Trellis decoders are a general decoding technique first applied to qubit-based quantum error correction codes by Ollivier and Tillich in 2006. Here, we improve the scalability and practicality of their theory, show that it has strong structure, extend the results using classical coding theory as a guide, and demonstrate a canonical form from which the structural properties of the decoding graph may be computed. The resulting formalism is valid for any prime-dimensional quantum system. The modified decoder works for any stabilizer code$S$and separates into two parts: 1) a one-time offline computation that builds a compact graphical representation of the normalizer of the code,$\mathcal {S}^{\perp}$and 2) a quick, parallel, online query of the resulting vertices using the Viterbi algorithm. We show the utility of trellis decoding by applying it to four high-density length-20 stabilizer codes for depolarizing noise and the well-studied Steane, rotated surface, and 4.8.8/6.6.6 color codes for$Z$only noise. Numerical simulations demonstrate a 20% improvement in the code-capacity threshold for color codes with boundaries by avoiding the mapping from color codes to surface codes. We identify trellis edge number as a key metric of difficulty of decoding, allowing us to quantify the advantage of single-axis ($X$or$Z$) decoding for Calderbank–Steane–Shor codes and block decoding for concatenated codes.
Trellis decoders are a general decoding technique first applied to qubit-based quantum error correction codes by Ollivier and Tillich in 2006. Here, we improve the scalability and practicality of their theory, show that it has strong structure, extend the results using classical coding theory as a guide, and demonstrate a canonical form from which the structural properties of the decoding graph may be computed. The resulting formalism is valid for any prime-dimensional quantum system. The modified decoder works for any stabilizer codeSand separates into two parts: 1) a one-time offline computation that builds a compact graphical representation of the normalizer of the code,S⊥and 2) a quick, parallel, online query of the resulting vertices using the Viterbi algorithm. We show the utility of trellis decoding by applying it to four high-density length-20 stabilizer codes for depolarizing noise and the well-studied Steane, rotated surface, and 4.8.8/6.6.6 color codes forZonly noise. Numerical simulations demonstrate a 20% improvement in the code-capacity threshold for color codes with boundaries by avoiding the mapping from color codes to surface codes. We identify trellis edge number as a key metric of difficulty of decoding, allowing us to quantify the advantage of single-axis (XorZ) decoding for Calderbank–Steane–Shor codes and block decoding for concatenated codes.
Quantum error correction is an essential ingredient for reliable quantum computation for theoretically provable quantum speedup. Topological color codes, one of the quantum error correction codes, have an advantage against the surface codes in that all Clifford gates can be implemented transversally. However, the hardness of decoding makes the color codes not suitable as the best candidate for experimentally feasible implementation of quantum error correction. Here we propose an Ising model formulation that enables highly accurate decoding of the color codes. In this formulation, we map stabilizer operators to classical spin variables to represent an error satisfying the syndrome. Then we construct an Ising Hamiltonian that counts the number of errors and formulate the decoding problem as an energy minimization problem of an Ising Hamiltonian, which is solved by simulated annealing. In numerical simulations on the (4.8.8) lattice, we find an error threshold of 10.36(5)% for bit-flip noise model, 18.47(5)% for depolarizing noise model, and 2.90(4)% for phenomenological noise model (bit-flip error is located on each of data and measurement qubits), all of which are higher than the thresholds of existing efficient decoding algorithms. Furthermore, we verify that the achieved logical error rates are almost optimal in the sense that they are almost the same as those obtained by exact optimizations by CPLEX with smaller decoding time in many cases. Since the decoding process has been a bottleneck for performance analysis, the proposed decoding method is useful for further exploration of the possibility of the topological color codes. Published by the American Physical Society 2024
Fault-tolerant quantum computation relies on scaling up quantum error correcting codes in order to suppress the error rate on the encoded quantum states. Topological codes, such as the surface code or color codes, are leading candidates for practical scalable quantum error correction and require efficient and scalable decoders. In this work, we propose and study the efficiency of a decoder for two-dimensional topological color codes on the 4.8.8 lattice (also known as the square-octagon code), by building on the work of Sarvepalli and Raussendorf [Phys. Rev. A 85, 022317 (2012)], for color codes on hexagonal lattices. The decoder is based on a rescaling approach, in which syndrome information on a part of the qubit lattice is processed locally, and then the lattice is rescaled iteratively to smaller sizes. We find a threshold of 6.0% for code capacity noise.
We study the algebraic compatibility between quantum homomorphic encryption and stabilizer quantum error correction in an honest-but-curious model, restricting the encryption to the transversal block-Pauli mask (XaZb)⊗n controlled by two secret key bits a and b. We prove a necessary and sufficient criterion for an [[n,1,d]] stabilizer code to remain compatible with this mask, namely that the transversal X and Z operators commute with every stabilizer generator, and we specialize it to a classical-code criterion for Calderbank--Shor--Steane codes. Combining code-space preservation with the logical action of the mask, every stabilizer code falls, with respect to each of the two transversal mask operators, into exactly one of three classes: incompatible, compatible but trivially acting, or compatible with a nontrivial logical Pauli action; the two operators need not fall into the same class. We verify this classification by exact symplectic computation for rotated and planar surface codes, toric codes, triangular and toric color codes, the [[15,1,3]] code and several small codes, and find that the class is fixed by the lattice size and the boundary presentation rather than by the topological phase: the surface-code family realizes all three classes, whereas color codes realize only the two compatible ones. For encrypted computation we show that the Shor code does not implement the logical T gate by a transversal T⊗9, whereas suitable triorthogonal codes do so up to a Clifford correction. We formulate logical-Pauli masking as a code-independent algebraic template. Syndrome extraction and recovery are key-independent whenever the mask preserves the code space, so that the client only needs the block size and the class of the code to decrypt.
The employment of quantum error correction codes (QECCs) within quantum computers potentially offers a reliability improvement for both quantum computation and communications tasks. However, the laws of quantum mechanics prevent us from directly invoking the mature family of classical error correction codes in the quantum domain. In order to circumvent the associated problems, the notion of quantum stabilizer codes (QSCs) was proposed in conjunction with syndrome-based decoding in the quantum regime. However, most of the powerful QSC schemes require long codewords for achieving a high performance, which potentially imposes additional challenges for implementation concerning their implementation, since their decoding may require longer than the quantum circuit's coherence time. Hence at the time of writing, QSCs exhibiting short to moderate codeword lengths are preferable. We commence by describing the pivotal problem encountered by classical error-correction codes, which also emerges when designing the QSCs, namely the intrinsic trade-off between the minimum distance versus coding rate. The complete formulation of this particular trade-off does not exist, but several lower and upper bounds can be found in the literature. It has been shown that a substantial gap can be observed between the upper and lower bound of the minimum distance, given the codeword length and the quantum coding rate. Hence, we propose an appealingly simple and invertible analytical approximation, for characterizing the trade-off between the quantum coding rate and the minimum distance of QSCs as well as their corresponding quantum bit error rate (QBER) performance upper-bound. For example, for a half-rate QSC having a codeword length of n = 128, the minimum distance is bounded by 11 < d < 22, while our approximation yields a minimum distance of d = 17 for the above-mentioned code. Next, we link this parametric study of the minimum distance versus quantum coding rate to the popular QSCs, namely to the family of quantum topological error correction codes (QTECCs). In order to construct the classical-to-quantum isomorphism, we conceive and investigate the family of classical topological error correction codes (TECCs), assuming that the bits of a codeword can be arranged in a lattice structure. We then present the classical-to-quantum isomorphism to pave the way for constructing their dual pairs in the quantum domain, which are the QTECCs. Finally, we characterize the performance of QTECCs in the face of the quantum depolarizing channel in terms of both their QBER and fidelity. Specifically, we demonstrate that for quantum coding rate rQ ~ 0, the threshold probability of the QBER below which the colour, rotated-surface, surface, and toric codes become capable of improving the uncoded QBER are given by 1.8 ×10-2, 1.3 ×10-2, 6.3 ×10-2 and 6.8 ×10-2, respectively. Furthermore, we also demonstrate that we can achieve beneficial fidelity improvements above the minimum fidelity of 0.94, 0.97 and 0.99 by employing the rQ = 1/7 colour code, the rQ = 1/9 rotated-surface code, and the rQ = 1/13 surface code, respectively. However, QSCs require additional quantum gates for their employment. Incorporating more quantum gates for performing error correction potentially introduces further sources of quantum decoherence into quantum computers. In this scenario, the primary challenge is to find the sufficient condition required by each of the quantum gates for beneficially employing QECCs in order to yield reliability improvements, given that the quantum gates utilized by the QECCs also introduce quantum decoherence. In this treatise, we approach this problem by firstly presenting the general framework of protecting quantum gates by the amalgamation of the transversal configuration of quantum gates and QSCs, which can be viewed as syndrome-based QECCs. Secondly, we provide examples of the advocated framework by invoking QTECCs for protecting both transversal Hadamard gates and controlled (CNOT) gates. Both our simulation and analytical results explicitly show that by utilizing QTECCs, the fidelity of the quantum gates can be beneficially improved, provided that quantum gates satisfying a certain minimum depolarization fidelity threshold (Fth) are available. For instance, for protecting transversal Hadamard gates, the minimum fidelity values required for each of the gates in order to attain fidelity improvements are 99.74%, 99.73%, 99.87%, and 99.86%, when they are protected by colour, rotated-surface, surface, and toric codes, respectively. Unfortunately, these specific Fth values can only be obtained for a very large number of physical qubits (n → ∞), when the quantum coding rate of the QTECCs approaches zero (rQ → 0). Finally, in order to conceive QSCs exhibiting a high quantum coding rate, we modify the construction of QTECCs for conceiving a low-complexity concatenated quantum turbocode (QTC). The above-mentioned high quantum coding rate is obtained by combining the quantum-domain version of short-block codes (SBCs) also known as single parity-check (SPC) codes as the outer codes and quantum unity-rate codes (QURCs) as the inner codes. Despite its design simplicity, the proposed QTC yields a near-hashing-bound error correction performance. For instance, compared to the best half-rate QTC known in the literature, namely the quantum irregular convolutional codes (QIrCCs) combined with the QURC scheme, which operates at the distance of D = 0.037 from the quantum hashing bound, our novel QSBC-QURC scheme can operate at the lower distance of D = 0:029. It is also worth mentioning that this is the first instantiation of QTCs capable of adjusting the quantum encoders according to the quantum coding rate required for mitigating the Pauli errors imposed by the time-variant depolarizing probabilities of the quantum channel.
Quantum error correction (QEC) represents a critical enabler for fault-tolerant quantum computing, protecting fragile quantum states from decoherence and operational errors. This paper provides a comprehensive analysis of leading quantum error correction schemes, focusing on surface codes and topological codes within the context of Noisy Intermediate-Scale Quantum (NISQ) era processors. We examine the theoretical foundations of stabilizer codes, compare surface code implementations with code distances ranging from d=3 to d=17, and analyze topological codes including toric codes and color codes. Our investigation encompasses syndrome extraction circuits, logical qubit encoding strategies, and threshold analysis for contemporary quantum hardware platforms. Performance metrics reveal that surface codes achieve error thresholds of approximately 1% with realistic noise models, while topological codes offer architectural advantages for specific qubit connectivity constraints. We discuss implementation challenges on current NISQ processors including IBM Quantum, Google Sycamore, and IonQ systems, addressing qubit overhead, gate fidelity requirements, and decoding algorithms. This analysis provides insights for selecting appropriate error correction schemes based on application requirements and hardware capabilities.
Quantum error correction (QEC) is considered essential for the development of scalable, fault-tolerant quantum computers in the medium- to long-term. By encoding quantum information across many physical qubits, carefully designed measurements can be taken to gain limited knowledge as to errors which may have occurred in the system, known as the error syndrome, with out causing quantum decoherence. In surface codes, qubits are arranged in a two-dimensional topological space, such as a torus or plane. In quantum low-density parity-check (qLDPC) codes, this is generalised to effectively arbitrary arrangements of qubits and syndrome checks. This work studies decoders: the algorithms responsible for inferring error correcting decisions from syndromes. In particular, the union–find decoder is studied in terms of its performance at scale. A simulation of a union–find decoding architecture is used to identify computational bottlenecks and pro pose improvements. In literature, it is assumed that the decoder runs in time scaling quadratically in the number of qubits under a naive implementation, but near-linearly when including two well-known optimisations. A key result is that, under independent and phenomenological noise models on surface codes, this complexity is strictly linear regardless of these optimisations. A supporting analytical argument is presented using percolation theory. Generalisation of the decoding problem to qLDPC codes is also studied, with approaches broadly relying on Gaussian elimination to find appropriate solutions. A strategy is proposed which uses a novel online variant of the Gaussian elimination algorithm to solve this linear system incrementally on growing local clusters, with accompanying complexity analysis and empir ical data demonstrating a reduction in runtime. An investigation into the use of metachecks is also presented, inspired by single-shot decoding, with implications for how such qLDPC decoders could be further improved.