Characterizing physical and logical errors in a transversal CNOT via cycle error reconstruction

TL;DR

This work introduces cycle error reconstruction (CER) as a scalable diagnostic to characterize physical error mechanisms relevant to fault-tolerant quantum operations. By learning Pauli eigenvalues via decays in randomized, interleaved easy cycles, CER yields detailed, steerable error descriptions for hard cycles like a transversal CNOT implemented with a 7-qubit Steane code on a 16-qubit trapped-ion processor. The authors extend the approach with marginalization and a Gibbs random field to build a compact, predictive model that connects component-level physics to system- and logical-level performance, including uncorrectable versus correctable errors under the Steane code. They demonstrate that CER, combined with GRF-based joint-distribution reconstruction, can predict logical error rates from physical error terms and identify actionable sources of coherent or miscalibrated noise, illustrating a path toward scalable QEC-aware diagnostics in larger quantum devices.

Abstract

The development of prototype quantum information processors has progressed to a stage where small instances of logical qubit systems perform better than the best of their physical constituents. Advancing towards fault-tolerant quantum computing will require an understanding of the underlying error mechanisms in logical primitives as they relate to the performance of quantum error correction. In this work we demonstrate the novel capability to characterize the physical error properties relevant to fault-tolerant operations via cycle error reconstruction. We illustrate this diagnostic capability for a transversal CNOT, a prototypical component of quantum logical operations, in a 16-qubit register of a trapped-ion quantum computer. Our error characterization technique offers three key capabilities: (i) identifying context-dependent physical layer errors, enabling their mitigation; (ii) contextualizing component gates in the environment of logical operators, validating the performance differences in terms of characterized component-level physics, and (iii) providing a scalable method for predicting quantum error correction performance using pertinent error terms, differentiating correctable versus uncorrectable physical layer errors. The methods with which our results are obtained have scalable resource requirements that can be extended with moderate overhead to capture overall logical performance in increasingly large and complex systems.

Figures (8)

• Figure 1: Cycle benchmarking circuit design erhard2019characterizing leveraged to learn individual Pauli eigenvalues to relative precision. Illustrated in this example is the simpler form that is sufficient for any Clifford hard cycle $\mathcal{H}$ where easy cycles are Pauli randomized cycles $P$carignandugas2023error. The easy cycle $B$ includes local rotations to prepare Pauli product states and define the measurement basis. The randomized Pauli cycle $A$ eliminates bias from measurement errors.
• Figure 2: Circuit representation of how a single Pauli error (red stars) propagates through a single cnot thereby changing its profile. Since errors occur probabilistically it is impossible to distinguish one type of error $P_i$ before the gate from the corresponding error $\mathcal{H}(P_i)$ after the gate.
• Figure 3: Optimization of experimental resources. a The number of distinct experiments to run for a single, two, and seven parallel cnots to obtain all unique orbits is 4 (the eigenstates of elements from the set $A$, for 1-cnot orbits), $4\times9 = 36$ (all combinations of $A$ and $B$, for 2-cnot orbits), and $36\times3 = 108$ (all tilings $C$ of elements from $A$ and $B$, for all possible 2-cnot orbits over 7 cnots), respectively. b Tradeoff between number of randomizations per initial state and shots per randomization for an eigenvalue estimate of $\lambda_{ZX}$. The color scale depicts the standard deviation in estimates of $\lambda_{ZX}$ at the chosen experimental parameters, estimated by subsampling over an experimental data set. In total the data set included 300 randomizations with 650 shots each. In our platform changing the randomization is more costly than repeating a shot. Time in minutes to estimate a fixed initial state using three values for $m$ is indicated by red contours. Total shots required is obtainable by the product of the shots per randomization, total randomizations, the distinct sequence lengths $m$, and the distinct initial states.
• Figure 4: Arrangement of a transversal cnot on a 16-ion register at different levels of abstraction. a An ion chain in a harmonic potential arranges with inter-ion distances growing towards the edges. Pairs of tightly focused laser beams (color gradients of same shade) are used to address ion pairs. Ions at the center are expected to see more crosstalk from neighbors due to tighter packing. Ions 7 and 8 (lighter shade) always idle. b Circuit layer abstraction of the chain with two qubit registers that are logically separate (shading) and are connected via a transversal cnot only. c Gibbs random field abstraction where qubits (blue spheres) possess error correlations only with others if they are connected via nodes in the graph (yellow squares).
• Figure 5: Noise profile of a single cnot embedded in a 16-ion register over different supports. a Heatmap of single-qubit marginal error rates over each idling ion for three distinct single cnot experiments. Qubits that the cnot is supported on are colored as in b. Red boxes highlight, before and after removal, a high $Z$ error on qubit 6 caused by incorrect quadrupole shift compensation. b Comparison of 2-qubit marginal error rates on the single cnot. Braces highlight error terms that are notably distinct across different supports, in particular the dominant error rates for cnot$_{8,9}$ are distinct from those found for cnot$_{1,14}$ which acts on non-adjacent qubits. We find that $ZX$ error terms are often associated with miscalibrations of gate power, as highlighted in Fig. \ref{['fig:Seven CNOT']}. A dominant $ZX$ error term on cnot$^{(*)}_{1,14}$ is reduced in the later experiment cnot$^{(**)}_{1,14}$.