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Kvantfelskorrigering med balanserade produktkoder

Edy Alberto Flores Leal

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

Quantum computing is one of the most promising technological revolutions of the last few decades. Despite its potential, one problem it suffers from is that quantum bits (or qubits), its basic units of information, are intrinsically susceptible to noise. To counter the effects of such noise, we need fault-tolerant quantum computing, and for that, we need a systematic method to protect quantum information. This is the role of quantum error correction (QEC). However, most QEC codes are built on the assumption that X- and Z-type errors occur with equal frequency, which is not always the case for some realistic quantum hardware platforms. Among the several QEC protocols currently developed, surface codes are the leading implementation due to their high thresholds and local connectivity. Previous results have shown that modified variants of these codes, such as the XZZX surface code, can perform well under biased noise. Despite being realizable in two-dimensional real architectures, surface codes have poor encoding rates, leading to large overheads that prevent them from being scalable in fault-tolerant quantum architectures. Quantum low-density parity-check (qLDPC) codes provide a promising alternative, since they can achieve higher encoding rates. In this thesis, we study the code-capacity performance of bias-tailored qLDPC codes under both X- and Z-bias Pauli noise. We construct bias-tailored qLDPC codes, including bivariate bicycle codes, generalized toric codes, and clustered-cyclic codes, by applying Hadamard deformations that transform the Calderbank-Shor-Steane (CSS) codes into non-CSS codes. Using Monte Carlo simulations with minimum-weight perfect matching and belief propagation with localized-statistics decoding, we evaluate the word error rate of these bias-tailored qLDPC codes across different bias regimes. We show that Hadamard-deformed qLDPC codes can yield substantial performance improvements under biased noise and can match or even surpass those performances obtained from XZZX surface code implementations. In particular, we find that generalized toric codes perform consistently the best under bias noise in comparison to the bivariate and clustered-cyclic codes. These results indicate that bias tailoring can be extended to high-rate qLDPC constructions, leading to future work to study their performance in more realistic settings, such as circuit-level noise.

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Quantum computing is one of the most promising technological revolutions of the last few decades. Despite its potential, one problem it suffers from is that quantum bits (or qubits), its basic units of information, are intrinsically susceptible to noise. To counter the effects of such noise, we need fault-tolerant quantum computing, and for that, we need a systematic method to protect quantum information. This is the role of quantum error correction (QEC). However, most QEC codes are built on the assumption that X- and Z-type errors occur with equal frequency, which is not always the case for some realistic quantum hardware platforms. Among the several QEC protocols currently developed, surface codes are the leading implementation due to their high thresholds and local connectivity. Previous results have shown that modified variants of these codes, such as the XZZX surface code, can perform well under biased noise. Despite being realizable in two-dimensional real architectures, surface codes have poor encoding rates, leading to large overheads that prevent them from being scalable in fault-tolerant quantum architectures. Quantum low-density parity-check (qLDPC) codes provide a promising alternative, since they can achieve higher encoding rates. In this thesis, we study the code-capacity performance of bias-tailored qLDPC codes under both X- and Z-bias Pauli noise. We construct bias-tailored qLDPC codes, including bivariate bicycle codes, generalized toric codes, and clustered-cyclic codes, by applying Hadamard deformations that transform the Calderbank-Shor-Steane (CSS) codes into non-CSS codes. Using Monte Carlo simulations with minimum-weight perfect matching and belief propagation with localized-statistics decoding, we evaluate the word error rate of these bias-tailored qLDPC codes across different bias regimes. We show that Hadamard-deformed qLDPC codes can yield substantial performance improvements under biased noise and can match or even surpass those performances obtained from XZZX surface code implementations. In particular, we find that generalized toric codes perform consistently the best under bias noise in comparison to the bivariate and clustered-cyclic codes. These results indicate that bias tailoring can be extended to high-rate qLDPC constructions, leading to future work to study their performance in more realistic settings, such as circuit-level noise.
Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topological codes that we call generalized toric codes (GTCs). We show that these XZZX codes are highly qubit efficient if tailored to biased noise. To characterize the code performance, we use the notion of effective distance, which generalizes code distance to the case of biased noise and constitutes a proxy for the logical failure rate. We find that the XZZX codes can achieve a favorable resource scaling by this metric under biased noise. We also show that the XZZX codes have remarkably high thresholds that reach what is achievable by random codes, and furthermore they can be efficiently decoded using matching decoders. Finally, by adding only one flag qubit, the XZZX codes can realize fault-tolerant QEC while preserving their large effective distance. In combination, our results show that tailored XZZX codes give a resource-efficient scheme for fault-tolerant QEC against biased noise.
We find that the substantial advantages predicted for bias-tailored quantum error correction (QEC) under code-capacity noise are strongly reduced once realistic syndrome extraction and circuit-level noise models are considered. We start by comparing XZZX codes to rectangular surface codes with a bias-dependent optimised anisotropy. Although code-capacity simulations predict an advantage of rectangular surface codes in the limit of high noise bias, this actually disappears under circuit-level noise, making the XZZX codes the preferred and simplest choice even for platforms that allow for a flexible variation of the code layout adapted to changes in noise calibration. Our results identify bias degradation during syndrome extraction under circuit-level noise as the central limitation of biased-tailored QEC. To partially mitigate this effect, we introduce a bias-filtering CNOT gadget that temporarily encodes the ancillary target qubit during syndrome extraction in a repetition code and, upon measurement and feed forward, manages to reduce the bias degradation. In a regime of high-bias and low-idle errors, this bias-filtering gadget yields a few-percent relative improvement of the XZZX code error threshold, demonstrating that lightweight bias-filtering strategies can recover part of the lost bias-tailoring advantage for realistic circuit-level noise.
Quantum computers are predicted to be able to solve problems impossible for a classical supercomputer. The quantum computers that we have now are however very sensitive to noise from their environment, which makes their results unreliable. This necessitates quantum error correction, the main topic of this thesis. Specifically, this thesis concerns quantum Low Density Parity Check (qLDPC) codes for superconducting qubits. Superconducting qubits are one of the ways to implement a quantum computer, used for example by Google and IBM. qLDPC codes are error-correcting codes whose classical counterpart is used in 5G and Wi-Fi, among other applications. They are a type of stabilizer code, meaning that the quantum state we wish to store is subjected to a sequence of non-destructive measurements, whose results can tell if an error has occurred or not. Explaining this more in-depth is the bulk of this thesis. The qLDPC code chosen as most fitting for superconducting qubits is the bivariate bicycle (BB) code. In it, each qubit interacts with its four nearest neighbours as well as with two qubits further away, which allows for fewer qubits to be needed for the same error-correcting performance. The results were collected in the following order: First, different decoders (classical algorithms that determine the most likely error based on the results of the non-destructive measurements mentioned above) are compared to determine which performs best. That turned out to be relay Belief Propagation (BP-relay), which is a version of standard BP that uses information from failed decoding rounds to initialize new rounds. BP-relay is then used to compare BB codes with the surface code, another error-correcting code, with the result that the BB code outperforms the surface code. In addition, the feasibility of implementing BB codes on current hardware is explored, with the error rates of IBM's latest chip showing promising performance, even for much worse non-local connections. This implies that qLDPC codes could be used in the near term to increase the fault tolerance of current quantum computers.
Bias-tailoring allows quantum error correction codes to exploit qubit noise asymmetry. Recently, it was shown that a modified form of the surface code, the XZZX code, exhibits considerably improved performance under biased noise. In this work, we demonstrate that quantum low density parity check codes can be similarly bias-tailored. We introduce a bias-tailored lifted product code construction that provides the framework to expand bias-tailoring methods beyond the family of 2D topological codes. We present examples of bias-tailored lifted product codes based on classical quasi-cyclic codes and numerically assess their performance using a belief propagation plus ordered statistics decoder. Our Monte Carlo simulations, performed under asymmetric noise, show that bias-tailored codes achieve several orders of magnitude improvement in their error suppression relative to depolarising noise.
Identifying the best families of quantum error correction (QEC) codes for near-term experiments is key to enabling fault-tolerant quantum computing. Ideally, such codes should have low overhead in qubit number, high physical error thresholds, and moderate requirements on qubit connectivity to simplify experiments, while allowing for high logical error suppression. Quantum low-density parity-check (LDPC) codes have been recently shown to provide a path towards QEC with low qubit overhead and small logical error probabilities. Here, we demonstrate that when the dominant errors are erasures, as can be engineered in different quantum computing architectures, quantum LDPC codes additionally provide high thresholds and even stronger logical error suppression in parameter regimes that are accessible to current experiments. Using large-scale QEC numerical simulations, we benchmark the performance of two families of high-rate quantum LDPC codes, namely the so-called Clifford-deformed La-cross codes and the bivariate bicycle codes, under a noise model strongly biased towards erasure errors. Both codes outperform the surface code by offering up to orders of magnitude lower logical error probabilities. Interestingly, we find that this decrease in the logical error probability may not be accompanied by an increase in the code threshold, as different QEC codes benefit differently from large erasure fractions. While here we focus on neutral atom qubits, the results also hold for other quantum platforms, such as trapped ions and superconducting qubits, for which erasure conversion has been demonstrated.
Realizing the full potential of quantum computation requires quantum error correction (QEC), with most recent breakthrough demonstrations of QEC using the surface code.QEC codes use multiple noisy physical qubits to encode information in fewer logical qubits, enabling the identification of errors through a decoding process.This process increases the logical fidelity (or accuracy) making the computation more reliable.However, most fast (efficient run-time) decoders neglect important noise characteristics, thereby reducing their accuracy.In this work, we introduce decoders that are both fast and accurate, and can be used with a wide class of QEC codes including the surface code.Our decoders, named belief-matching and belief-find, exploit all noise information and thereby unlock higher accuracy demonstrations of QEC.Using the surface code threshold as a performance metric, we observe a threshold at 0.94% error probability for our decoders, outperforming the 0.82% threshold for a standard minimum-weight perfect matching decoder.We also test our belief-matching decoders in a theoretical case study of codes tailored to a biased noise model.We find that the decoders lead to a much higher threshold and lower qubit overhead in the tailored surface code with respect to the standard, square surface code.Surprisingly, in the wellbelow-threshold regime, the rectangular surface code becomes more resource efficient than the tailored surface code due to a previously unnoticed phenomenon that we call "fragile boundaries."Our decoders outperform all other fast decoders in terms of threshold and accuracy, enabling better results in current quantum-error-correction experiments and opening up new areas for theoretical case studies.