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arXiv:2605.24262

Quantum non-demolition measurements as a practical primitive for fault-tolerant computation against biased noise

Christophe Vuillot, Diego Ruiz, Jérémie Guillaud, Mazyar Mirrahimi

quant-phSubmitted May 22, 2026• Announced May 26, 2026• 0 citations
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Abstract

Leveraging noise bias, where phase-flip errors dominate over bit-flips, can drastically reduce the hardware overhead of fault-tolerant quantum computation, but existing approaches require bias-preserving CNOT gates whose implementation remains experimentally challenging and is provably impossible for strictly two-dimensional systems. We show that high-fidelity quantum non-demolition (QND) multi-qubit Pauli $Z$ measurements provide an equally powerful yet more accessible primitive. We demonstrate that such measurements can fully replace bias-preserving CNOT gates for compiling all operations required by bias-tailored error correction, including stabilizer measurements for repetition codes, XZZX surface codes, and LDPC codes. We propose concrete physical implementations of this primitive for two platforms: solid-state nuclear spins coupled to electron spin ancillas, and dissipatively stabilized superconducting cat qubits. Through circuit-level numerical simulations, we show that an asymmetric XZZX surface code implemented with weight-four QND $Z$ measurements achieves a phase-flip threshold of $\sim\!1.25\%$ and provides a qubit overhead reduction of up to $6\times$ compared to a bias-unaware surface code at noise bias $η= 10^4$. In the regime of very large bias, a repetition code with QND $Z$ measurements attains a threshold of $\sim\!2.3\%$ and achieves overhead comparable to that of a bias-preserving CNOT scheme, without requiring such a gate. Our results establish QND multi-$Z$ measurements as a practical and hardware-efficient route to fault-tolerant quantum computation for a broad class of biased-noise platforms.

Comments
18 pages, 20 figures, 2 tables

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Abstract We show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favorable sub-threshold resource scaling that can be obtained by specializing a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation. We finally suggest some small-scale experiments that could exploit noise bias to reduce qubit overhead in two-dimensional architectures

•Oct 23, 2020•1 citation
Leveraging noise bias, where phase-flip errors dominate over bit-flips, can drastically reduce the hardware overhead of fault-tolerant quantum computation, but existing approaches require bias-preserving CNOT gates whose implementation remains experimentally challenging and is provably impossible for strictly two-dimensional systems. We show that high-fidelity quantum non-demolition (QND) multi-qubit Pauli ZZZ measurements provide an equally powerful yet more accessible primitive. We demonstrate that such measurements can fully replace bias-preserving CNOT gates for compiling all operations required by bias-tailored error correction, including stabilizer measurements for repetition codes, XZZX surface codes, and LDPC codes. We propose concrete physical implementations of this primitive for two platforms: solid-state nuclear spins coupled to electron spin ancillas, and dissipatively stabilized superconducting cat qubits. Through circuit-level numerical simulations, we show that an asymmetric XZZX surface code implemented with weight-four QND ZZZ measurements achieves a phase-flip threshold of ∼ ⁣1.25%\sim\!1.25\%∼1.25% and provides a qubit overhead reduction of up to 6×6\times6× compared to a bias-unaware surface code at noise bias η=104η= 10^4η=104. In the regime of very large bias, a repetition code with QND ZZZ measurements attains a threshold of ∼ ⁣2.3%\sim\!2.3\%∼2.3% and achieves overhead comparable to that of a bias-preserving CNOT scheme, without requiring such a gate. Our results establish QND multi-ZZZ measurements as a practical and hardware-efficient route to fault-tolerant quantum computation for a broad class of biased-noise platforms.
Qubits with strongly biased noise, in which phase-flip errors are orders of magnitude more frequent than bit-flips, arise both naturally, as in electron and nuclear spins, and by engineering, as in stabilized cat qubits. This noise structure holds the promise of reducing the daunting hardware overhead of fault-tolerant quantum computing, but exploiting it requires physical operations that do not convert frequent phase-flips into rare bit-flips. In this review, we analyze the most prominent fault-tolerant protocols for biased-noise qubits, organized according to the available set of such bias-preserving operations. When this set is restricted to the CZ gate together with preparation and measurement in the X basis, we show that the complexity of the required syndrome extraction gadgets essentially cancels the benefit of the noise bias: at experimentally relevant error rates, one may as well ignore the bias and rely on standard error correction designed for depolarizing noise. The situation changes drastically when a bias-preserving CX gate is available: the hierarchy of errors can then be reflected in the structure of the code, with frequent phase-flips corrected by a dedicated high-threshold code and rare bit-flips by concatenation with a high-rate code. The same hierarchy also enables hardware-efficient preparation of magic states. Finally, as a bias-preserving CX is forbidden in naturally biased platforms and challenging in engineered ones, we present a measurement-based architecture in which a high-fidelity quantum non-demolition readout of multi-qubit Pauli Z operators takes its place, extending these overhead reductions to a much broader range of physical platforms.
Biased-noise qubits, in which one type of error (e.g. XXX- and YYY-type errors) is significantly suppressed relative to the other (e.g. ZZZ-type errors), can significantly reduce the overhead of quantum error correction. Codes such as the rectangular surface code or XZZX code substantially reduce the qubit overhead under biased noise, but they still face challenges. The rectangular surface code suffers from a relatively low threshold, while the XZZX code requires twice as many physical qubits to maintain the same code distance as the surface code. In this work, we introduce a 2D local code construction that outperforms these codes for noise biases η≥7×104η\ge 7\times10^{4}η≥7×104, reducing the qubit overhead by over 50% at pZ=10−3p_Z=10^{-3}pZ​=10−3 and η=2×106η= 2 \times 10^6η=2×106 to achieve a logical error rate of 10−1210^{-12}10−12. Our construction relies on the concatenation of two classical codes. The inner codes are repetition phase-flip codes while the outer codes are high-rate bit-flip codes enabled by their implementation at the logical level, which circumvents device connectivity constraints. These results indicate that under sufficiently biased noise, it is advantageous to address phase-flip and bit-flip errors at different layers of the coding scheme. The inner code should prioritize a high threshold for phase-flip errors, while the bit-flip outer code should optimize for encoding rate efficiency. In the strong biased-noise regime, high-rate outer codes keep the overhead for correcting residual bit-flip errors comparable to that of the repetition code itself, meaningfully lower than that required by earlier approaches.
The requirements for fault-tolerant quantum error correction can be simplified by leveraging structure in the noise of the underlying hardware.In this work, we identify a new type of structured noise motivated by neutral-atom qubits, biased erasure errors, which arises when qubit errors are dominated by detectable leakage from only one of the computational states of the qubit.We study the performance of this model using gate-level simulations of the XZZX surface code.Using the predicted erasure fraction and bias of metastable 171 Yb qubits, we find a threshold of 8.2% for two-qubit gate errors, which is 1.9 times higher than the threshold for unbiased erasures and 7.5 times higher than the threshold for depolarizing errors.Surprisingly, the improved threshold is achieved without bias-preserving controlled-NOT gates and, instead, results from the lower noise entropy in this model.We also introduce an XZZX cluster state construction for measurement-based error correction, hybrid fusion, that is optimized for this noise model.By combining fusion operations and deterministic entangling gates, this construction preserves the intrinsic symmetry of the XZZX code, leading to a higher threshold of 10.3% and enabling the use of rectangular codes with fewer qubits.We discuss a potential physical implementation using a single plane of atoms and movable tweezers.
Noise-biased qubits have bit-flip errors that are exponentially suppressed relative to phase-flip errors, and offer a promising route toward fault-tolerant quantum computing. However, this bias can be compromised during gate operations with non-biased control qubits. To address this limitation, we propose a "0-n" Fock qubit architecture that maintains the noise bias by encoding information in the ground state and n-th excited state of a nonlinear multi-level system, such as a transmon. This encoding is achieved via a dissipative stabilization that acts as decay and gain for lower and upper intermediate levels, respectively. We first analytically demonstrate that bit-flip errors are exponentially suppressed with the number of levels. Then, we present a practical implementation using a multi-mode lossy filter to achieve the frequency-selective dissipation. Finally, we numerically demonstrate that bit-flip probabilities approaching 10−810^{-8}10−8 are achievable for controlled-X gates on cat qubits using the 0-n qubit as an ancilla with n≥9n \geq 9n≥9 (i.e. ten or more levels), for realistic experimental parameters. Building on this, we simulate syndrome extraction in a repetition code, achieving logical error rates in the megaquop regime with only a distance of d=9d=9d=9.
Bosonic cat qubits stabilized by two-photon driven dissipation benefit from exponential suppression of bit-flip errors and an extensive set of gates preserving this protection. These properties make them promising building blocks of a hardware-efficient and fault-tolerant quantum processor. In this paper, we propose a performance optimization of the repetition cat code architecture using fast but noisy CNOT gates for stabilizer measurements. This optimization leads to high thresholds for the physical figure of merit, given as the ratio between intrinsic single-photon loss rate of the bosonic mode and the engineered two-photon loss rate, as well as an improved scaling below threshold of the required overhead, to reach an expected level of logical error rate. Relying on the specific error models for cat qubit operations, this optimization exploits fast parity measurements, using accelerated low-fidelity CNOT gates, combined with fast ancilla parity-check qubits. The significant enhancement in the performance is explained by: 1- the highly asymmetric error model of cat qubit CNOT gates with a major component on control (ancilla) qubits, and 2- the robustness of the repetition cat code error correction performance in presence of the leakage induced by fast operations. In order to demonstrate these performances, we develop a method to sample the repetition code under circuit-level noise that also takes into account cat qubit state leakage.
Abstract We show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favorable sub-threshold resource scaling that can be obtained by specializing a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation. We finally suggest some small-scale experiments that could exploit noise bias to reduce qubit overhead in two-dimensional architectures