Multi-Layer Cycle Benchmarking for high-accuracy error characterization

TL;DR

This work tackles the fundamental problem of learnability in Pauli-noise characterizations for scalable quantum devices. It introduces Multi-Layer Cycle Benchmarking (MLCB), which jointly analyzes multiple Clifford layers to impose high-accuracy constraints on otherwise unlearnable Pauli eigenvalues within sparse Pauli-Lindblad noise models. Through both a simple two-layer example and general multi-layer constructions on open and closed chains, MLCB leverages multi-layer orbits to recover ratios of unlearnable eigenvalues, achieving a reported $\sim$75% reduction in unlearnable degrees of freedom and enabling more accurate noise models. The approach is validated experimentally on a 20-qubit IQM Garnet device and supported by numerical simulations showing improved error characterization and enhanced performance of noise-aware error mitigation techniques like probabilistic error cancellation. Overall, MLCB is presented as a scalable, practical tool for precise noise characterization with direct benefits for error mitigation in near-term quantum processors.

Abstract

Accurate noise characterization is essential for reliable quantum computation. Effective Pauli noise models have emerged as powerful tools, offering detailed description of the error processes with a manageable number of parameters, which guarantees the scalability of the characterization procedure. However, a fundamental limitation in the learnability of Pauli fidelities impedes full high-accuracy characterization of both general and effective Pauli noise, thereby restricting e.g., the performance of noise-aware error mitigation techniques. We introduce Multi-Layer Cycle Benchmarking (MLCB), an enhanced characterization protocol that improves the learnability associated with effective Pauli noise models by jointly analyzing multiple layers of Clifford gates. We show a simple experimental implementation and demonstrate that, in realistic scenarios, MLCB can reduce unlearnable noise degrees of freedom by up to $75\%$, improving the accuracy of sparse Pauli-Lindblad noise models and boosting the performance of error mitigation techniques like probabilistic error cancellation. Our results highlight MLCB as a scalable, practical tool for precise noise characterization and improved quantum computation.

Figures (11)

• Figure 1: Standard CB protocol: In the state preparation stage (left brown area) an eigenstate of a Pauli operator is prepared. The Clifford layer $C$ subject to characterization (in this case, consisting of several parallel CZ gates shown in blue) is then implemented $d$ times, alternating it with layers of single-qubit gates (white boxes) which are needed to implement RC. This can be interpreted as the implementation of $d$ dressed Clifford layers. After a final layer of single-qubit gates which ensures that RC does not modify the unitary associated with the circuit, the measurement stage (right brown box), which also incorporates readout-twirling, provide us with the expectation value of a Pauli operator (typically the same operator as considered in the state preparation stage).
• Figure 2: A with square topology and four layers of parallel CZ gates, identified with blue, green, red and orange colors, that covers all nearest-neighbour connections between the qubits.
• Figure 3: protocol: (a) The structure is analogous to the one shown in Fig. \ref{['fig:SLCB']} with the important difference that, here, the "building block" which is repeated $d$ times is not a single Clifford layer but a combination of more layers (in this specific case two, shown in blue and green). Single-qubit gates used for twirling are indicated in white. (b) Simple scenario where allows one to measure with high accuracy the product $f^B_{XIX}f^G_{XZX}$ of two eigenvalues, belonging to two different layers, which cannot be accessed with conventional CB protocols. For the sake of simplicity, single-qubit gates used for twirling are now explicitly shown. (c) Simple system consisting of three qubits and two layers.
• Figure 4: (a) An example of several disconnected graphs $G_i$ obtained by considering edges associated with two layers, blue and green, of parallel two-qubit gates. The graphs can be divided into open and closed chains, as shown by the black rectangles. (b) Analysis of open chains showing the conjugations of specific Pauli strings $\alpha^{(m)}$ (with $m=0, \dots 4$) under four subsequent layers, in the pattern BGBG, that constitute the building block of protocols. Different chains, featuring four and five qubits, are depicted on the left (each with a specific qubit numbering). By looking at the corresponding qubits (and gates) on the right side one can identify the circuit blocks that allow to measure specific products of eigenvalues. The latter can be then used to determine the ratios $\mu_q^{BG}$ of unlearnable eigenvalues associated with the qubit $q$ (highlighted in yellow). Gray-shaded areas shows scenarios in which that the presence of additional qubit is trivial. (c) Analysis of closed chains consisting of four qubits. As for the previous panel, the circuits shows how MLCB can measure a product of four eigenvalues, which can then be used to determine the ratio $\mu_q^{BG}$ (with $q=0$ highlighted in yellow).
• Figure 5: (a) Layout of the IQM GarnetTMAbdurakhimov2024 quantum processor, highlighting the four characterized layers: blue (B), green (G), red (R), and orange (O). (b) Bootstrapped distributions of the difference $\Delta_{11}^{BR}$ between products of high-weight Pauli eigenvalues, as measured by , for three consecutive experimental runs (purple) and their average (gray). (c) All measured differences $\Delta_q^{BR}$ for the BR layer pair. (d) Q-Q plot of the ratios $\Omega_q^{L_1L_2}$ (defined in Eq. \ref{['eq:ratio_Omega']}) for all layer pairs $\{\rm BR, BO, BG, RO, RG, OG\}$ and a reference standard normal distribution (black dashed line).