Quantum Error Correction Research Articles

A Bridge-based Algorithm for Simultaneous Primal and Dual Defects Compression on Topologically Quantum-error-corrected Circuits

Topological quantum error correction (TQEC) using the surface code is among the most promising techniques for fault-tolerant quantum circuits. The required resource of a TQEC circuit can be modeled as a space-time volume of a three-dimensional diagram by describing the defect movement along the time axis. For large-scale complex problems, it is crucial to minimize the space-time volume for a quantum algorithm with a reasonable physical qubit number and computation time. Previous work proposed an automated tool for bridge compression on a large-scale TQEC circuit. However, the existing automated bridge compression is only for dual defects and not for primal defects. This paper presents an algorithm to simultaneously perform bridge compression on primal and dual defects. In addition, the automatic compression algorithm performs initialization/measurement simplification and flipping to improve the compression. Compared with the state-of-the-art work, experimental results show that our proposed algorithm can averagely reduce space-time volumes by 53%.

Statistical Testing of Quantum Programs via Fixed-Point Amplitude Amplification

We present a new technique for accelerating quantum program testing. Given a quantum circuit with an input/output specification, our goal is to check whether executing the program on the input state produces the expected output. In quantum computing, however, it is impossible to directly check the equivalence of the two quantum states. Instead, we rely on statistical testing, which involves repeated program executions, state measurements, and subsequent comparisons with the specified output. To guarantee a high level of assurance, however, this method requires an extensive number of measurements. In this paper, we propose a solution to alleviate this challenge by adapting Fixed-Point Amplitude Amplification (FPAA) for quantum program testing. We formally present our technique, demonstrate its ability to reduce the required number of measurements as well as runtime cost without sacrificing the original statistical guarantee, and showcase its runtime effectiveness through case studies.

Quantum Cellular Automata for Quantum Error Correction and Density Classification.

Quantum cellular automata are alternative quantum-computing paradigms to quantum Turing machines and quantum circuits. Their working mechanisms are inherently automated, therefore measurement free, and they act in a translation invariant manner on all cells or qudits of a register, generating a global rule that updates cell states locally, i.e., based solely on the states of their neighbors. Although desirable features in many applications, it is generally not clear to which extent these fully automated discrete-time local updates can generate and sustain long-range order in the (noisy) systems they act upon. In particular, whether and how quantum cellular automata can perform quantum error correction remain open questions. We close this conceptual gap by proposing quantum cellular automata with quantum-error-correction capabilities. We design and investigate two (quasi)one dimensional quantum cellular automata based on known classical cellular-automata rules with density-classification capabilities, namely the local majority voting and the two-line voting. We investigate the performances of those quantum cellular automata as quantum-memory components by simulating the number of update steps required for the logical information they act upon to be afflicted by a logical bit flip. The proposed designs pave a way to further explore the potential of new types of quantum cellular automata with built-in quantum-error-correction capabilities.

CONTENTS OF TRAINING IN CONSTRUCTING CONTROLLED QUANTUM GATES

The article makes a selection and subsequent structuring of the content of training in the construction of controlled quantum gates using twelve decompositions of single-qubit quantum gates (Z-Y, Z-X, X-Y, Y-Z, X-Z, Y-X; Z-Y-X, Z-X-Y, X-Y-Z, X-Z-Y, Y-X-Z, Y-Z-X) used in the process of teaching the section “Quantum Computing and Programming in the QCL Language” of the training course “Applied Programming in High-Level Languages” to students of the Institute of Information Technology and Technological Education (3rd year, Herzen State Pedagogical University of Russia).

GeQuPI: Quantum Program Improvement with Multi-Objective Genetic Programming

Processing quantum information poses novel challenges regarding the debugging of faulty quantum programs. Notably, the lack of accessible information on intermediate states during quantum processing, renders traditional debugging techniques infeasible. Moreover, even correct quantum programs might not be processable, as current quantum computers are limited in computation capacity. Thus, quantum program developers have to consider trade-offs between accuracy (i.e., probabilistically correct functionality) and computational cost of the proposed solutions. Manually finding sufficiently accurate and efficient solutions is a challenging task, even for quantum computing experts. To tackle these challenges, we propose a quantum program improvement framework for an automated generation of accurate and efficient solutions, coined Genetic Quantum Program Improver (GeQuPI). In particular, we focus on the tasks of debugging and optimization of quantum programs. Our framework uses techniques from quantum information theory and applies multi-objective genetic programming, which can be further hybridized with quantum-aware optimizers. To demonstrate the benefits of GeQuPI, it is applied to 47 quantum programs reused from literature and openly published libraries. The results show that our approach is capable of correcting faulty programs and optimize inefficient ones for the majority of the studied cases, showing average optimizations of 35% with respect to computational cost.

An exploratory study on the usage of quantum programming languages

As in the classical computing realm, quantum programming languages in quantum computing allow one to instruct a quantum computer to perform certain tasks. In the last 25 years, many imperative, functional, and multi-paradigm quantum programming languages with different features and goals have been developed. However, to the best of our knowledge, no study has investigated who uses quantum languages, how practitioners learn a quantum language, how experience are practitioners with quantum languages, what is the most used quantum languages, in which context practitioners use quantum languages, what are the challenges faced by quantum practitioners while using quantum languages, are program written with quantum languages tested, and what are quantum practitioners' perspectives on the variety of quantum languages and the potential need for new languages. In this paper, we first conduct a systematic survey to find and collect all quantum languages proposed in the literature and/or by organizations. Secondly, we identify and describe 37 quantum languages. Thirdly, we survey 251 quantum practitioners to answer several research questions about their quantum language usage. Fourthly, we conclude that (i) 58.2% of all practitioners are 25–44 years old, 63.0% have a master's or doctoral degree, and 86.2% have more than five years of experience using classical languages. (ii) 60.6% of practitioners learn quantum languages from the official documentation. (iii) Only 16.3% of practitioners have more than five years of experience with quantum languages. (iv) Qiskit (Python) is the most used quantum language, followed by Cirq (Python) and QDK (Q#). (v) 42.8% use quantum languages for research. (vi) Lack of documentation and usage examples are practitioners' most challenging issues. Practitioners prefer open-source quantum languages with an easy-to-learn syntax (e.g., based on an existing classical language), available documentation and examples, and an active community. (vii) 76.4% of all participants test their quantum programs, and 42.6% test them automatically. (viii) A standard quantum language, perhaps high-level language, for quantum computation could accelerate the development of quantum programs. Finally, we present a set of suggestions for developers and researchers on the development of new quantum languages or enhancement of existing ones.

Weight-Reduced Stabilizer Codes with Lower Overhead

Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are measured in order to infer information about errors that may have occurred. In typical settings, measuring these operators is itself a noisy process and the noise strength scales with the number of qubits involved in a given parity check, or its weight. Hastings has proposed a method for reducing the weights of the parity checks of a stabilizer code, though it has previously only been studied in the asymptotic regime. Here, we instead focus on the regime of small to medium-size codes suitable for quantum computing hardware. We both provide a fully explicit description of Hastings’s method and propose a substantially simplified weight-reduction method that is applicable to the class of quantum product codes. Our simplified method allows us to reduce the check weights of hypergraph- and lifted-product codes to at most six, while preserving the number of logical qubits and at least retaining (in fact, often increasing) the code distance. The price we pay is an increase in the number of physical qubits by a constant factor but we find that our method is much more efficient than Hastings’s method in this regard. We benchmark the performance of our codes on a simulated photonic quantum computing architecture based on Gottesman-Kitaev-Preskill qubits and passive linear optics, finding that our weight-reduction method substantially improves code performance. Published by the American Physical Society 2024

Correction to: Quantum error-correcting codes from the quantum construction X

Correction to: Quantum error-correcting codes from the quantum construction X

The impact of quantum computing on the development of algorithms and software

Introduction: There is a great potential that the quantum computing can change the way of algorithms and software development more than classical computers. Thus, this article will try to focus on how algorithm design and software development can be affected by quantum computing as well as what possibilities could appear when quantum principles are implemented into traditional paradigms. This paper aims at identifying the impact of quantum computing on algorithm and software advancement, through a discussion of essential quantum algorithms, quantum languages, as well as the opportunities and challenges of quantum technologies. Method: An extensive literature review and theoretical investigation was also performed to investigate the foundational concepts of quantum computing and subsequent effects on algorithm and software engineering. Some of the research questions included exploring the contrast between classical and quantum algorithms, reviewing current literature on quantum programming languages, and delving into examples of real-life deployments of quantum algorithms cross numerous domains. Results: This paper shows that quantum computing brings qualitatively new paradigms in the algorithm design and function while the quantum algorithms such as Shor’s and Grover’s perform exponentially faster certain problems. Software development for quantum has brought the need to devise new frameworks of coding in light of probability in quantum circuit. It is also comforting to note that there is still effort being made although in its most embryonic form to create quantum programming languages like Qiskit and Cirq. Some of challenges include quantum decoherence; limited number of quantum hardware; and need for strong error correction processes.Conclusion: While there are currently relatively few quantum algorithms it is believed that the findings in this field have the ability to revolutionize algorithm and software design and subjects like cryptography, optimization and AI. However, trends in quantum computing show that the constraints to computational capabilities are likely to be lifted to allow creativity to develop the most powerful software solutions

Phase diagram of the three-dimensional subsystem toric code

Subsystem quantum error-correcting codes typically involve measuring a sequence of noncommuting parity check operators. They can sometimes exhibit greater fault tolerance than conventional codes, which use commuting checks. However, unlike subspace codes, it is unclear if subsystem codes—in particular their advantages—can be understood in terms of ground-state properties of a physical Hamiltonian. In this paper, we address this question for the three-dimensional subsystem toric code (3D STC), as recently constructed by Kubica and Vasmer [], which exhibits single-shot error correction. Motivated by a conjectured relation between single-shot properties and thermal stability, we study the zero- and finite-temperature phases of an associated noncommuting Hamiltonian. By mapping the Hamiltonian model to a pair of 3D Z2 gauge theories coupled by a kinetic constraint, we find various phases at zero temperature, all separated by first-order transitions: There are 3D toric code-like phases with deconfined point-like excitations in the bulk, and there are phases with a confined bulk supporting a 2D toric code on the surface when appropriate boundary conditions are chosen. The latter is similar to the surface topological order present in 3D STC. However, the similarities between the single-shot correction in 3D STC and the confined phases are only partial: they share the same sets of degrees of freedom, but they are governed by different dynamical rules. Instead, we argue that the process of single-shot error correction can more suitably be associated with a path (rather than a point) in the zero-temperature phase diagram, a perspective, which inspires alternative measurement sequences enabling single-shot error correction. Moreover, since none of the above-mentioned phases survives at nonzero temperature, the single-shot error-correction property of the code does not imply thermal stability of the associated Hamiltonian phase. Published by the American Physical Society 2024

Quantum error correction using multiple nitrogen-vacancy center qubits

Abstract Quantum error correction (QEC) is crucial for protecting quantum information from the decoherence caused by the interaction between the system and the environment. Many QEC techniques and algorithms have been proposed and demonstrated in various physical platforms at low temperatures, such as superconducting circuits, Rydberg’s atoms, and trapped ions. At room temperature, the QEC realization with nitrogen-vacancy (NV) centers in diamond has become very attractive due to the promising nature of the centers that have a relatively long spin coherence time and can be initialized and read out optically. Here, we investigate the potential realization of a simple repetitive three-qubit QEC scheme in which three NVs are coupled via dipolar coupling. A single NV qubit has been protected using two other coupled NVs which act as ancilla qubits. In this configuration of three NVs, a single NV qubit is protected from bit or phase-flip errors. This work paves the way for realizing five-qubit QEC with NVs at room temperature to preserve a qubit against any arbitrary single-qubit error.

Quantum Codes for Asymmetric Channels: ZZZY Surface Codes

Quantum Codes for Asymmetric Channels: ZZZY Surface Codes

Algebraic and Geometric Methods for Construction of Topological Quantum Codes from Lattices

Current work provides an algebraic and geometric technique for building topological quantum codes. From the lattice partition derived of quotient lattices Λ′/Λ of index m combined with geometric technique of the projections of vector basis Λ′ over vector basis Λ, we reproduce surface codes found in the literature with parameter [[2m2,2,|a|+|b|]] for the case Λ=Z2 and m=a2+b2, where a and b are integers that are not null, simultaneously. We also obtain a new class of surface code with parameters [[2m,2,|a|+|b|]] from the Λ=A2-lattice when m can be expressed as m=a2+ab+b2, where a and b are integer values. Finally, we will show how this technique can be extended to the construction of color codes with parameters [[18m,4,6(|a|+|b|)]] by considering honeycomb lattices partition A2/Λ′ of index m=9(a2+ab+b2) where a and b are not null integers.

Quantifying noncovariance of quantum channels with respect to groups

A quantum channel is covariant with respect to a group if it commutes with the action of the group. In general, a quantum channel may not be covariant with respect to a given group. The degree of noncovariance can vary between different channels, and it is desirable to have a quantitative characterization for the degree of channel noncovariance. In this work, we propose a measure based on the Hilbert-Schmidt norm to quantify noncovariance of quantum channels with respect to a group and demonstrate that it satisfies several desirable properties. Compared with the existing measures of channel noncovariance, our measure applies to not only compact Lie groups but also finite groups, and it is easy to evaluate. Using this measure and its modified version together with two existing measures, we evaluate and analyze channel noncovariance through an example, finding that these measures of channel noncovariance are closely related but differ from each other. They capture different perspectives of noncovariance of quantum channels. As applications, we provide a relation between channel noncovariance and approximate quantum error correction using our measures of channel noncovariance.

High-fidelity teleportation of a logical qubit using transversal gates and lattice surgery

Quantum state teleportation is commonly used in designs for large-scale quantum computers. Using Quantinuum’s H2 trapped-ion quantum processor, we demonstrate fault-tolerant state teleportation circuits for a quantum error correction code—specifically the Steane code. The circuits use up to 30 qubits at the physical level and employ real-time quantum error correction. We conducted experiments on several variations of logical teleportation circuits using both transversal gates and lattice surgery. We measured the logical process fidelity to be 0.975 ± 0.002 for the transversal teleportation implementation and 0.851 ± 0.009 for the lattice surgery teleportation implementation as well as 0.989 ± 0.002 for an implementation of Knill-style quantum error correction.

Safeguarding Oscillators and Qudits with Distributed Two-Mode Squeezing

Recent advancements in multi-mode Gottesman-Kitaev-Preskill (GKP) codes have shown great promise in enhancing the protection of both discrete and analog quantum information. This broadened range of protection brings opportunities beyond quantum computing to benefit quantum sensing by safeguarding squeezing — the essential resource in many quantum metrology protocols. However, the potential for quantum sensing to benefit quantum error correction has been less explored. In this work, we provide a unique example where techniques from quantum sensing can be applied to improve multi-mode GKP codes. Inspired by distributed quantum sensing, we propose the distributed two-mode squeezing (dtms) GKP codes that offer benefits in error correction with minimal active encoding operations. Indeed, the proposed codes rely on a single (active) two-mode squeezing element and an array of beamsplitters that effectively distributes continuous-variable correlations to many GKP ancillae, similar to continuous-variable distributed quantum sensing. Despite this simple construction, the code distance achievable with dtms-GKP qubit codes is comparable to previous results obtained through brute-force numerical search \cite{lin2023closest}. Moreover, these codes enable analog noise suppression beyond that of the best-known two-mode codes \cite{noh2020o2o} without requiring an additional squeezer. We also provide a simple two-stage decoder for the proposed codes, which appears near-optimal for the case of two modes and permits analytical evaluation.

Improving Threshold for Fault-Tolerant Color-Code Quantum Computing by Flagged Weight Optimization

Color codes are promising quantum error-correction (QEC) codes because they have an advantage over surface codes in that all Clifford gates can be implemented transversally. However, the thresholds of color codes under circuit-level noise are relatively low, mainly because measurements of their high-weight stabilizer generators cause an increase in the circuit depth and, thus, substantial errors are introduced. This makes color codes not the best candidate for fault-tolerant quantum computing. Here, we propose a method to suppress the impact of such errors by optimizing weights of decoders using conditional error probabilities conditioned on the measurement outcomes of flag qubits. In numerical simulations, we improve the threshold of the (4.8.8) color code under circuit-level noise from 0.14% to around 0.27%, which is calculated by using an integer programming decoder. Furthermore, in the (6.6.6) color code, we achieve a circuit-level threshold of around 0.36%, which is almost the same value as the highest value in the previous studies employing the same noise model. In both cases, the effective code distance is also improved compared to a conventional method that uses a single ancilla qubit for each stabilizer measurement. Thereby, the achieved logical error rates at low physical error rates are almost one order of magnitude lower than those of the conventional method with the same code distance. Even when compared to the single-ancilla method with a higher code distance, considering the increased number of qubits used in our method, we achieve lower logical error rates in most cases. This method can also be applied to other weight-based decoders, making the color codes more promising as candidates for the experimental implementation of QEC. Furthermore, one can utilize this approach to improve a threshold of wider classes of QEC codes, such as high-rate quantum low-density parity-check codes. Published by the American Physical Society 2024

Domain Wall Color Code.

We introduce the domain wall color code, a new variant of the quantum error-correcting color code that exhibits exceptionally high code-capacity error thresholds for qubits subject to biased noise. In the infinite bias regime, a two-dimensional color code decouples into a series of repetition codes, resulting in an error-correcting threshold of 50%. Interestingly, at finite bias, our color code demonstrates thresholds identical to those of the noise-tailored XZZX surface code for all single-qubit Pauli noise channels. The design principle of the code is that it introduces domain walls which permute the code's excitations upon domain crossing. For practical implementation, we supplement the domain wall code with a scalable restriction decoder based on a matching algorithm. The proposed code is identified as a comparably resource-efficient quantum error-correcting code highly suitable for realistic noise.

Clifford orbits and stabilizer states

Abstract Stabilizer states serve as “classical objects” in the stabilizer formalism of quantum theory, and play an important role in quantum error correction, fault-tolerant quantum computation, and quantum communication. They provide an efficient description of many basic features of quantum theory and exhibit a rich structure. For prime dimensional systems, they may be defined by two quite different yet equivalent ways: The first is via stabilizer groups (maximal Abelian subgroups of the discrete Heisenberg-Weyl group). The second is via the orbits of the Clifford group acting on any computational basis state. However, in a general dimensional system, this equivalence breaks down, and consequently, it is desirable to classify the difference and relation between the above two approaches to stabilizer states. In this work, we show that these two approaches are equivalent if and only if the system dimension is square-free (i.e., has no square factor). Furthermore, we completely clarify the relation between the Clifford orbits and stabilizer states in an arbitrary dimensional system, derive the explicit expressions of the Clifford orbits and determine their cardinalities.

Constacyclic codes over F q S T and their applications

Let q = p r , where p is an odd prime and r is a positive integer. Consider a ring F q S T , where S = F q + u F q + v F q + uv F q with u 2 = 1 , v 2 = 1 , uv = vu and T = F q + u F q + v F q + w F q + uv F q + uw F q + vw F q + uvw F q with u 2 = 1 , v 2 = 1 , w 2 = 1 , uv = vu , uw = wu , vw = wv . In this paper, we examine the algebraic structure of constacyclic codes over the ring F q S T of block length ( α , β , γ ) . Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over F q S T is also given. Moreover, we derive a number of new QECCs.