Measurement-free quantum error correction optimized for biased noise

TL;DR

The paper develops measurement-free quantum error correction tailored to biased Z-dominated noise—relevant for neutral-atom platforms with Rydberg gates—by using the $[[7,1,3]]$ Steane code and a measurement-free T-gate via magic-state injection. It introduces a reduced error-correction cycle that targets Z-errors (convertible to X-errors via logical Hadamards) and demonstrates fault-tolerance under biased noise, with robustness to imperfect bias. The work provides detailed logical-noise modeling, extended-rectangle analysis, and an implementation path on neutral atoms, showing improved break-even points compared to measurement-based schemes. These building blocks aim to push logical error rates below the threshold of scalable QEC and offer a practical route toward universal, fault-tolerant computation with measurement-free protocols.

Abstract

In this paper, we derive optimized measurement-free protocols for quantum error correction and the implementation of a universal gate set optimized for an error model that is noise biased . The noise bias is adapted for neutral atom platforms, where two- and multi-qubit gates are realized with Rydberg interactions and are thus expected to be the dominating source of noise. Careful design of the gates allows to further reduce the noise model to Pauli-Z errors. In addition, the presented circuits are robust to arbitrary single-qubit gate errors, and we demonstrate that the break-even point can be significantly improved compared to fully fault-tolerant measurement-free schemes. The obtained logical qubits with their suppressed error rates on logical gate operations can then be used as building blocks in a first step of error correction in order to push the effective error rates below the threshold of a fully fault-tolerant and scalable quantum error correction scheme.

Figures (11)

• Figure 1: (a) Exemplary circuit. Our set of basis gates includes controlled-phase gates (in particular the CZ-gate and CS$^\dagger$-gate), H-, X- and T-gates and multi-controlled Z-gates (for example the CCZ-gate). Initially, we assume the noise model defined in \ref{['eq:biasednoise']}, where only Z-type errors appear after two- and multi-qubit gates. (b) The $[[7,1,3]]$-qubit Steane code. The code has distance $d=3$ and is designed to correct bit-flip errors and phase errors on up to one qubit. The code is defined by the stabilizer group generated by the operators shown. The logical $X_L$ and $Z_L$ operators act on one side of the triangle.
• Figure 2: Circuits for error correction and T-gate. (a) Schematic circuit for the error correction scheme. For the second round of stabilizer extraction an additional stabilizer $X_0X_2X_4X_6$ (gray gates) has to be mapped out to avoid combinations of X- and Z-errors. However, this is only required before performing logical T- and S-gates. Otherwise, the extraction of an overcomplete set is not required and the feedback for correcting Z-errors is similar to the feedback for correcting $X$ errors (first green box). We refer to this as reduced error correction cycle. (b) Detailed error correction circuit. (c) Magic state injection. (d) Logical encoding of the magic state. (e) Logical CS-gate with subsequent reset on the control. The detailed circuits for the encoding of the magic state and the logical CS-gate with subsequent reset are shown in \ref{['app:circuits']}. The operation $R$ describes qubit reset.
• Figure 3: Characterization of logical noise model for the error correction cycle, encoding of $\ket{0_L}$, a logical CZ-gate and a logical T-gate. Here and in the following, the dashed line represents the physical error rate $p$, to indicate the break-even point. For the encoding and the logical gates the dark lines refer to the bare gate and the light lines to the gate surrounded by error correction cycles (extended rectangles). The points show the numerically simulated data and lines are the analytical estimates for the logical error rates (see \ref{['app:lognoisemodel']}). The bare logical CZ- and T-gate can only give rise to Z-errors. In principle, for the logical CZ-gate and the T-gate surrounded by QEC one also finds X- and Y-type errors. X- and Y-type errors for the extended CZ- and T-gate and the noise model for extended H- and S-gate are shown in \ref{['app:lognoisemodel']}.
• Figure 4: Comparison of the measurement-free QEC-cycle and the T-gate with measurement based schemes. In the green region, the measurement-free schemes have lower logical error rates than the measurement-based version. In the red region measurement-based schemes perform better. The error rates are obtained by summing over the error rates of input states $\ket{0_L}$ and $\ket{+_L}$.
• Figure 5: (a) Exemplary determination of the break-even points for the QEC-circuit. (b) Break-even points for the QEC-circuit and the T-gate for different error rates $\alpha=p_1/p$ of the single-qubit gates. The lines correspond to fits of the form of \ref{['eq:break_even']}.