in each layer (Hockings et al., 2024).

Circuit eigenvalues are estimated experimentally by preparing Pauli eigenstates, applying the noisy circuit, and measuring in the corresponding Pauli basis. The NISQ-device implementation uses the “difference trick”: all $$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$6 and $$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$7 eigenstates of the input Pauli are prepared, the circuit is run, and the output Pauli expectation values are combined to estimate the circuit eigenvalue (Pelaez et al., 2024). Pauli eigenstates are prepared with simple gate sequences: $$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$8 and $$\rho \rightarrow \sum_{P_a \in \mathbb{P}^n} p_a P_a \rho P_a^\dagger,$$9 eigenstates are made with Hadamard and $P_a$0 gates, and $P_a$1 is the computational basis (Pelaez et al., 2024).

Randomized compiling and twirling play a central role in several versions of ACES. The original formulation leverages randomized compiling so that, after averaging, the effective noise becomes Pauli and local to each gate (Flammia, 2021). For NISQ hardware, one study reported that simulations used Pauli twirling to average errors into the stochastic Pauli channel, but hardware runs on IBM Algiers and Osaka did not use twirling; ACES there directly characterized the device’s native error channel as a Pauli channel rather than twirling it into one (Pelaez et al., 2024). In a superconducting two-qubit experiment, the protocol was explicitly tailored with Pauli twirling and additional treatment of measurement errors, and its reconstructions were compared against interleaved randomized benchmarking (Palomaki et al., 2 Oct 2025).

After circuit eigenvalues are measured, gate eigenvalues are inferred by solving the linear system. The standard route is least-squares or weighted least-squares estimation (Pelaez et al., 2024, Hockings et al., 2024). In large-scale syndrome-extraction settings, experimental outcomes for multiple compatible Pauli eigenvalues are sampled simultaneously, circuit-eigenvalue estimators are formed by sample averaging, variances and covariances are modeled explicitly, and gate eigenvalues are inferred through

$P_a$2

with the weight matrix estimated from the data (Hockings et al., 2024). The resulting Pauli probabilities are then obtained by inverse Walsh-Hadamard transform and projected onto the probability simplex (Hockings et al., 2024). A closely related formulation projects reconstructed probabilities into the physical probability simplex using Mahalanobis distance rather than Euclidean distance (Hockings et al., 28 Feb 2025).

4. Scalability and experimental resource profile

Scalability is one of the defining claims of ACES. The original paper demonstrated by numerical simulations that ACES simultaneously and precisely estimates all the Pauli error rates on every gate and measurement in a 100 qubit quantum device using fewer than 20 relatively shallow Clifford circuits and an experimentally feasible number of samples (Flammia, 2021). That study simulated a 100-qubit device with 5070 parameters and used only 19 Clifford circuits (Flammia, 2021).

Subsequent work gave a more explicit resource analysis. For one- and two-qubit gates in a nearest-neighbour architecture, the NISQ-device implementation states that the number of circuits required is

$P_a$3

assuming Pauli probe weight at most $P_a$4, while the number of experimental shots per probe scales as $P_a$5 for additive error $P_a$6 (Pelaez et al., 2024). In simulations with purely Pauli noise channels, that study reported average eigenvalue absolute error under $P_a$7 and total variation distance under $P_a$8 with $P_a$9 shots per circuit using $n$0 circuits of depth $n$1 (Pelaez et al., 2024).

The syndrome-extraction variant strengthens the scalability claim. It develops a figure of merit for expected normalized root-mean-squared error in estimated gate eigenvalues and uses it to optimize tuple design and shot allocation (Hockings et al., 2024). In numerical simulations of the entire noise-characterization procedure for the syndrome-extraction circuit of a distance-25 surface code with over 1000 qubits, the number of experiments required to estimate all Pauli error probabilities remains essentially constant as system size scales (Hockings et al., 2024). A specific example is that 261 experiment types suffice for a circuit of 1249 qubits, using 31 carefully optimized tuples to characterize all 51,576 gate eigenvalues across the circuit (Hockings et al., 2024). The same work reports that all simulations for 1200+ qubits were run in under 5 hours on a laptop (Hockings et al., 2024).

In error-correction calibration studies, ACES was presented as practical at large scales required for fault tolerance. One study states that these noise-characterization experiments could be performed and processed in seconds for superconducting quantum computers, and gives a concrete example in which collecting $n$2 shots for an ACES experiment requires about 2 seconds of device time while classical processing takes less than 4 seconds on a conventional laptop for a distance-25 surface code (Hockings et al., 28 Feb 2025). In the heavy-hex dynamic-compass-code experiment, the entire characterization for all circuit elements in a distance-5 code implementation was performed in approximately 30 minutes of QPU time (Lee et al., 15 Apr 2026).

5. Use in quantum error correction and decoding

ACES has become closely associated with circuit-level noise characterization for quantum error correction. In the heavy-hex dynamic-compass-code experiment, ACES was performed to acquire detailed context-dependent error information on all elements of the syndrome extraction process (Lee et al., 15 Apr 2026). The protocol outputs the relevant Pauli channel for each gate in each layer of non-overlapping gates, and for measurements it provides both bit-flip error probabilities as well as the probability of a classical read-out error (Lee et al., 15 Apr 2026). These data are used to construct a shot-independent base detector error model, which is then supplied to a matching decoder for performing hard decoding (Lee et al., 15 Apr 2026).

The same study used ACES together with soft information from measurement devices and leakage detection through post-selection. The resulting noise-informed approach yielded up to $n$3 improvement in the logical error rate of a distance-5 implementation of the dynamic compass code in experiment (Lee et al., 15 Apr 2026). More specifically, ACES-informed decoding led to $n$4 improvement in the $n$5 basis and $n$6 in the $n$7 basis relative to a per-qubit calibration model, and to $n$8 in the $n$9 basis and $p_a$0 in the $p_a$1 basis relative to a global calibration model (Lee et al., 15 Apr 2026). The paper attributes this to the fact that ACES produces context-dependent error models derived from characterization of the syndrome extraction circuits themselves (Lee et al., 15 Apr 2026).

A related surface-code study formalized this role as calibration of the decoder prior. There, ACES is described as a scalable, practical method for circuit-level Pauli noise characterization that enables decoder calibration at the scales required for fault-tolerant quantum computation and near-optimal decoding when compared to the true noise model (Hockings et al., 28 Feb 2025). In that study, noise-aware decoding calibrated by ACES increased the error suppression factor of the surface code, with reductions in the logical error rate that increase exponentially with the code distance (Hockings et al., 28 Feb 2025). For random log-normal Pauli noise, the reported error suppression factors were $p_a$2, $p_a$3, $p_a$4, and $p_a$5; at distance $p_a$6, the corresponding logical error rates were $p_a$7, $p_a$8, $p_a$9, and $\mathcal{E}^P(P_b) = \lambda_b P_b,$0 (Hockings et al., 28 Feb 2025). The same work states that for larger code distances up to $\mathcal{E}^P(P_b) = \lambda_b P_b,$1, noise-aware decoding halves the logical error rate and reduces qubit overhead by up to $\mathcal{E}^P(P_b) = \lambda_b P_b,$2 (Hockings et al., 28 Feb 2025).

This suggests that ACES is not merely a characterization primitive but also a calibration layer connecting device physics to decoder weights. A plausible implication is that its value grows with code distance and circuit complexity because the mismatch between simplistic calibration snapshots and circuit-contextual noise becomes increasingly consequential.

6. Limitations, assumptions, and related developments

ACES relies on structural assumptions. The original formulation assumes Pauli-twirled noise and Clifford circuits for the resource-efficient construction, although it emphasizes that the required assumption is weaker than gate-independent noise over the full Clifford group (Flammia, 2021). The NISQ-device implementation identifies several practical limitations: no separation of SPAM from gate errors in hardware, the assumption of a Pauli channel even when real device noise may not be strictly Pauli, overheads associated with twirling, and the open problem of extending the approach to arbitrary or non-Clifford gatesets (Pelaez et al., 2024). The superconducting two-qubit study similarly emphasizes that accurate reconstruction was obtained by tailoring the noise via Pauli twirling and by accounting for measurement errors (Palomaki et al., 2 Oct 2025).

A recurrent point is that ACES can characterize native error channels as Pauli channels without explicit twirling, but the applicability of the Pauli approximation in the absence of twirling remains subject to further scrutiny (Pelaez et al., 2024). In the same vein, worst-case discrepancies between reconstructed infidelities and IBM-reported error rates on hardware were attributed to the fact that ACES did not separately account for SPAM errors, which were instead lumped into gate-error estimates (Pelaez et al., 2024).

Related developments extend or adapt the ACES philosophy. “Scalable noise characterization of syndrome-extraction circuits with averaged circuit eigenvalue sampling” develops explicit optimization criteria, tuple-selection heuristics, and a deterministic packing algorithm for compatible Pauli measurements, and releases these methods in the Julia package QuantumACES (Hockings et al., 2024). “Characterizing Superconducting Qubits using Averaged Circuit Eigenvalue Sampling” applies ACES to two coupled superconducting qubits, verifies reconstructed fidelities against interleaved randomized benchmarking, and shows that injected phase errors in two-qubit gates can be identified through reconstructed Pauli channels (Palomaki et al., 2 Oct 2025). “Fermionic Averaged Circuit Eigenvalue Sampling” generalizes the same simultaneous, self-consistent learning strategy to fermionic linear optical gates, using FLO twirling and Kravchuk transformations rather than the Walsh-Hadamard structure of Pauli channels (Chapman et al., 2 Apr 2025).

The literature also clarifies a possible source of confusion: ACES concerns sampling circuit eigenvalues associated with noisy quantum channels in Clifford or related circuit families, not eigenvalue estimation of arbitrary non-unitary matrices in the sense of iterative phase estimation. The paper “A Universal Quantum Circuit Scheme For Finding Complex Eigenvalues” addresses a different problem—estimating complex eigenvalues of non-unitary matrices using iterative phase estimation—and its use of “eigenvalue” is unrelated to the circuit-level noise-metrology framework of ACES (Daskin et al., 2013).

In aggregate, the published record describes ACES as a framework for simultaneously estimating gate-resolved Pauli noise parameters from carefully designed circuit ensembles, with demonstrated applications to NISQ characterization, large-scale syndrome-extraction circuits, and noise-aware decoding in quantum error correction (Flammia, 2021, Pelaez et al., 2024, Hockings et al., 2024, Hockings et al., 28 Feb 2025, Lee et al., 15 Apr 2026).