arXiv subjects

Christophe Vuillot

Publications and source records attributed to Christophe Vuillot.

At least 19 recordsLinked to original sources

arXiv2026

Algebraic structure of Tiger codes

Tiger codes form a family of multimode bosonic quantum codes that unify several previously known constructions, including cat, paircat, and the two-mode binomial code. In this work, we give a rigorous algebraic treatment of these codes. Starting from a kernel definition of the codespace, we prove that the annihilation-type constraints admit a finite generating set, construct an explicit orthonormal basis, and show that the logical structure of the code is governed by the homology of an underlying chain complex, as expected in the original work on Tiger codes of Xu et al. We then develop a Fourier transform over the codespace to prove that the span of phase-rotated projected coherent states is dense therein, and to yield dual $X$- and $Z$-type descriptions of the code. We further extend the framework to non-linear number constraints, encompassing codes such as the four-legged cat or the repetition cat code. Finally, we investigate the implementation of logical operations. We first generalise the construction of logical Pauli operators proposed by Xu et al. to arbitrary logical spaces, and then construct non-Clifford gates using physical polynomial phase rotations of the form $e^{iP(\hat{\boldsymbol{n}})}$. We derive criteria on the real polynomial $P$ which, for positive single-logical-qubit Tiger codes satisfying an additional sign assumption, such as the paircat code, characterise the polynomials $P$ that preserve the codespace by decomposing them into a family of univariate polynomials. Through this decomposition, we relate the degrees of the resulting components to the induced logical action in the Clifford hierarchy. These results establish Tiger codes as a mathematically robust framework for describing a broad class of bosonic encodings.

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arXiv2026

Biased-noise qubits: a guide to efficient fault-tolerance using the hierarchy of errors

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.

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arXiv2026

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

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 $\eta = 10^4$. 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.

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arXiv2025

Unfolded distillation: very low-cost magic state preparation for biased-noise qubits

Magic state distillation enables universal fault-tolerant quantum computation by implementing non-Clifford gates via the preparation of high-fidelity magic states. However, it comes at the cost of substantial logical-level overhead in both space and time. In this work, we propose a very low-cost magic state distillation scheme for biased-noise qubits. By leveraging the noise bias, our scheme enables the preparation of a magic state with a logical error rate of $3 \times 10^{-7}$, using only 53 qubits and 5.5 error correction rounds, under a noise bias of $\eta \gtrsim 5 \times 10^6$ and a phase-flip noise rate of $0.1\%$. This reduces the circuit volume by more than one order of magnitude relative to magic state cultivation for unbiased-noise qubits and by more than two orders of magnitude relative to standard magic state distillation. Moreover, our scheme provides three key advantages over previous proposals for biased-noise qubits. First, it only requires nearest-neighbor two-qubit gates on a 2D lattice. Second, the logical fidelity remains nearly identical even at a more modest noise bias of $\eta \gtrsim 80$, at the cost of a slightly increased circuit volume. Third, the scheme remains effective even at high physical phase-flip rates, in contrast to previously proposed approaches whose circuit volume grows exponentially with the error rate. Our construction is based on unfolding the $X$ stabilizer group of the Hadamard 3D quantum Reed-Muller code in 2D, enabling distillation at the physical level rather than the logical level, and is therefore referred to as $\textit{unfolded}$ distillation.

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arXiv2025

High-performance local decoders for defect matching in 1D

Local decoders, also known as cellular-automaton decoders, offer a promising path toward real-time quantum error correction by replacing centralized classical decoding, with inherent hardware constraints, by a natively parallel and streamlined architecture from a simple local transition rule. We propose two new types of local decoders for the quantum repetition code in one dimension. The signal-rule decoders interpret odd parities between neighboring qubits as defects, attracted to each other via the exchange of classical point-like excitations, represented by a few bits of local memory. We prove the existence of a threshold in the code-capacity model and present numerical evidence of exponential logical error suppression under a phenomenological noise model, with data and measurement errors at each error correction cycle. Compared to previously known local decoders that suffer from sub-optimal threshold and scaling, our construction significantly narrows the gap with global decoders for practical system sizes and error rates. Implementation requirements can be further reduced by eliminating the need for local classical memories, with a new rule defined on two rows of qubits. This shearing-rule works well at relevant system sizes making it an appealing short-term solution. When combined with biased-noise qubits, such as cat qubits, these decoders enable a fully local quantum memory in one dimension.

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arXiv2025

A graph-state based synthesis framework for Clifford isometries

We tackle the problem of Clifford isometry compilation, i.e, how to synthesize a Clifford isometry into an executable quantum circuit. We propose a simple framework for synthesis that only exploits the elementary properties of the Clifford group and one equation of the symplectic group. We highlight the versatility of our framework by showing that several normal forms of the literature are natural corollaries. We recover the state of the art two-qubit gate depth necessary for the execution of a Clifford circuit on an LNN architecture, concomitantly with another work. We also propose practical synthesis algorithms for Clifford isometries with a focus on Clifford operators, graph states and codiagonalization of Pauli rotations. Benchmarks show that in all three cases we improve the 2-qubit gate count and depth of random instances compared to the state-of-the-art methods. We also improve the execution of practical quantum chemistry experiments.

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arXiv2024

Fault-Tolerant Constant-Depth Clifford Gates on Toric Codes

We propose and simulate the performance of a set of fault-tolerant and constant-depth logical gates on 2D toric codes. This set combines fold-transversal gates, Dehn twists and single-shot logical Pauli measurements and generates the full Clifford group.

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arXiv2024

Homological Quantum Rotor Codes: Logical Qubits from Torsion

We formally define homological quantum rotor codes which use multiple quantum rotors to encode logical information. These codes generalize homological or CSS quantum codes for qubits or qudits, as well as linear oscillator codes which encode logical oscillators. Unlike for qubits or oscillators, homological quantum rotor codes allow one to encode both logical rotors and logical qudits in the same block of code, depending on the homology of the underlying chain complex. In particular, a code based on the chain complex obtained from tessellating the real projective plane or a Möbius strip encodes a qubit. We discuss the distance scaling for such codes which can be more subtle than in the qubit case due to the concept of logical operator spreading by continuous stabilizer phase-shifts. We give constructions of homological quantum rotor codes based on 2D and 3D manifolds as well as products of chain complexes. Superconducting devices being composed of islands with integer Cooper pair charges could form a natural hardware platform for realizing these codes: we show that the $0$-$π$-qubit as well as Kitaev's current-mirror qubit -- also known as the Möbius strip qubit -- are indeed small examples of such codes and discuss possible extensions.

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arXiv2024

Letting the tiger out of its cage: bosonic coding without concatenation

Continuous-variable cat codes are encodings into a single photonic or phononic mode that offer a promising avenue for hardware-efficient fault-tolerant quantum computation. Protecting information in a cat code requires measuring the mode's occupation number modulo two, but this can be relaxed to a linear occupation-number constraint using the alternative two-mode pair-cat encoding. We construct multimode codes with similar linear constraints using any two integer matrices satisfying a CSS-like homological condition of a quantum rotor code. Just like the pair-cat code, syndrome extraction can be performed in tandem with stabilizing dissipation using current superconducting-circuit designs. The framework includes codes with various finite- or infinite-dimensional codespaces, and codes with finite or infinite Fock-state support. It encompasses two-component cat, pair-cat, dual-rail, two-mode binomial, various bosonic repetition codes, and aspects of chi-squared encodings while also yielding codes from homological products, lattices, generalized coherent states, and algebraic varieties. Among our examples are analogues of repetition codes, the Shor code, and a surface-like code that is not a concatenation of a known cat code with the qubit surface code. Codewords are coherent states projected into a Fock-state subspace defined by an integer matrix, and their overlaps are governed by Gelfand-Kapranov-Zelevinsky hypergeometric functions.

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arXiv2024

Optimal number of parametrized rotations and Hadamard gates in parametrized Clifford circuits with non-repeated parameters

We present an efficient algorithm to reduce the number of non-Clifford gates in quantum circuits and the number of parametrized rotations in parametrized quantum circuits. The method consists in finding rotations that can be merged into a single rotation gate. This approach has already been considered before and is used as a pre-processing procedure in many optimization algorithms, notably for optimizing the number of Hadamard gates or the number of $T$ gates in Clifford$+T$ circuits. Our algorithm has a better complexity than similar methods and is particularly efficient for circuits with a low number of internal Hadamard gates. Furthermore, we show that this approach is optimal for parametrized circuits composed of Clifford gates and parametrized rotations with non-repeated parameters. For the same type of parametrized quantum circuits, we also prove that a previous procedure optimizing the number of Hadamard gates and internal Hadamard gates is optimal. This procedure is notably used in our low-complexity algorithm for optimally reducing the number of parametrized rotations.

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arXiv2024

Robust sparse IQP sampling in constant depth

Between NISQ (noisy intermediate scale quantum) approaches without any proof of robust quantum advantage and fully fault-tolerant quantum computation, we propose a scheme to achieve a provable superpolynomial quantum advantage (under some widely accepted complexity conjectures) that is robust to noise with minimal error correction requirements. We choose a class of sampling problems with commuting gates known as sparse IQP (Instantaneous Quantum Polynomial-time) circuits and we ensure its fault-tolerant implementation by introducing the tetrahelix code. This new code is obtained by merging several tetrahedral codes (3D color codes) and has the following properties: each sparse IQP gate admits a transversal implementation, and the depth of the logical circuit can be traded for its width. Combining those, we obtain a depth-1 implementation of any sparse IQP circuit up to the preparation of encoded states. This comes at the cost of a space overhead which is only polylogarithmic in the width of the original circuit. We furthermore show that the state preparation can also be performed in constant depth with a single step of feed-forward from classical computation. Our construction thus exhibits a robust superpolynomial quantum advantage for a sampling problem implemented on a constant depth circuit with a single round of measurement and feed-forward.

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arXiv2024

Optimal Hadamard gate count for Clifford$+T$ synthesis of Pauli rotations sequences

The Clifford$+T$ gate set is commonly used to perform universal quantum computation. In such setup the $T$ gate is typically much more expensive to implement in a fault-tolerant way than Clifford gates. To improve the feasibility of fault-tolerant quantum computing it is then crucial to minimize the number of $T$ gates. Many algorithms, yielding effective results, have been designed to address this problem. It has been demonstrated that performing a pre-processing step consisting of reducing the number of Hadamard gates in the circuit can help to exploit the full potential of these algorithms and thereby lead to a substantial $T$-count reduction. Moreover, minimizing the number of Hadamard gates also restrains the number of additional qubits and operations resulting from the gadgetization of Hadamard gates, a procedure used by some compilers to further reduce the number of $T$ gates. In this work we tackle the Hadamard gate reduction problem, and propose an algorithm for synthesizing a sequence of $π/4$ Pauli rotations with a minimal number of Hadamard gates. Based on this result, we present an algorithm which optimally minimizes the number of Hadamard gates lying between the first and the last $T$ gate of the circuit.

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arXiv2024

LDPC-cat codes for low-overhead quantum computing in 2D

Quantum low-density parity-check (qLDPC) codes are a promising construction for drastically reducing the overhead of fault-tolerant quantum computing (FTQC) architectures. However, all of the known hardware implementations of these codes require advanced technologies, such as long-range qubit connectivity, high-weight stabilizers, or multi-layered chip layouts. An alternative approach to reduce the hardware overhead of fault-tolerance is to use bosonic cat qubits where bit-flip errors are exponentially suppressed by design. In this work, we combine both approaches and propose an architecture based on cat qubits concatenated in classical LDPC codes correcting for phase-flips. We find that employing such phase-flip LDPC codes provides two major advantages. First, the hardware implementation of the code can be realised using short-range qubit interactions in 2D and low-weight stabilizers, which makes it readily compatible with current superconducting circuit technologies. Second, we demonstrate how to implement a fault-tolerant universal set of logical gates with a second layer of cat qubits while maintaining the local connectivity. We conduct a numerical brute force optimisation of these classical codes to find the ones with the best encoding rate for algorithmically relevant code distances. We discover that some of the best codes benefit from a cellular automaton structure. This allows us to define families of codes with high encoding rates and distances. Finally, we numerically assess the performance of our codes under circuit-level noise. Assuming a physical phase-flip error probability $\epsilon \approx 0.1\%$, our $[165+8\ell, 34+2\ell, 22]$ code family allows to encode $100$ logical qubits with a total logical error probability (including both logical phase-flip and bit-flip) per cycle and per logical qubit $\epsilon_L \leq 10^{-8}$ on a $758$ cat qubit chip.

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arXiv2022

Quantum XYZ Product Codes

We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved. Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as $Θ(N^{2/3})$. While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.

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arXiv2022

Quantum Pin Codes

We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a generalization of quantum color codes and Reed-Muller codes and share a lot of their structure and properties. Pin codes have gauge operators, an unfolding procedure and their stabilizers form so-called $\ell$-orthogonal spaces meaning that the joint overlap between any $\ell$ stabilizer elements is always even. This last feature makes them interesting for devising magic-state distillation protocols, for instance by using puncturing techniques. We study examples of these codes and their properties.

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arXiv2021

Planar Floquet Codes

A protocol called the "honeycomb code", or generically a "Floquet code", was introduced by Hastings and Haah in \cite{hastings_dynamically_2021}. The honeycomb code is a subsystem code based on the honeycomb lattice with zero logical qubits but such that there exists a schedule for measuring two-body gauge checks leaving enough room at all times for two protected logical qubits. In this work we show a way to introduce boundaries to the system which curiously presents a rotating dynamics but has constant distance and is therefore not fault-tolerant.

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arXiv2020

Towards Scalable Bosonic Quantum Error Correction

We review some of the recent efforts in devising and engineering bosonic qubits for superconducting devices, with emphasis on the Gottesman-Kitaev-Preskill (GKP) qubit. We present some new results on decoding repeated GKP error correction using finitely-squeezed GKP ancilla qubits, exhibiting differences with previously studied stochastic error models. We discuss circuit-QED ways to realize CZ gates between GKP qubits and we discuss different scenario's for using GKP and regular qubits as building blocks in a scalable superconducting surface code architecture.

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arXiv2018

Code Deformation and Lattice Surgery Are Gauge Fixing

The large-scale execution of quantum algorithms requires basic quantum operations to be implemented fault-tolerantly. The most popular technique for accomplishing this, using the devices that can be realised in the near term, uses stabilizer codes which can be embedded in a planar layout. The set of fault-tolerant operations which can be executed in these systems using unitary gates is typically very limited. This has driven the development of measurement-based schemes for performing logical operations in these codes, known as lattice surgery and code deformation. In parallel, gauge fixing has emerged as a measurement-based method for performing universal gate sets in subsystem stabilizer codes. In this work, we show that lattice surgery and code deformation can be expressed as special cases of gauge fixing, permitting a simple and rigorous test for fault-tolerance together with simple guiding principles for the implementation of these operations. We demonstrate the accuracy of this method numerically with examples based on the surface code, some of which are novel.

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