Cycle Error Reconstruction in Quantum Processors

• CER is a cycle-centric error characterization protocol that uses randomized compiling and Pauli twirling to transform noisy cycles into effective stochastic Pauli channels for estimating marginal error probabilities.
• It employs folded cycle schemes with adjustable hyperparameters to separate coherent and incoherent error contributions, refining error diagnostics in quantum circuits.
• CER supports scalable reconstruction of low-weight Pauli marginals, bridging physical-layer noise diagnostics with logical-level fault-tolerance through targeted calibration and mitigation strategies.

Searching arXiv for papers on Cycle Error Reconstruction and related work. Cycle Error Reconstruction (CER) is a cycle-centric error-characterization protocol for quantum processors that leverages randomized compiling and Pauli twirling to estimate the error profile of a selected computing cycle, typically a fixed-schedule layer of native instructions or a layer of one- and two-qubit gates executed in parallel. In the CER literature, a “cycle” is the scheduled list of instructions over an arbitrarily large fraction of the chip, partitioned into “easy cycles” of single-qubit gates and a fixed “hard cycle” whose error is the target of reconstruction. Under randomized compiling, the noisy implementation of a dressed cycle is converted, on average, into an effective dressed cycle with a stochastic Pauli error model, enabling estimation of Pauli fidelities, Pauli eigenvalues, and low-weight marginals of the associated Pauli error distribution. In the extended folded variant, CER introduces repeated application of the hard cycle before twirling, which exposes the split between coherent and incoherent contributions to the error profile. The protocol is also referred to as K-body Noise Reconstruction (KNR) in work emphasizing local marginals and polynomial scaling under locality assumptions (Carignan-Dugas et al., 2023, Carignan-Dugas et al., 2023).

1. Definition and scope

CER is designed to reconstruct the error distribution associated with a selected effective cycle rather than to characterize isolated gates in abstraction from scheduling context. This emphasis is motivated by the observation that error distributions are considerably influenced by the precise gate scheduling across the entire quantum processing unit, so the natural diagnostic object is a cycle rather than a standalone primitive. In the general formalism, a cycle is a fixed-schedule layer of native instructions applied to all qubits in the processor, and two consecutive layers of an arbitrarily chosen “easy” cycle and a fixed “hard” cycle define a dressed cycle (Carignan-Dugas et al., 2023).

Within this framework, CER targets either the full Pauli-twirled error model of the cycle or structured reductions of it. One formulation states the goal as estimating, for every Pauli supported on at most $k$ of the parallel gates in the hard cycle, its marginal error probability

$\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$

thereby reconstructing all $k$-qubit reduced Pauli distributions. Another formulation emphasizes estimation of Pauli eigenvalues $\lambda_P$, followed by inversion to Pauli error probabilities $\chi_P$. A further extension, obtained by folding the hard cycle multiple times before twirling, aims not only at total error rates but at a detailed estimate of the coherent contribution to the error profile of a hard computing cycle (Carignan-Dugas et al., 2023, Carignan-Dugas et al., 2023).

The scope of CER has expanded from physical-layer diagnostics on IBM-Q 5-qubit devices to coherent-versus-incoherent decomposition on IBM chips, to characterization of a 16-qubit transversal CNOT in trapped ions, and to decoder-informed learning of the dominant Pauli rates relevant to fault-tolerant quantum computation. This suggests that CER functions both as a benchmarking primitive and as a bridge between device-level noise characterization and logical-level performance analysis (Fazio et al., 16 Apr 2025, Iyer et al., 11 Jul 2025).

2. Relation to cycle benchmarking and KNR

CER is structurally close to Cycle Benchmarking (CB). CB uses Pauli twirling and randomized compiling to turn an arbitrary cycle error into a stochastic Pauli channel, then extracts Pauli fidelities by fitting an exponential decay in the number of cycle repetitions. In this sense, CER inherits the CB circuit structure, the reliance on repeated-cycle survival signals, and the use of Pauli observables as the relevant diagnostic coordinates. However, CB reports total error rates but does not distinguish coherent versus incoherent parts, and in its standard form it is not organized around reconstructing marginal Pauli distributions (Carignan-Dugas et al., 2023).

CER departs from CB in two principal ways. First, it uses the same randomized-compiled, Pauli-twirled setting to estimate many Pauli-sector quantities sufficient to reconstruct marginals of the Pauli error distribution. Second, in the folded extension it introduces an extra hyperparameter $x=k$, meaning that the hard cycle is folded back on itself $k$ times before the final Pauli twirl. Varying this hyperparameter allows one to expose the quadratic buildup of coherent errors versus the linear buildup of incoherent errors, thereby generalizing the standard fidelity-decay model (Carignan-Dugas et al., 2023).

KNR is described as essentially equivalent to CER when the objective is to learn marginal Pauli error probabilities over small qubit subsets. The common idea is that locality constrains the number of significant parameters, so one can learn $k$-body marginals or top-weight terms without reconstructing the exponentially large full Pauli distribution. A common misconception is therefore that CER necessarily requires exhaustive estimation of all $4^n$ Pauli rates. The literature instead distinguishes exact CER, which yields all $4^n$ rates but costs $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$0 data, from scalable variants that exploit locality or sparsity to recover only low-weight marginals or the largest $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$1 Pauli error rates (Iyer et al., 11 Jul 2025).

3. Experimental protocol and circuit structure

The basic CER protocol begins by separating the target schedule into easy and hard cycles. The hard cycle $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$2 is the layer of entangling gates whose error is to be characterized, while the easy cycle is a parallel single-qubit layer used for random Pauli dressings. Under randomized compiling, each noisy cycle $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$3 is twirled into an effective stochastic Pauli channel $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$4 followed by the ideal hard cycle. Repeating randomized compiling and averaging makes $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$5 a diagonal Pauli channel (Carignan-Dugas et al., 2023).

For each experimental setting one chooses a Pauli basis $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$6 to prepare and measure, a random randomized-compiling dressing string $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$7, and a number $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$8 of dressed-cycle repetitions. In the folded variant one also chooses the fold number $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$9. The full circuit is described as

$k$0

By construction this implements, up to a final known Pauli, a stochastic Pauli channel so that each shot returns “$k$1” outcomes from which one estimates a Pauli fidelity $k$2 (Carignan-Dugas et al., 2023).

In the more general CER formulation based on CB circuits, the experiment consists of selecting random Pauli dressings $k$3, inserting them before and after repeated applications of the cycle, and wrapping the repeated cycle between a state-preparation Pauli basis and a measurement Pauli basis. The trapped-ion realization makes this explicit through state preparation $k$4, Pauli randomization $k$5, repeated compiled Pauli cycles $k$6, and final basis rotation $k$7. For even $k$8, $k$9, so the ideal map is identity and one always measures in a Pauli basis (Carignan-Dugas et al., 2023, Fazio et al., 16 Apr 2025).

Data collection is organized over repeated choices of $\lambda_P$0, random dressings, and, when coherent-versus-incoherent separation is desired, fold numbers $\lambda_P$1. Averaging over many random dressing strings gives

$\lambda_P$2

and the variation with $\lambda_P$3 is used to reveal coherent buildup. In the top-$\lambda_P$4 learning setting, the same Pauli-twirled cycle-benchmarking machinery is used to identify and learn the largest few Pauli error rates $\lambda_P$5, with $\lambda_P$6 under plausible sparsity assumptions (Carignan-Dugas et al., 2023, Iyer et al., 11 Jul 2025).

4. Mathematical reconstruction

The central algebraic step in CER is the reduction of a general Markovian noise channel to a Pauli channel under twirling. In one formulation,

$\lambda_P$7

where $\lambda_P$8 is the Pauli error probability of $\lambda_P$9. The experimentally accessible quantity is a Pauli-transfer-matrix eigenvalue $\chi_P$0, extracted from the survival law

$\chi_P$1

These eigenvalues are related to the Pauli probabilities by a Walsh–Hadamard transform,

$\chi_P$2

which provides the inversion from observed decay parameters to Pauli error rates (Iyer et al., 11 Jul 2025).

The general CER formulation on small supports uses Pauli fidelities

$\chi_P$3

for a Pauli channel $\chi_P$4, together with character orthogonality to obtain

$\chi_P$5

and, for marginals on a support $\chi_P$6,

$\chi_P$7

When the hard cycle normalizes the Pauli group on the chosen support, CER exploits conjugation orbits and reconstructs orbit-marginals from orbit-averaged fidelities rather than from each Pauli separately (Carignan-Dugas et al., 2023).

The folded extension refines this picture by introducing a small-error expansion for the fidelity of the folded cycle: $\chi_P$8 Here $\chi_P$9 and $x=k$0 are the marginal decoherent and coherent infidelities of the hard-cycle component that commutes with $x=k$1. Repeating the folded cycle $x=k$2 times yields

$x=k$3

where $x=k$4 absorbs SPAM errors and $x=k$5 is a small linear-in-$x=k$6 cross-term arising from residual interactions with the easy cycles. In practice one fits

$x=k$7

with $x=k$8 revealing the coherent contribution and $x=k$9 the incoherent one (Carignan-Dugas et al., 2023).

In the trapped-ion transversal-CNOT study, the corresponding observable is an orbital eigenvalue, because Paulis decay through an orbit under the hard cycle. The measured decay of $k$0 yields $k$1, and an inverse Walsh–Hadamard transform over the marginal subset produces the corresponding orbit-marginal probabilities. A constrained least-squares projection is then used to enforce nonnegativity and normalization when sampling noise would otherwise produce slight negatives (Fazio et al., 16 Apr 2025).

5. Assumptions, scaling, and limitations

CER rests on a set of explicit approximations. The folded coherent-error analysis assumes Markovian, time-homogeneous errors generated by a time-independent Lindbladian; error locality, so that Hamiltonian and Lindblad operators are $k$2-local on the device connectivity graph; small error rates, allowing truncation of Taylor and Baker–Campbell–Hausdorff expansions at second order; and a perfect or idealized Pauli twirl via randomized compiling, so that residual coherent cross-terms remain small. The general marginal-reconstruction formulation similarly assumes that after randomized compiling each effective dressed cycle has purely Pauli-stochastic error, that different errors are uncorrelated across cycles up to negligible second-order terms, and that a reduced-weight error model applies (Carignan-Dugas et al., 2023, Carignan-Dugas et al., 2023).

Under these assumptions, the protocol is scalable. One resource estimate states that the number of Pauli marginals $k$3 grows at most polynomially in $k$4 under local noise, the total number of fit parameters is $k$5, and the numbers of sequence lengths $k$6 and folds $k$7 are independent of $k$8, $k$9. The number of random Pauli-compiled circuits per $k$0 is $k$1 to achieve concentration, and the number of shots per circuit only needs to deliver $k$2–$k$3 precision, again independent of $k$4. In the low-weight-marginal formulation, the total number of runs scales as

$k$5

which is polynomial in the number of supports $k$6 for fixed $k$7 (Carignan-Dugas et al., 2023, Carignan-Dugas et al., 2023).

The literature also spells out the limitations. Strong time-correlations such as $k$8 noise can bias estimates under the Markovian-repeat assumption. Additive estimation of Pauli fidelities within each orbit is less robust to SPAM than CB-style orbital averages. Highly non-Pauli or coherent leakage errors require larger orbited groups or alternative twirling sets, such as $k$9 dihedral groups. In the decoder-oriented formulation, exact CER yields all $4^n$0 rates but costs $4^n$1 data; top-$4^n$2 learning reduces this burden only under sparsity assumptions. A plausible implication is that CER is most effective when the experimental regime already supports high-quality randomized compiling and when the dominant noise admits a compact Pauli description (Carignan-Dugas et al., 2023, Iyer et al., 11 Jul 2025).

6. Physical-layer and logical-layer demonstrations

Numerical and hardware demonstrations establish the range of phenomena CER can resolve. In the folded coherent-error study, numerical benchmarks simulated a single spectator qubit idling next to a train of CNOTs with $4^n$3 from ibmq_montreal and an imposed coherent $4^n$4-rotation of rate $4^n$5 per CNOT. Using $4^n$6 and $4^n$7, the method recovered both incoherent rates $4^n$8 and coherent rates $4^n$9 to within their statistical uncertainties. Proof-of-concept experiments on ibmq_guadalupe, ibmq_montreal, and ibmq_manila targeted a spectator ancilla adjacent to a CNOT hard cycle and measured a pronounced coherent $4^n$0-biased error, $4^n$1, versus $4^n$2 coherent rates $4^n$3; up to $4^n$4–$4^n$5 of the bias was attributed to coherent $4^n$6-coupling crosstalk (Carignan-Dugas et al., 2023).

The earlier CER implementation on IBM-Q 5-qubit devices applied the protocol to “burlington,” “essex,” “ourense” and “vigo,” reconstructing single-support and nearest-neighbor marginals across different topologies. The reported findings included low entangler error, with dominant two-qubit cycles having marginal Pauli error $4^n$7; spectator crosstalk, where idle qubits adjacent to a busy two-qubit gate routinely showed weight-1 $4^n$8 errors at the $4^n$9–$\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$00 level; and negligible higher-weight errors on disjoint pairs or triples. These data were then used to guide stochastic calibration, discussed below (Carignan-Dugas et al., 2023).

A later application characterized a 16-qubit transversal CNOT in a trapped-ion quantum computer. There, the hard cycle was

$\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$01

executed serially due to the single motional bus but logically treated as one “parallel” cycle. CER learned all 1-CNOT marginals and 2-CNOT marginals across two disjoint pairs. Typical single-CNOT marginals on the nonadjacent pair $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$02 were

$\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$03

Adjacent pairs showed up to twice larger correlated $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$04 and $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$05 orbits due to optical crosstalk. For the full transversal CNOT, the 1-CNOT marginals increased mainly in the dephasing orbits, with $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$06 and $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$07, while weight-2 orbits remained $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$08. Using a nearest-neighbor Gibbs Random Field built from CER marginals, the reconstructed 14-qubit joint distribution yielded an uncorrectable logical-error probability

$\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$09

compared to a total gate infidelity of $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$10; a consistency check via standard cycle-benchmarking sampling of random eigenvalues gave $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$11 (Fazio et al., 16 Apr 2025).

These demonstrations collectively show that CER is not confined to average error-rate reporting. It can isolate spectator crosstalk, expose coherent $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$12-bias, identify short-range correlations, and separate correctable from uncorrectable contributions in a logical primitive. This suggests a role for CER as a context-sensitive diagnostic for both physical calibration and logical benchmarking (Carignan-Dugas et al., 2023, Fazio et al., 16 Apr 2025).

7. Calibration, mitigation, and decoder-informed use

CER has been used directly to drive calibration. In the IBM-Q 5-qubit study, the dominant spectator $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$13 errors identified by CER defined a calibration target for stochastic calibration (SC), a fast compilation-based calibration method intended to identify and suppress local coherent error sources occurring in an effective cycle of interest. Three local virtual-$\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$14 phases on idle qubits were tuned by maximizing

$\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$15

with each curve fit to a quadratic to find the optimum phases. After SC, spectator $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$16 marginals dropped by up to a factor of $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$17, and overall cycle infidelity improved by $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$18. The broader work reports up to a $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$19-fold improvement of circuit performance (Carignan-Dugas et al., 2023).

CER has also been proposed as an input to mitigation and calibration strategies that distinguish coherent from incoherent error sources. Because the folded extension quantifies coherent and incoherent contributions separately, it can guide targeted calibration of coherent $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$20-coupling crosstalk while leaving incoherent components to other suppression tools. The source text explicitly notes potential applications ranging from benchmarking NISQ devices, calibrating pulse-level corrections for coherent crosstalk, to designing hybrid mitigation strategies that attack incoherent errors via zero-noise extrapolation and coherent errors via unitary compensation (Carignan-Dugas et al., 2023).

A separate line of work uses CER-derived Pauli rates for maximum-likelihood decoding. There, conventional decoders based only on the average infidelity $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$21 are contrasted with decoders informed by the top-$\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$22 CER rates plus the Uncorrelated Split Search (USS) heuristic, which recursively fills in missing Pauli probabilities from a restricted dataset. For the level-2 concatenated Steane code ($\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$23), representative results in a CG1D ensemble show that $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$24, corresponding to $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$25 of the Pauli error rates, yields median gains of $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$26, $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$27, $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$28, and $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$29 across bins with $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$30 and $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$31, respectively. The paper further states that with CER data constituting merely $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$32 of the Pauli error rates in the system, decoding achieves a $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$33 gain in performance compared to the case where decoding is based solely on the fidelity of the underlying noise process, and that a typical violin plot shows most instances achieving $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$34–$\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$35 gain when $\mu(P_A)=\sum_{Q\in\mathcal P_n:\,Q|_A=P_A} p(Q),$36. The corresponding total-variation-distance data show that USS reduces the model mismatch fivefold on average (Iyer et al., 11 Jul 2025).

Taken together, these developments place CER at the intersection of characterization, calibration, and fault-tolerance analysis. It provides a cycle-level description of effective Pauli noise, can be extended to estimate coherent and incoherent components separately, and supplies structured noise information that can be propagated upward to logical primitives and decoders. A plausible implication is that CER is valuable precisely because it preserves the schedule dependence of noise while still admitting scalable reconstruction under locality or sparsity assumptions (Fazio et al., 16 Apr 2025, Iyer et al., 11 Jul 2025).