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New Journal of Physics

Multi-echo evaluation of non-invertible errors in target quantum computations

Jados P dos Santos, Ivan Henao, Raam Uzdin

Published Sep 30, 2026• 0 citations
Publisher

Abstract

Abstract This work introduces a method for quantifying errors in quantum computations using multiple echoes of a target circuit. Our approach addresses a gap in the range of available benchmarking methods: unlike existing approaches that evaluate the average performance of a device, circuit, or gate, our method estimates the error in a target computation using the same initial state and circuit as the computation itself. It therefore captures how noise manifests in the specific computation of interest. We present two variants of the approach. The first estimates incoherent errors through purity loss, while the second estimates the echo infidelity, which captures both incoherent errors and non-invertible coherent errors, such as crosstalk. The purity-loss method applies to low-error circuits, or layers of gates, regardless of system size. The echo-infidelity method applies to circuits of arbitrary size and remains effective even at high error rates. Because our approach does not rely on an explicit noise model, it can complement methods that provide more detailed error information by testing the validity of the sparsity assumptions underlying those models. Finally, we discuss platforms on which our method can be implemented immediately.

DOI
10.1088/1367-2630/aeae65

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quant-pharXiv:2609.04132•Sep 4, 2026•0 citations
Abstract This work introduces a method for quantifying errors in quantum computations using multiple echoes of a target circuit. Our approach addresses a gap in the range of available benchmarking methods: unlike existing approaches that evaluate the average performance of a device, circuit, or gate, our method estimates the error in a target computation using the same initial state and circuit as the computation itself. It therefore captures how noise manifests in the specific computation of interest. We present two variants of the approach. The first estimates incoherent errors through purity loss, while the second estimates the echo infidelity, which captures both incoherent errors and non-invertible coherent errors, such as crosstalk. The purity-loss method applies to low-error circuits, or layers of gates, regardless of system size. The echo-infidelity method applies to circuits of arbitrary size and remains effective even at high error rates. Because our approach does not rely on an explicit noise model, it can complement methods that provide more detailed error information by testing the validity of the sparsity assumptions underlying those models. Finally, we discuss platforms on which our method can be implemented immediately.
Quantum processors can already execute tasks beyond the reach of classical simulation, albeit for artificial problems. At this point, it is essential to design error metrics that test the experimental accuracy of quantum algorithms with potential for a practical quantum advantage. The distinction between coherent errors and incoherent errors is crucial, as they often involve different error suppression tools. The first class encompasses miscalibrations of control signals and crosstalk, while the latter is usually related to stochastic events and unwanted interactions with the environment. We introduce the incoherent infidelity as a measure of incoherent errors and present a scalable method for measuring it. This method is applicable to generic quantum evolutions subjected to time-dependent Markovian noise. Moreover, it provides an error quantifier for the target circuit, rather than an error averaged over many circuits or quantum gates. The estimation of the incoherent infidelity is suitable to assess circuits with sufficiently low error rates, regardless of the circuit size, which is a natural requirement to run useful computations.
Noise mechanisms in quantum systems can be broadly characterized as either coherent (i.e., unitary) or incoherent. For a given fixed average error rate, coherent noise mechanisms will generally lead to a larger worst-case error than incoherent noise. We show that the coherence of a noise source can be quantified by the unitarity , which we relate to the average change in purity averaged over input pure states. We then show that the unitarity can be efficiently estimated using a protocol based on randomized benchmarking that is efficient and robust to state-preparation and measurement errors. We also show that the unitarity provides a lower bound on the optimal achievable gate infidelity under a given noisy process.
The performance of quantum gates is often assessed using some form of randomized benchmarking. However, the existing methods become infeasible for more than approximately five qubits. Here we show how to use a simple and customizable class of circuits-randomized mirror circuits-to perform scalable, robust, and flexible randomized benchmarking of Clifford gates. We show that this technique approximately estimates the infidelity of an average many-qubit logic layer, and we use simulations of up to 225 qubits with physically realistic error rates in the range 0.1%-1% to demonstrate its scalability. We then use up to 16 physical qubits of a cloud quantum computing platform to demonstrate that our technique can reveal and quantify crosstalk errors in many-qubit circuits.
Crosstalk and several forms of coherent noise are invisible when a qubit or a gate is calibrated or benchmarked in isolation. These are unlocked during the execution of a full quantum circuit applying entangling gates to several qubits simultaneously. Unitary crosstalk noise, such as an unwanted Z-Z coupling, limits the state fidelity during the execution of cross-resonance controlled-not (cnot) gates in superconductor quantum computers. This work presents (1) a method of tracing coherent errors by exploiting their sensitivity to the arrangement of cnot gates in the circuit and (2) a correction scheme that modifies the original circuit by inserting carefully the chosen compensating gates (single- or two-qubit) to possibly undo coherent errors. On two vastly different types of IBMQ processors offering quantum volume 8 to 32, our experimental results show up to 25% reduction in the infidelity of [[7,1,3]][[7,1,3]][[7,1,3]] code $|+\ensuremath{\rangle}$ state (Clifford circuits) and five- to 15-qubit WWW states (non-Clifford circuits). Our experimental circuits aggressively deploy forced commutation of cnot gates to obtain low-noise state-preparation circuits. An encoded state initialized with fewer errors marks an important step towards successful demonstration of fault-tolerant quantum computers.
Two schemes are presented that mitigate the effect of errors and decoherence in short-depth quantum circuits. The size of the circuits for which these techniques can be applied is limited by the rate at which the errors in the computation are introduced. Near-term applications of early quantum devices, such as quantum simulations, rely on accurate estimates of expectation values to become relevant. Decoherence and gate errors lead to wrong estimates of the expectation values of observables used to evaluate the noisy circuit. The two schemes we discuss are deliberately simple and do not require additional qubit resources, so to be as practically relevant in current experiments as possible. The first method, extrapolation to the zero noise limit, subsequently cancels powers of the noise perturbations by an application of Richardson's deferred approach to the limit. The second method cancels errors by resampling randomized circuits according to a quasiprobability distribution.
Quantum benchmarks provide compact measures of performance that are important for evaluating and comparing quantum systems. Circuit-level benchmarks are particularly valuable because they capture the accumulated effects of noise across interacting operations, but existing approaches may require structured gate sets and costly compilation, classical simulation of reference outputs, or subsystem decompositions that do not capture full-register behavior. We introduce Error Per Circuit Layer (EPCL), an overlap-based circuit-level benchmark that estimates an effective layer polarization by applying identical random circuits to two disjoint quantum registers and measuring the overlap between their output states as a function of circuit depth. EPCL avoids classical simulation of ideal output distributions and recovery to a known reference state, and is compatible with arbitrary gate sets, including non-Clifford gates. We derive the expected overlap decay under an ensemble-averaged depolarizing model and identify the assumptions under which the fitted decay parameter represents an effective layer polarization. Numerical simulations show that EPCL recovers the predicted polarization under weak local stochastic noise and remains well described by a single-exponential decay at stronger stochastic noise levels. The simulations further show that coherent errors associated with fixed entangling layers may require Pauli twirling or randomized compiling to produce the expected decay, while inter-register correlations contribute an additional covariance term to the measured overlap. Finally, experiments on IBM quantum hardware demonstrate clear EPCL decay in 8- and 16-qubit implementations. These results support EPCL as a method for measuring aggregate register performance without requiring classical simulation of ideal circuit outputs or restriction to structured gate sets.