Quantum Error Correction Research Articles

High-threshold and low-overhead fault-tolerant quantum memory

The accumulation of physical errors1–3 prevents the execution of large-scale algorithms in current quantum computers. Quantum error correction4 promises a solution by encoding k logical qubits onto a larger number n of physical qubits, such that the physical errors are suppressed enough to allow running a desired computation with tolerable fidelity. Quantum error correction becomes practically realizable once the physical error rate is below a threshold value that depends on the choice of quantum code, syndrome measurement circuit and decoding algorithm5. We present an end-to-end quantum error correction protocol that implements fault-tolerant memory on the basis of a family of low-density parity-check codes6. Our approach achieves an error threshold of 0.7% for the standard circuit-based noise model, on par with the surface code7–10 that for 20 years was the leading code in terms of error threshold. The syndrome measurement cycle for a length-n code in our family requires n ancillary qubits and a depth-8 circuit with CNOT gates, qubit initializations and measurements. The required qubit connectivity is a degree-6 graph composed of two edge-disjoint planar subgraphs. In particular, we show that 12 logical qubits can be preserved for nearly 1 million syndrome cycles using 288 physical qubits in total, assuming the physical error rate of 0.1%, whereas the surface code would require nearly 3,000 physical qubits to achieve said performance. Our findings bring demonstrations of a low-overhead fault-tolerant quantum memory within the reach of near-term quantum processors.

Deep Quantum Error Correction

Quantum error correction codes (QECC) are a key component for realizing the potential of quantum computing. QECC, as its classical counterpart (ECC), enables the reduction of error rates, by distributing quantum logical information across redundant physical qubits, such that errors can be detected and corrected. In this work, we efficiently train novel end-to-end deep quantum error decoders. We resolve the quantum measurement collapse by augmenting syndrome decoding to predict an initial estimate of the system noise, which is then refined iteratively through a deep neural network. The logical error rates calculated over finite fields are directly optimized via a differentiable objective, enabling efficient decoding under the constraints imposed by the code. Finally, our architecture is extended to support faulty syndrome measurement, by efficient decoding of repeated syndrome sampling. The proposed method demonstrates the power of neural decoders for QECC by achieving state-of-the-art accuracy, outperforming for small distance topological codes, the existing end-to-end neural and classical decoders, which are often computationally prohibitive.

On the duals of quasi-cyclic codes and their application to quantum codes

On the duals of quasi-cyclic codes and their application to quantum codes

Near MDS and near quantum MDS codes via orthogonal arrays

Near maximum distance separable (NMDS) codes are closely related to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. But there are many unsolved problems about construction of NMDS codes. In this paper, by using symmetrical orthogonal arrays (OAs), we construct a lot of NMDS, m-MDS and almost extremal NMDS codes. Quantum error-correcting codes (QECCs) play a central role in quantum information processing and can protect quantum information from various quantum noises. We present a general method for constructing QECCs over mixed alphabets through asymmetrical OAs. Since quantum maximum distance separable (QMDS) codes over mixed alphabets with the dimension equal to one have not been found in all the literature so far, the definition of a near QMDS code over mixed alphabets is proposed. By using asymmetrical OAs, we obtain many such codes.

Construction of some new entanglement-assisted quantum error-correcting codes of large lengths

The study of entanglement-assisted quantum error-correcting codes (EAQECCs) is currently a hot topic in coding theory and quantum information theory. As far as we know, there are few constructions of q-ary EAQECCs of large lengths, and most of the q-ary EAQECCs constructed in the literature have lengths less than or equal to q2+1. As a result, it will be interesting to construct q-ary EAQECCs of large lengths. The goal of this paper is to construct some good and new q-ary EAQECCs of large lengths from q2-ary matrix-product codes associated with the so-called non-singular by columns (NSC) quasi-unitary matrices. To this end, we first propose an effective method and a corresponding algorithm for constructing NSC quasi-unitary matrices. We then present an effective method for constructing EAQECCs of large lengths from matrix-product codes associated with NSC quasi-unitary matrices. Subsequently, we give the explicit construction of q-ary EAQECCs of lengths kn with n≤q2+1, k=k0 for q≥k0, where k0=3,4,5,7,8,9,11,13,16,17,19, and k=q+2, q2−1, q2 for 3≤q≤19. Finally, some new q-ary EAQECCs of large lengths such as lengths 3(q2−1), 3q2, 4(q2−1), 4q2, q3, q(q2+1), q2(q+2), (q2+1)(q+2), q2(q2−1), q4−1, q4, q2(q2+1) for some q are constructed, which are not covered by previously known ones.

Optimizing quantum gates towards the scale of logical qubits

A foundational assumption of quantum error correction theory is that quantum gates can be scaled to large processors without exceeding the error-threshold for fault tolerance. Two major challenges that could become fundamental roadblocks are manufacturing high-performance quantum hardware and engineering a control system that can reach its performance limits. The control challenge of scaling quantum gates from small to large processors without degrading performance often maps to non-convex, high-constraint, and time-dynamic control optimization over an exponentially expanding configuration space. Here we report on a control optimization strategy that can scalably overcome the complexity of such problems. We demonstrate it by choreographing the frequency trajectories of 68 frequency-tunable superconducting qubits to execute single- and two-qubit gates while mitigating computational errors. When combined with a comprehensive model of physical errors across our processor, the strategy suppresses physical error rates by ~3.7× compared with the case of no optimization. Furthermore, it is projected to achieve a similar performance advantage on a distance-23 surface code logical qubit with 1057 physical qubits. Our control optimization strategy solves a generic scaling challenge in a way that can be adapted to a variety of quantum operations, algorithms, and computing architectures.

Decoding topological XYZ 2 codes with reinforcement learning based on attention mechanisms

Quantum error correction, a technique that relies on the principle of redundancy to encode logical information into additional qubits to better protect the system from noise, is necessary to design a viable quantum computer. For this new topological stabilizer code-XYZ 2 code defined on the cellular lattice, it is implemented on a hexagonal lattice of qubits and it encodes the logical qubits with the help of stabilizer measurements of weight six and weight two. However topological stabilizer codes in cellular lattice quantum systems suffer from the detrimental effects of noise due to interaction with the environment. Several decoding approaches have been proposed to address this problem. Here, we propose the use of a state-attention based reinforcement learning decoder to decode XYZ 2 codes, which enables the decoder to more accurately focus on the information related to the current decoding position, and the error correction accuracy of our reinforcement learning decoder model under the optimisation conditions can reach 83.27% under the depolarizing noise model, and we have measured thresholds of 0.18856 and 0.19043 for XYZ 2 codes at code spacing of 3–7 and 7–11, respectively. our study provides directions and ideas for applications of decoding schemes combining reinforcement learning attention mechanisms to other topological quantum error-correcting codes.

Tensor Network Message Passing.

When studying interacting systems, computing their statistical properties is a fundamental problem in various fields such as physics, applied mathematics, and machine learning. However, this task can be quite challenging due to the exponential growth of the state space as the system size increases. Many standard methods have significant weaknesses. For instance, message-passing algorithms can be inaccurate and even fail to converge due to short loops, while tensor network methods can have exponential computational complexity in large graphs due to long loops. In this Letter, we propose a new method called "tensor network message passing." This approach allows us to compute local observables like marginal probabilities and correlations by combining the strengths of tensor networks in contracting small subgraphs with many short loops and the strengths of message-passing methods in globally sparse graphs, thus addressing the crucial weaknesses of both approaches. Our algorithm is exact for systems that are globally treelike and locally dense-connected when the dense local graphs have a limited tree width. We have conducted numerical experiments on synthetic and real-world graphs to compute magnetizations of Ising models and spin glasses, and have demonstrated the superiority of our approach over standard belief propagation and the recently proposed loopy message-passing algorithm. In addition, we discuss the potential applications of our method in inference problems in networks, combinatorial optimization problems, and decoding problems in quantum error correction.

Hermitian self-orthogonal matrix product codes and their applications to quantum codes

Hermitian self-orthogonal matrix product codes and their applications to quantum codes

Compiling Quantum Circuits for Dynamically Field-Programmable Neutral Atoms Array Processors

Dynamically field-programmable qubit arrays (DPQA) have recently emerged as a promising platform for quantum information processing. In DPQA, atomic qubits are selectively loaded into arrays of optical traps that can be reconfigured during the computation itself. Leveraging qubit transport and parallel, entangling quantum operations, different pairs of qubits, even those initially far away, can be entangled at different stages of the quantum program execution. Such reconfigurability and non-local connectivity present new challenges for compilation, especially in the layout synthesis step which places and routes the qubits and schedules the gates. In this paper, we consider a DPQA architecture that contains multiple arrays and supports 2D array movements, representing cutting-edge experimental platforms. Within this architecture, we discretize the state space and formulate layout synthesis as a satisfiability modulo theories problem, which can be solved by existing solvers optimally in terms of circuit depth. For a set of benchmark circuits generated by random graphs with complex connectivities, our compiler OLSQ-DPQA reduces the number of two-qubit entangling gates on small problem instances by 1.7x compared to optimal compilation results on a fixed planar architecture. To further improve scalability and practicality of the method, we introduce a greedy heuristic inspired by the iterative peeling approach in classical integrated circuit routing. Using a hybrid approach that combined the greedy and optimal methods, we demonstrate that our DPQA-based compiled circuits feature reduced scaling overhead compared to a grid fixed architecture, resulting in 5.1X less two-qubit gates for 90 qubit quantum circuits. These methods enable programmable, complex quantum circuits with neutral atom quantum computers, as well as informing both future compilers and future hardware choices.

On dual-containing, almost dual-containing matrix-product codes and related quantum codes

Matrix-product (MP) codes are a type of long codes formed by combining several commensurate constituent codes with a defining matrix. In this paper, we study the MP code when the defining matrix A satisfies the condition that AA⊤ is (D,τ)-monomial. We give an explicit formula for calculating the dimension of the hull of a MP code. We present the necessary and sufficient conditions for a MP code to be dual-containing (DC), almost dual-containing (ADC), self-orthogonal (SO) and almost self-orthogonal (ASO), respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code that is DC, ADC, SO and ASO, respectively. We give alternative necessary and sufficient conditions for a MP code to be ADC and ASO, respectively, and show several cases where a MP code is not ADC or ASO. We give the construction methods of DC and ADC MP codes, including those with optimal minimum distance lower bounds. We introduce the notation of τ-optimal defining (τ-OD) matrices and provide the criteria for determining whether two types of k×k matrices are τ-OD matrices at k=3 and k=4, respectively. We give many examples of DC and ADC MP codes involving τ-OD matrices, some of which are optimal or almost optimal according to the Database [11]. By applying the generalized Steane's enlargement procedure to these DC MP codes, we obtain some good quantum codes that improve those available in the Database [7].

Scalable Fault-Tolerant Quantum Technologies with Silicon Color Centers

The scaling barriers currently faced by both quantum networking and quantum computing technologies ultimately amount to the same core challenge of distributing high-quality entanglement at scale. In this Perspective, a novel quantum information-processing architecture based on optically active spins in silicon is proposed that offers a combined single technological platform for scalable fault-tolerant quantum computing and networking. The architecture is optimized for overall entanglement distribution and leverages color-center spins in silicon (T centers) for their manufacturability, photonic interface, and high-fidelity information-processing properties. Silicon nanophotonic optical circuits allow for photonic links between T centers, which are networked via telecom-band optical photons in a highly connected graph. This high connectivity unlocks the use of low-overhead quantum error-correcting codes, significantly accelerating the time line for modular scalable fault-tolerant quantum repeaters and quantum processors. Published by the American Physical Society 2024

Unveiling the Non-Abelian Statistics of D(S_{3}) Anyons Using a Classical Photonic Simulator.

Simulators can realize novel phenomena by separating them from the complexities of a full physical implementation. Here, we put forward a scheme that can simulate the exotic statistics of D(S_{3}) non-Abelian anyons with minimal resources. The qudit lattice representation of this planar code supports local encoding of D(S_{3}) anyons. As a proof-of-principle demonstration, we employ a classical photonic simulator to encode a single qutrit and manipulate it to perform the fusion and braiding properties of non-Abelian D(S_{3}) anyons. The photonic technology allows us to perform the required nonunitary operations with much higher fidelity than what can be achieved with current quantum computers. Our approach can be directly generalized to larger systems or to different anyonic models, thus enabling advances in the exploration of quantum error correction and fundamental physics alike.

Silicon spin qubit noise characterization using real-time feedback protocols and wavelet analysis

Recently, several groups have demonstrated two-qubit gate fidelities in semiconductor spin qubit systems above 99%. Achieving this regime of fault-tolerant compatible high fidelities is nontrivial and requires exquisite stability and precise control over the different qubit parameters over an extended period of time. This motivates the search for the efficient calibration of qubit control parameters against different sources of micro- and macroscopic noise and methods for noise analysis. Here, we present several single- and two-qubit parameter feedback protocols, optimized for and implemented in the state-of-the-art fast field-programmable gate array hardware. Furthermore, we present a wavelet-based analysis on feedback data collected over a ∼9 h time frame to gain insight into the different sources of noise in the system. Scalable feedback is an outstanding challenge and the presented implementation and analysis gives insight into the benefits and drawbacks of qubit parameter feedback, as feedback related overhead increases. This work demonstrates a pathway toward robust qubit parameter feedback and systematic noise analysis, crucial for mitigation strategies toward systematic high-fidelity qubit operation compatible with quantum error correction protocols.

Protecting entanglement between logical qubits via quantum error correction

Protecting entanglement between logical qubits via quantum error correction

A degressive quantum convolutional neural network for quantum state classification and code recognition

With the rapid development of quantum computing, a variety of quantum convolutional neural networks (QCNNs) are proposed. However, only features of an n-qubits input are transferred to the next layer in a quantum pooling layer, which results in the accuracy reduction. To solve this problem, a QCNN with a degressive circuit is proposed. In order to enhance the ability of extracting global features, we remove the parameters sharing strategy in the quantum convolutional layer and design a quantum convolutional kernel with global eyesight. In addition, to prevent a sharp feature reduction, a degressive parameterized quantum circuit is adopted to construct the pooling layer. Then the Z-basis measurement is only performed on the first qubit to control the operations on other qubits. Compared with the state-of-the-art QCNN, i.e., hybrid quantum-classical convolutional neural network, the accuracy of our model increased by 0.9%, 1%, and 3%, respectively, in three tasks: quantum state classification, binary code recognition, and quaternary code recognition.

Single-Shot Decoding of Good Quantum LDPC Codes

Quantum Tanner codes constitute a family of quantum low-density parity-check codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.

MDS, Hermitian almost MDS, and Gilbert–Varshamov quantum codes from generalized monomial-Cartesian codes

We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in m variables. When m=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=1$$\\end{document} our codes are MDS, and when m=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=2$$\\end{document} and our lower bound for the minimum distance is 3, the codes are at least Hermitian almost MDS. For an infinite family of parameters, when m=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=2$$\\end{document} we prove that our codes beat the Gilbert–Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.

Local disentanglers for the equivalence of two-dimensional topological quantum codes

Local disentanglers for the equivalence of two-dimensional topological quantum codes

Quantum-enhanced Chaotic Image Encryption: Strengthening Digital Data Security With 1-D Sine-based Chaotic Maps and Quantum Coding

The proposed work delves into the utilization of chaotic maps along with the quantum mechanisms to strengthen the security of the digital images along with addressing the limitations posed by conventional 1-D chaotic systems characterized by their pseudorandom and periodic attributes. Leveraging the inherent uncertainty in quantum theory, the proposed research introduces a new image encryption scheme. This method is built upon the concepts of two-dimensional quantum coding and the 1-D sine-based chaotic map (1-D SBCM). In the proposed encryption scheme, initially, a random sequence is created using 1-D SBCM by varying the seed parameters. This sequence is subsequently utilized for the purpose of scrambling. After that, a pseudorandom number generator (PRNG) is meticulously devised, drawing inspiration from the concepts of quantum coherence. This PRNG generates a confidential code stream that defies predictability and remains incongruent with the plaintext image, thereby contributing to heightened security levels. Subsequently, the research employs the novel enhanced quantum representation (NEQR) model, capitalizing on its advanced capabilities. Within this framework, the study introduces a quantum right cyclic shift operator, as well as a quantum XOR operator. These operators play a vital role in the formation of highly robust encrypted images. The application of these quantum operators not only enhances the security measures but also increases the complexity of the encryption procedure Statistical evidence derived from comprehensive experimentation corroborates the efficacy of the proposed image encryption scheme. Through rigorous analysis, it becomes evident that the system exhibits a robust performance in terms of strong security. The integration of quantum principles and quantum coding demonstrates that the proposed encryption scheme is resilient against potential threats.