Is error detection helpful on IBM 5Q chips ?

This paper reports on experiments realized on several IBM 5Q chips which show evidence for the advantage of using error detection and fault-tolerant design of quantum circuits. We show an average improvement of the task of sampling from states that can be fault-tolerantly prepared in the $[[4,2,2]]$ code, when using a fault-tolerant technique well suited to the layout of the chip. By showing that fault-tolerant quantum computation is already within our reach, the author hopes to encourage this approach.Comment: 17 pages, 13 figures, 6 table

Sparse spectral approximations for computing polynomial functionals

We give a new fast method for evaluating sprectral approximations of nonlinear polynomial functionals. We prove that the new algorithm is convergent if the functions considered are smooth enough, under a general assumption on the spectral eigenfunctions that turns out to be satisfied in many cases, including the Fourier and Hermite basis

Quantum Pin Codes

arXiv: 1906.11394We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a vast generalization of quantum color codes and Reed-Muller codes. A lot of the structure and properties of color codes carries over to pin codes. Pin codes have gauge operators, an unfolding procedure and their stabilizers form multi-orthogonal spaces. This last feature makes them interesting for devising magic-state distillation protocols. We study examples of these codes and their properties

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

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

Quantum Error Correction with the Toric-GKP Code

We examine the performance of the single-mode GKP code and its concatenation with the toric code for a noise model of Gaussian shifts, or displacement errors. We show how one can optimize the tracking of errors in repeated noisy error correction for the GKP code. We do this by examining the maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean path-integral modeling a particle in a random cosine potential. We demonstrate the efficiency of a minimum-energy decoding strategy as a proxy for the path integral evaluation. In the second part of this paper, we analyze and numerically assess the concatenation of the GKP code with the toric code. When toric code measurements and GKP error correction measurements are perfect, we find that by using GKP error information the toric code threshold improves from $10\%$ to $14\%$. When only the GKP error correction measurements are perfect we observe a threshold at $6\%$. In the more realistic setting when all error information is noisy, we show how to represent the maximum likelihood decoding problem for the toric-GKP code as a 3D compact QED model in the presence of a quenched random gauge field, an extension of the random-plaquette gauge model for the toric code. We present a new decoder for this problem which shows the existence of a noise threshold at shift-error standard deviation $\sigma_0 \approx 0.243$ for toric code measurements, data errors and GKP ancilla errors. If the errors only come from having imperfect GKP states, this corresponds to states with just 4 photons or more. Our last result is a no-go result for linear oscillator codes, encoding oscillators into oscillators. For the Gaussian displacement error model, we prove that encoding corresponds to squeezing the shift errors. This shows that linear oscillator codes are useless for quantum information protection against Gaussian shift errors.Comment: 50 pages, 14 figure

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, depending on the homology of the underlying chain complex. In particular, such a code based on the chain complex obtained from tessellating the real projective plane or a M\"{o}bius 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 $0$-$\pi$-qubit as well as Kitaev's current-mirror qubit -- also known as the M\"{o}bius strip qubit -- are indeed small examples of such codes and discuss possible extensions.Comment: 47 pages, 10 figures, 2 table

Optimal Hadamard gate count for Clifford+T+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 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

Simulation fidèle du bruit généré par un jet supersonique chaud avec déclenchement de la turbulence et propagation acoustique non-linéaire

International audienceA procedure to accurately simulate a free hot supersonic jet and its associated noise, which uses simultaneously a turbulence tripping method and a two-way coupling between a flow solver and a nonlinear acoustic solver, is proposed in this study. A Mach 3.1 overexpanded hot jet is computed via a large-eddy simulation by solving the filtered Navier-Stokes equations with a finite volume method on unstructured grids. The resulting noise is propagated in the far field by solving the full Euler equations with a high-order discontinuous Galerkin methodon unstructured grids. The full convergent-divergent nozzle is explicitly included in the computational domain thanks to the unstructured flow solver. Both a refined grid and a geometrical boundary layer tripping in the convergent are used to get highly disturbed turbulent conditions at the nozzle lips. The flow field appears to agree with the expected turbulence behavior and the available experimental data. The jet development shows significant improvement compared to similar past simulations. The far field acoustic levels are finely recovered at most of observation angles. An analysis of the acoustic near and far fields is then performed. The studied conditions lead to strong shock-associated noise and Mach wave emission. The spatio-frequency and azimuthal content of the acoustic field is described in order to identify the main noise properties. A particular noise component, different from screech tones and radiating upstream like Mach waves, is highlighted. Nonlinear propagation effects are finally quantified through specific metrics. They are found significant in both the near and the far fields which justifies the use of a nonlinear acoustic solver.Une procédure de simulation fidèle d'un jet libre supersonique chaud et du bruit associé, utilisant simultanément une méthode de déclenchement de la turbulence et un couplage fort entre un solveur Navier-Stokes pour l'écoulement et un solveur Euler pour l'acoustique non-linéaire, est proposée dans cette étude. Un jet chaud sur-détendu à Mach 3.1 est calculé au moyen d'une simulation aux grandes échelles par résolution des équations de Navier-Stokes filtrées avec une méthode de type volumes finis sur maillage non-structuré. Le bruit résultant est propagé en champ lointain par résolution des équations d'Euler complètes avec une méthode de Galerkin discontinue d'ordre élevé sur maillage non-structuré. La tuyère convergente-divergente est explicitement incluse dans le domaine de calcul et son maillage est facilité par l'approche non-structurée. Pour obtenir des conditions d'écoulement fortement perturbées au niveau des lèvres de la tuyère, on utilise à la fois un maillage raffiné et une méthode de déclenchement géométrique de la couche limite dans le convergent. Le champ aérodynamique obtenu correspond au comportement turbulent attendu et aux données expérimentales disponibles. Le développement du jet présente une amélioration significative par rapport aux simulations précédentes similaires. Les niveaux acoustiques en champ lointain sont retrouvés avec précision à la plupart des angles d'observation. Une analyse des champs acoustiques proche et lointain est ensuite effectuée. Les conditions étudiées conduisent à un fort bruit de choc et à l'émission d'ondes de Mach. Le contenu spatio-fréquentiel et azimutal du champ acoustique est décrit afin d'identifier les principales propriétés du bruit. Une composante de bruit particulière, distincte du screech et rayonnant en amont via un mécanisme similaire aux ondes de Mach, est mise en évidence. Les effets de propagation non-linéaires sont finalement quantifiés par des métriques spécifiques. Leurs variations significatives à la fois en champ proche et en champ lointain justifient l'utilisation d'un solveur acoustique non-linéaire