QEC (Quantum Error Correction) assumes that qubit states are disturbed by errors primarily through interactions with their local environment states. The problem is that, when quantum states are superpositioned, QEC improperly assumes all the superpositioned terms share same local environment states and resulting errors though they are, in general, different term by term to which QEC with the fixed number of syndrome qubits is, except for some slightly superpositioned states such as two terms ones, not applicable. Further, fundamental difficulty to scale quantum parallelism regardless of implementation and physical details is derived from theory of Shannon.
QEC (Quantum Error Correction) assumes that qubit states are disturbed by errors primarily through interactions with their local environment states. The problem is that, when quantum states are superpositioned, QEC improperly assumes all the superpositioned terms share same local environment states and resulting errors though they are, in general, different term by term to which QEC with the fixed number of syndrome qubits is, except for some slightly superpositioned states such as two terms ones, not applicable. Further, fundamental difficulty to scale quantum parallelism regardless of implementation and physical details is derived from theory of Shannon.
An introduction to quantum error correction (QEC) is given, and some recent developments are described. QEC consists of two parts: the physics of error processes and their reversal, and the construction of quantum error–correcting codes. Errors are caused both by imperfect quantum operations, and by coupling between the quantum system and its environment. Any such process can be analysed into a sum of ‘error operators’, which are tensor products of Pauli spin operators. These are the analogues of classical error vectors. A quantum error correcting code is a set of orthogonal states, ‘quantum codewords’, which behave in a certain useful way under the action of the most likely error operators. A computer or channel which only uses such states can be corrected by measurements which determine the error while yielding no information about which codeword or superposition of codewords is involved. Powerful codes can be found using a construction based on classical error–correcting codes. An analysis which allows even the corrective operations themselves to be imperfect leads to powerful and counter–intuitive results: the quantum coherence of a long quantum computation can be preserved even though every qubit in the computer relaxes spontaneously many times before the computation is complete.
1. IntroductionQuantum system is vulnerable to disturbance from the external environment, which leads todecoherence in the system. We have to overcome this difficulty to realize a working quanutumcomputer and a dependable quantum information processing. Quantum error correction (QEC)[1, 2, 3] is one of the most promising candidates for overcoming decoherence.QEC proposals to date are separated roughly into two classes, one employs extra ancilla qubitsfor error syndrome readout while the other, called operator quantum error correction (OQEC)employs a higher rank projection operator. However, the schemes have not been built in actualquantum computing or quantum information processing so far. This is dueto the following obstaclesinherent in these QECs; the syndrome must be read out by introducing ancilla qubits duringcomputing/information processing in the former case, while realization of a high-rank projectionoperator is physically challenging in the latter case.It was shown in [4] that for some quantum channels there exist different QECs in which nosyndrome measurements, no ancillas and no projection operators were required. Recovery operationand decoding operation are combined into a single unitary operation and the output state is a directproductof a decoded qubitstate and an ancilla state. Aqubit state isreproducedwithout recoveringthe codeword and, moreover, the projection operation is automatically built in our output state.The purpose of this paper is to extend the results in [4] to general quantum channels. We showthat for any quantum channel there is a unitary recovery operation, with which the output stateis a tensor product of decoded qubit state and an ancilla state. As a result, a decoding scheme canbe done by a unitary gate followed by a partial trace operation.The rest of the paper is organized as follows. We introduce the basic notions of OQEC andthen prove the main theorem in Section 2. We also give simple examples demonstrating our resultand a simplified proof of a theorem given in [2] illustrating that our recovery channel can be usedto do correction for many other channels related to ours. Section 3 is devoted to summary anddiscussions.
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.
Quantum error correction (QEC) is required in quantum computers to mitigate the impact of errorson physical qubits. The goal is to optimize the neural network for high decoding performance whilemaintaining a minimalistic hardware implementation. The errors associated with decoherence can bereduced by adopting QEC schemes that encode multiple imperfect physical qubits into a logical quantumstate, similar to classical error correction. The relevance of these studies lies in the mathematicaland software modeling and implementation of corrective codes to correct several types of quantumerrors in the development and implementation of quantum algorithms for solving classes of problems ofa classical nature. The scientific novelty of this direction is expressed in the elimination of one of theshortcomings of the quantum computing process. The development of the theory and principles for constructingmodeling systems that are resistant to external interference (dependence of data distortion onnoise, dependence of the error of a quantum computing process on the measure and purity of entanglement)for modeling quantum computing is a dynamic area, as evidenced by a large number of existingmodels reflecting certain quantum computational processes and phenomena (quantum teleportation,parallelism, entanglement of quantum states) and scientific papers. Although quantum computing is notyet ready to move from theory to practice, it is nevertheless possible to reasonably guess what form aquantum computer might take, or, more importantly for programming language design, what interface itwould be possible to interact with such a quantum computer. It is natural to apply the lessons learnedfrom the programming of classical computing to quantum computing. The analysis of works in this areashowed that a new qualitative level has now been reached, which opens up promising opportunities forthe implementation of multi-qubit quantum computing. Prospects for implementation and developmentare associated not only with technological capabilities, but also with the solution of issues of buildingeffective quantum systems for solving actual mathematical problems, cryptography problems and control(optimization) problems.
Abstract Quantum error correction (QEC) codes are usually designed to correct errors regardless of their physical origins. In large-scale devices, this is an essential feature. In smaller-scale devices, however, the main error sources are often understood, and this knowledge could be exploited for more efficient error correction. Optimizing the QEC protocol is therefore a promising strategy in smaller devices. Typically, this involves tailoring the protocol to a given decoherence channel by solving an appropriate optimization problem. Here we introduce a new optimization-based approach, which maximizes the robustness to faults in the recovery. Our approach is inspired by recent experiments, where such faults have been a significant source of logical errors. We illustrate this approach with a three-qubit model, and show how near-term experiments could benefit from more robust QEC protocols.
Fault-tolerant protocol assumes the application of error correction after every quantum gate. However, correcting errors is costly in terms of time and number of qubits. Here we demonstrate that quantum error correction (QEC) can be applied significantly less often with only a minimal loss of fidelity. This is done by simulating the implementation of 50 encoded, single-qubit, quantum gates within the [[7,1,3]] QEC code in a noisy, nonequiprobable Pauli error environment with QEC being applied at different intervals. We find that applying QEC after every gate is rarely optimal and even applying QEC only once after all 50 gates sacrifices only a slight amount of fidelity with the benefit of 50-fold saving of resources. In addition, we find that in cases where bit-flip errors are dominant, it is best not to apply QEC at all.