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arXiv:2509.07288

Compressing Syndrome Measurement Sequences

Benjamin Anker, Milad Marvian

Published Sep 8, 2025• 0 citations
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Abstract

In this work, we analyze a framework for constructing fault-tolerant measurement schedules of varying lengths by combining stabilizer generators, and prove results about the distance of such schedules by combining according to classical codes. Using this framework, we produce explicit measurement schedules sufficient for fault-tolerant error correction of quantum codes of distance $d$ with $r$ independent stabilizer generators using only $O(d \log{r})$ measurements if the code is LDPC, and $O(d \log d \log r)$ measurements if the code is produced via concatenating a smaller code with itself $O(\log d)$ times. In both of these cases the number of measurements can be asymptotically fewer than the number of stabilizer generators which define the code. Although optimizing our construction to use the fewest measurements produces high-weight stabilizers, we also show that we can reduce the number of measurements used for specific examples while maintaining low-weight stabilizer measurements. We numerically examine the performance of our construction on the surface code under several noise models and demonstrate the exponential error suppression with increasing distance which is characteristic of weak fault tolerance.

DOI
10.48550/arxiv.2509.07288

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In this work, we analyze a framework for constructing fault-tolerant measurement schedules of varying lengths by combining stabilizer generators, and prove results about the distance of such schedules by combining according to classical codes. Using this framework, we produce explicit measurement schedules sufficient for fault-tolerant error correction of quantum codes of distance ddd with rrr independent stabilizer generators using only O(dlog⁡r)O(d \log{r})O(dlogr) measurements if the code is LDPC, and O(dlog⁡dlog⁡r)O(d \log d \log r)O(dlogdlogr) measurements if the code is produced via concatenating a smaller code with itself O(log⁡d)O(\log d)O(logd) times. In both of these cases the number of measurements can be asymptotically fewer than the number of stabilizer generators which define the code. Although optimizing our construction to use the fewest measurements produces high-weight stabilizers, we also show that we can reduce the number of measurements used for specific examples while maintaining low-weight stabilizer measurements. We numerically examine the performance of our construction on the surface code under several noise models and demonstrate the exponential error suppression with increasing distance which is characteristic of weak fault tolerance.
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We propose the use of certain low-density generator-matrix (LDGM) codes as syndrome measurement (SM) codes for quantum low-density parity check (QLDPC) codes. We use an efficient progressive-edge-growth-like algorithm to create LDGM SM codes with column and row weights that result in measured stabilizers that have constant weight, thus preserving the desirable properties of the underlying QLDPC code. This process allows for control over stabilizer weights and SM code distance, resulting in significantly better performance than repeated syndrome extraction and allowing for both higher distances and fewer syndrome measurements. We implement these SM codes on a distance-5 rotated surface code, and show that this procedure results in a lower probability of logical error. As syndrome measurements performed are a reasonable metric for the time a circuit takes to implement, we conclude that these LDGM codes allow for improved implementation of QLDPC codes without sacrificing the low weights of the syndrome measurements performed.
We reduce the extra qubits needed for two fault-tolerant quantum computing protocols: error correction, specifically syndrome bit measurement, and cat state preparation. For fault-tolerant syndrome extraction, we show an exponential reduction in qubit overhead over the previous best protocol. For a weight-w stabilizer, we demonstrate that stabilizer measurement tolerating one fault (distance-three) needs at most ⌈ log₂ w ⌉ + 1 ancillas. If qubits reset quickly, four ancillas suffice. We also study the preparation of cat states, simple yet versatile entangled states. We prove that the overhead needed for distance-three fault tolerance is only logarithmic in the cat state size. These results could be useful both for near-term experiments with a few qubits, and for the general study of the asymptotic resource requirements of syndrome measurement and state preparation. For 'a' measured flag bits, there are 2^a possible flag patterns that can identify faults. Hence our results come from solving a combinatorial problem: the construction of maximal-length paths in the a-dimensional hypercube, corresponding to maximal-weight stabilizers or maximal-weight cat states.
The high overhead of fault-tolerant measurement sequences (FTMSs) poses a major challenge for implementing quantum stabilizer codes. Here, we address this problem by constructing efficient FTMSs for the class of quantum Hamming codes [ ⁣[2r−1,2r−1−2r,3] ⁣][\![2^r-1, 2^r-1-2r, 3]\!][[2r−1,2r−1−2r,3]] with r=3k+1r=3k+1r=3k+1 (k∈Z+k \in \mathbb{Z}^+k∈Z+). Our key result demonstrates that the sequence length can be reduced to exactly 2r+12r+12r+1-only one additional measurement beyond the original non-fault-tolerant sequence, establishing a tight lower bound. The proposed method leverages cyclic matrix transformations to systematically combine rows of the initial stabilizer matrix and preserving a self-dual CSS-like symmetry analogous to that of the original quantum Hamming codes. This induced symmetry enables hardware-efficient circuit reuse: the measurement circuits for the first rrr stabilizers are transformed into circuits for the remaining rrr stabilizers simply by toggling boundary Hadamard gates, eliminating redundant hardware. For distance-3 fault-tolerant error correction, our approach simultaneously reduces the time overhead via shorting the FTMS length and the hardware overhead through symmetry-enabled circuit multiplexing. These results provide an important advance towards the important open problem regarding the design of minimal FTMSs for quantum Hamming codes and may shed light on similar challenges in other quantum stabilizer codes.
Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requiresΩ(rd2)\Omega (r d^{2})Ω(rd2)parity check measurements for a distance-dddcode defined byrrrparity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using onlyrrrmeasurements. In this work, we consider a phenomenological noise model for parity check measurements assuming that each bit of a codeword and the measurement outcome suffer from independent bit flips with some error rateppp. For this model, we demonstrate that fault-tolerant quantum error correction can be achieved usingO(dlog⁡(d))O(d \log (d))O(dlog(d))measurements for any code with distanced≥Ω(nα)d \geq \Omega (n^\alpha)d≥Ω(nα)for some constantα>0\alpha > 0α>0. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer thanrrrmeasurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code. The short measurement sequences constructed generally have high weight and our phenomenological noise model is not realistic in this regime. Our error correction strategy could find applications to small codes and LDPC codes if one can manage to keep the weight of the measured parity checks low.