Quantum Error Correction Research Articles

Testing Multi-Subroutine Quantum Programs: From Unit Testing to Integration Testing

Quantum computing has emerged as a promising field with the potential to revolutionize various domains by harnessing the principles of quantum mechanics. As quantum hardware and algorithms continue to advance, developing high-quality quantum software has become crucial. However, testing quantum programs poses unique challenges due to the distinctive characteristics of quantum systems and the complexity of multi-subroutine programs. This article addresses the specific testing requirements of multi-subroutine quantum programs. We begin by investigating critical properties by surveying existing quantum libraries and providing insights into the challenges of testing these programs. Building upon this understanding, we focus on testing criteria and techniques based on the whole testing process perspective, spanning from unit testing to integration testing. We delve into various aspects, including IO analysis, quantum relation checking, structural testing, behavior testing, integration of subroutine pairs, and test case generation. We also introduce novel testing principles and criteria to guide the testing process. We conduct comprehensive testing on typical quantum subroutines, including diverse mutants and randomized inputs, to evaluate our proposed approach. The analysis of failures provides valuable insights into the effectiveness of our testing methodology. Additionally, we present case studies on representative multi-subroutine quantum programs, demonstrating the practical application and effectiveness of our proposed testing principles and criteria.

Quantum error correction for heavy hexagonal code using deep reinforcement learning with policy reuse

Quantum error correction for heavy hexagonal code using deep reinforcement learning with policy reuse

The black hole interior from non-isometric codes and complexity

Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of “non-isometric codes protected by computational complexity”. We show that many previous ideas, such as the existence of a large number of “null states”, a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, “state-dependent/state-specific” operator reconstruction, and the “simple entropy” approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.

Quantum error-correction using humming sparrow optimization based self-adaptive deep cnn noise correction module

The error correction model’s main purpose in heavy hexagonal quantum codes is to improve their reliability for quantum computing applications. Existing challenges include finding the optimal decoder for quantum error correction in heavy hexagonal codes. This research propels the frontier of quantum error correction, with a specific focus on tailoring topological quantum error-correcting codes for the unique challenges posed by superconducting qubits in quantum computers. In response, this research harnesses the power of deep learning, presenting a Humming sparrow optimization based self-adaptive deep CNN (HSO-based SADCNN) model designed for heavy hexagonal codes. This decoder incorporates a Self-adaptive Deep CNN (SADCNN) Noise Correction Module, a sophisticated component to refine error correction. The proposed decoder’s efficacy is rigorously evaluated across varying code distances (three, five, and seven) using the Humming Sparrow Optimization (HSO) algorithm. HSO, intricately designed to fine-tune the SADCNN decoder, significantly enhances its error correction capabilities for heavy hexagonal quantum codes. The algorithm seamlessly integrates advantageous characteristics of herding and tracing from Humming Bird optimization and Sparrow search optimization, representing a critical stride in advancing the reliability of quantum computing applications, particularly within the intricate domain of heavy hexagonal quantum codes. Based upon the achievements, the Training Percentage (TP) 90 metrics demonstrate significant progress, boasting a commendable accuracy of 97.35%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$97.35\\%$$\\end{document}, coupled with reduced logical error probability and a diminished bit error rate, marked at 5.51 and 3.72, respectively.

Early Fault-Tolerant Quantum Computing

In recent years, research in quantum computing has largely focused on two approaches: near-term intermediate-scale quantum (NISQ) computing and future fault-tolerant quantum computing (FTQC). A growing body of research into early fault-tolerant quantum computing (EFTQC) is exploring how to utilize quantum computers during the transition between these two eras. However, without agreed-upon characterizations of this transition, it is unclear how best to utilize EFTQC architectures. We argue for the perspective that this transition period will be characterized by a law of diminishing returns in quantum error correction (QEC), where the ability of the architecture to maintain quality operations at scale determines the point of diminishing returns. Two challenges emerge from this picture: how to model this phenomenon of diminishing return of QEC as the performance of devices is continually improving and how to design algorithms to make the most use of these devices. To address these challenges, we present models for the performance of EFTQC architectures, capturing the diminishing returns of QEC. We then use these models to elucidate the regimes in which algorithms suited to such architectures are advantageous. As a concrete example, we show that for the canonical task of phase estimation, in a regime of moderate scalability and using just over one million physical qubits, the “reach” of the quantum computer can be extended (compared to the standard approach) from 90-qubit instances to over 130-qubit instances using a simple early fault-tolerant quantum algorithm, which reduces the number of operations per circuit by a factor of 100 and increases the number of circuit repetitions by a factor of 10 000. This clarifies the role that such algorithms might play in the era of limited-scalability quantum computing. Published by the American Physical Society 2024

Near-Optimal Performance of Quantum Error Correction Codes.

The Knill-Laflamme conditions distinguish exact quantum error correction codes, and they have played a critical role in the discovery of state-of-the-art codes. However, the family of exact codes is a very restrictive one and does not necessarily contain the best-performing codes. Therefore, it is desirable to develop a generalized and quantitative performance metric. In this Letter, we derive the near-optimal channel fidelity, a concise and optimization-free metric for arbitrary codes and noise. The metric provides a narrow two-sided bound to the optimal code performance, and it can be evaluated with exactly the same input required by the Knill-Laflamme conditions. We demonstrate the numerical advantage of the near-optimal channel fidelity through multiple qubit code and oscillator code examples. Compared to conventional optimization-based approaches, the reduced computational cost enables us to simulate systems with previously inaccessible sizes, such as oscillators encoding hundreds of average excitations. Moreover, we analytically derive the near-optimal performance for the thermodynamic code and the Gottesman-Kitaev-Preskill code. In particular, the Gottesman-Kitaev-Preskill code's performance under excitation loss improves monotonically with its energy and converges to an asymptotic limit at infinite energy, which is distinct from other oscillator codes.

Quantum Programming Methods: Approaches to the Formation of Ontology and Formalization of the Modern Computing Paradigm in the Educational Process

The article confirms the increasing level of involvement of the international community in the development of quantum technologies. In the scientific space in the domestic segment, the authors have identified a significant lag between the level and quality of existing educational programs of higher educational institutions in relation to international practical achievements in the field of development and programming of quantum computing devices. In order to eliminate the existing scientific and methodological gap, it is recommended to introduce relevant subjects into the educational programs of enlarged groups of training areas 09.00.00 and 10.00.00 as separate disciplines, or into already existing [disciplines], and expand them to study the possibilities of quantum programming methods. Based on their own teaching experience, the authors propose a possible strategy for making appropriate changes, as well as options for adjusting the work programs of disciplines that include tasks for programming quantum computing devices.

Fold-Transversal Clifford Gates for Quantum Codes

We generalize the concept of folding from surface codes to CSS codes by considering certain dualities within them. In particular, this gives a general method to implement logical operations in suitable LDPC quantum codes using transversal gates and qubit permutations only.To demonstrate our approach, we specifically consider a [[30, 8, 3]] hyperbolic quantum code called Bring's code. Further, we show that by restricting the logical subspace of Bring's code to four qubits, we can obtain the full Clifford group on that subspace.

Cross-Cap Defects and Fault-Tolerant Logical Gates in the Surface Code and the Honeycomb Floquet Code

We consider the Z2 toric code, surface code, and Floquet code defined on a nonorientable surface, which can be considered as families of codes extending Shor’s nine-qubit code. We investigate the fault-tolerant logical gates of the Z2 toric code in this setup, which corresponds to e↔m exchanging symmetry of the underlying Z2 gauge theory. We find that nonorientable geometry provides a new way for the emergent symmetry to act on the code space, and discover the new realization of the fault-tolerant Hadamard gate of the two-dimensional surface code with a single cross cap connecting the vertices nonlocally along a slit, dubbed a nonorientable surface code. This Hadamard gate can be realized by a constant-depth local unitary circuit modulo nonlocality caused by a cross cap. Via folding, the nonorientable surface code can be turned into a bilayer local quantum code, where the folded cross cap is equivalent to a bilayer twist terminated on a gapped boundary and the logical Hadamard only contains local gates with intralayer couplings when being away from the cross cap, as opposed to the interlayer couplings on each site needed in the case of the folded surface code. We further obtain the complete logical Clifford gate set for a stack of nonorientable surface codes and similarly for codes defined on Klein-bottle geometries. We then construct the honeycomb Floquet code in the presence of a single cross cap, and find that the period of the sequential Pauli measurements acts as a HZ logical gate on the single logical qubit, where the cross cap enriches the dynamics compared with the orientable case. We find that the dynamics of the honeycomb Floquet code is precisely described by a condensation operator of the Z2 gauge theory, and illustrate the exotic dynamics of our code in terms of a condensation operator supported at a nonorientable surface. Published by the American Physical Society 2024

Studying phonon coherence with a quantum sensor

Nanomechanical oscillators offer numerous advantages for quantum technologies. Their integration with superconducting qubits shows promise for hardware-efficient quantum error-correction protocols involving superpositions of mechanical coherent states. Limitations of this approach include mechanical decoherence processes, particularly two-level system (TLS) defects, which have been widely studied using classical fields and detectors. In this manuscript, we use a superconducting qubit as a quantum sensor to perform phonon number-resolved measurements on a piezoelectrically coupled phononic crystal cavity. This enables a high-resolution study of mechanical dissipation and dephasing in coherent states of variable size (n¯≃1−10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{n}\\simeq 1-10$$\\end{document} phonons). We observe nonexponential relaxation and state size-dependent reduction of the dephasing rate, which we attribute to TLS. Using a numerical model, we reproduce the dissipation signatures (and to a lesser extent, the dephasing signatures) via emission into a small ensemble (N = 5) of rapidly dephasing TLS. Our findings comprise a detailed examination of TLS-induced phonon decoherence in the quantum regime.

Group-theoretic error mitigation enabled by classical shadows and symmetries

Estimating expectation values is a key subroutine in quantum algorithms. Near-term implementations face two major challenges: a limited number of samples required to learn a large collection of observables, and the accumulation of errors in devices without quantum error correction. To address these challenges simultaneously, we develop a quantum error-mitigation strategy called symmetry-adjusted classical shadows, by adjusting classical-shadow tomography according to how symmetries are corrupted by device errors. As a concrete example, we highlight global U(1) symmetry, which manifests in fermions as particle number and in spins as total magnetization, and illustrate their group-theoretic unification with respective classical-shadow protocols. We establish rigorous sampling bounds under readout errors obeying minimal assumptions, and perform numerical experiments with a more comprehensive model of gate-level errors derived from existing quantum processors. Our results reveal symmetry-adjusted classical shadows as a low-cost strategy to mitigate errors from noisy quantum experiments in the ubiquitous presence of symmetry.

Deterministic Preparation of Optical Squeezed Cat and Gottesman-Kitaev-Preskill States.

Large-amplitude squeezed cat states and high-quality Gottesman-Kitaev-Preskill (GKP) states are essential for effective quantum error correction, yet their optical preparation has been hindered by challenges such as low success probabilities, small amplitudes, and insufficient squeezing. Addressing these limitations, our research introduces scalable optical schemes for the deterministic preparation of large-amplitude squeezed cat states from photon-number states. Fock states have the benefit of producing consistent cat states across all measurement outcomes and intrinsically provides a degree of squeezing. Notably, these squeezed cat states facilitate the deterministic generation of high-quality approximate GKP states via "breeding," showing that GKP error correction in optics is technically feasible in near-term experiments. Our schemes allow fault-tolerant quantum computation through the use of GKP error correction.

An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes (C1,C2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C_1, C_2)$$\\end{document} such that C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document} contains C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document} is even, and the shortening of the dual of C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document} with respect to the support of each codeword of C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document} is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes (C1,C2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C_1, C_2)$$\\end{document} is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.

Flying-cat parity checks for quantum error correction

Long range, multiqubit parity checks have applications in both quantum error correction and measurement-based entanglement generation. Such parity checks could be performed using qubit-state-dependent phase shifts on propagating pulses of light described by coherent states |α〉 of the electromagnetic field. We consider “flying-cat” parity checks based on an entangling operation that is quantum nondemolition for Schrödinger's cat states |α〉±|−α〉. This operation encodes parity information in the phase of maximally distinguishable coherent states |±α〉, which can be read out using a phase-sensitive measurement of the electromagnetic field. In contrast to many implementations, where single-qubit errors and measurement errors can be treated as independent, photon loss during flying-cat parity checks introduces errors on physical qubits at a rate that is anticorrelated with the probability for measurement errors. We analyze this trade-off for three-qubit parity checks, which are a requirement for universal fault-tolerant quantum computing with the subsystem surface code. We further show how a six-qubit entangled “tetrahedron” state can be prepared using these three-qubit parity checks. The tetrahedron state can be used as a resource for controlled quantum teleportation of a two-qubit state or as a source of shared randomness with potential applications in three-party quantum key distribution. Finally, we provide conditions for performing high-quality flying-cat parity checks in a state-of-the-art circuit QED architecture, accounting for qubit decoherence, internal cavity losses, and finite-duration pulses, in addition to transmission losses. Published by the American Physical Society 2024

Hermitian hull of constacyclic codes over a class of non-chain rings and new quantum codes

Hermitian hull of constacyclic codes over a class of non-chain rings and new quantum codes

Analyzing Quantum Error Resilience for Quantum Communication in Guided and Unguided Media

AbstractThis research examines quantum key distribution and its applications in guided and unguided quantum communication. The importance of secure communication in the quantum era requires a thorough exploration of both guided and unguided quantum communication strategies. The research aims to address the challenges posed by guided channels, such as fiber optics, and unguided channels, such as free‐space quantum communication. This study addresses existing knowledge gaps in quantum error resilience management and signal processing techniques in unguided quantum communication. Advanced quantum gate analysis, environmental noise analysis, and quantum channel modeling techniques are employed. The research presents key findings on the impact of gate imperfections on quantum error resilience in guided media, the influence of noise‐induced errors in unguided media, and a unified metric for assessing various error sources in guided channels. Additionally, the study analyses stabilizer codes and surface codes for error mitigation through quantum error correction strategies. Simulation results provide a benchmark for theoretical predictions and guide the refinement of quantum communication protocols. In this context, machine learning‐based error prediction is introduced as a cutting‐edge approach to enhance the robustness of quantum communication systems.

Efficient Initialization of Fluxonium Qubits based on Auxiliary Energy Levels.

Fast and high-fidelity qubit initialization is crucial for low-frequency qubits such as fluxonium, and in applications of many quantum algorithms and quantum error correction codes. In a circuit quantum electrodynamics system, the initialization is typically achieved by transferring the state between the qubit and a short-lived cavity through microwave driving, also known as the sideband cooling process in atomic system. Constrained by the selection rules from the parity symmetry of the wave functions, the sideband transitions are only enabled by multiphoton processes which require multitone or strong driving. Leveraging the flux tunability of fluxonium, we circumvent this limitation by breaking flux symmetry to enable an interaction between a noncomputational qubit transition and the cavity excitation. With single-tone sideband driving, we realize qubit initialization with a fidelity exceeding 99% within a duration of 300ns, robust against the variation of control parameters. Furthermore, we show that our initialization scheme has a built-in benefit in simultaneously removing the second-excited state population of the qubit, and can be easily incorporated into a large-scale fluxonium processor.

Channel-optimized quantum communication scheme based on combination code

In this paper, a channel-optimized quantum communication scheme with imperfect Bell states is being considered, based on quantum teleportation and error correction codes technology. In this scheme, a novel “combination code” technique is proposed to protect the pre-shared Bell states from storage errors and to safeguard the information-encoded states from noisy channels. Ultimately, it is revealed by Monte Carlo simulation results that the channel fidelity of the proposed channel-optimized quantum communication scheme with imperfect Bell states is significantly advantageous over standard quantum error-correcting code (QECC) and entanglement-assisted quantum error-correcting code (EAQECC) communication schemes.

Analog Information Decoding of Bosonic Quantum Low-Density Parity-Check Codes

Quantum error correction is crucial for scalable quantum information-processing applications. Traditional discrete-variable quantum codes that use multiple two-level systems to encode logical information can be hardware intensive. An alternative approach is provided by bosonic codes, which use the infinite-dimensional Hilbert space of harmonic oscillators to encode quantum information. Two promising features of bosonic codes are that syndrome measurements are natively analog and that they can be concatenated with discrete-variable codes. In this work, we propose novel decoding methods that explicitly exploit the analog syndrome information obtained from the bosonic qubit readout in a concatenated architecture. Our methods are versatile and can be generally applied to any bosonic code concatenated with a quantum low-density parity-check (QLDPC) code. Furthermore, we introduce the concept of quasi-single shot protocols as a novel approach that significantly reduces the number of repeated syndrome measurements required when decoding under phenomenological noise. To realize the protocol, we present the first implementation of time-domain decoding with the overlapping window method for general QLDPC codes and a novel analog single-shot decoding method. Our results lay the foundation for general decoding algorithms using analog information and demonstrate promising results in the direction of fault-tolerant quantum computation with concatenated bosonic-QLDPC codes. Published by the American Physical Society 2024

Deterministic Generation of Qudit Photonic Graph States from Quantum Emitters

We propose and analyze deterministic protocols to generate qudit photonic graph states from quantum emitters. We show that our approach can be applied to generate any qudit graph state and we exemplify it by constructing protocols to generate one- and two-dimensional qudit cluster states, absolutely maximally entangled states, and logical states of quantum error-correcting codes. Some of these protocols make use of time-delayed feedback, while others do not. The only additional resource requirement compared to the qubit case is the ability to control multilevel emitters. These results significantly broaden the range of multiphoton entangled states that can be produced deterministically from quantum emitters. Published by the American Physical Society 2024