A simple and unifying method to show the perfect error-correcting condition is provided based on the quantum mutual information. The one-to-one parameterization of quantum operations and the properties of the quantum relative entropy are used effectively in this paper, where the equivalence between the subspace transmission and the entanglement transmission is clearly presented. We also revisit a variant of the no-cloning and no-deleting theorem based on an information-theoretical tradeoff between two parties for the reversibility of quantum operations, and demonstrate that the no-cloning and no-deleting theorem leads to the perfect error-correcting condition on Kraus operators.
Large-scale universal quantum computing requires the implementation of quantum error correction (QEC). While the implementation of QEC has already been demonstrated for quantum memories, reliable quantum computing requires also the application of nontrivial logical gate operations to the encoded qubits. Here, we present examples of such operations by implementing, in addition to the identity operation, the NOT and the Hadamard gate to a logical qubit encoded in a five qubit system that allows correction of arbitrary single qubit errors. We perform quantum process tomography of the encoded gate…
We provide a systematic way of constructing entanglement-assisted quantum error-correcting codes via graph states in the scenario of preexisting perfectly protected qubits. It turns out that the preexisting entanglement can help beat the quantum Hamming bound and can enhance (not only behave as an assistance) the performance of the quantum error correction. Furthermore we generalize the error models to the case of not-so-perfectly-protected qubits and introduce the quantity infidelity as a figure of merit and show that our code outperforms also the ordinary quantum error-correcting codes.
We expand the class of holographic quantum error correcting codes by developing the notion of block perfect tensors, a wider class that includes previously defined perfect tensors. The relaxation of this constraint opens up a range of other holographic codes. We demonstrate this by introducing the self-dual CSS heptagon holographic code, based on the 7-qubit Steane code. Finally we show promising thresholds for the erasure channel by applying a straightforward, optimal erasure decoder to the heptagon code and benchmark it against existing holographic codes.
We devise a scheme that protects quantum coherent states of light from probabilistic losses, thus achieving the first continuous-variable quantum erasure-correcting code. If the occurrence of erasures can be probed, then the decoder enables, in principle, a perfect recovery of the original light states. Otherwise, if supplemented with postselection based on homodyne detection, this code can be turned into an efficient erasure-filtration scheme. The experimental feasibility of the proposed protocol is carefully addressed.
Recent studies of globally controlled structures have culminated in a theoretical demonstration that fault-tolerant quantum computation can be carried out on a one--dimensional chain with control over two global fields only. This required some patterns of classical states to localise operations, which were stabilised with the Zeno effect. However, it is impossible to achieve perfect stabilisation using this method, so error correction of these states is desirable, and is the focus this paper.
We develop a family of perfect quantum error correcting codes that correct for phase errors that arise on any qubit, at any time, during a perfect state transfer experiment. These ensure that we find the optimal operating regime for corrected state transfer. For a specific class of system, we further show that while dephasing noise can be corrected, depolarising noise cannot.
Mitchison and Jozsa recently suggested that the "chained-Zeno" counterfactual computation protocol recently proposed by Hosten et al. is counterfactual for only one output of the computer. This claim was based on the existing abstract algebraic definition of counterfactual computation, and indeed according to this definition, their argument is correct. However, a more general definition (physically adequate) for counterfactual computation is implicitly assumed by Hosten et. al. Here we explain in detail why the protocol is counterfactual and how the "history tracking" method of the existing de…
Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate t…
The possibility of using the two-fold topological degeneracy of spin-1/2 chiral spin liquid states on the torus to construct quantum error correcting codes is investigated. It is shown that codes constructed using these states on finite periodic lattices do not meet the necessary and sufficient conditions for correcting even a single qubit error with perfect fidelity. However, for large enough lattice sizes these conditions are approximately satisfied, and the resulting codes may therefore be viewed as approximate quantum error correcting codes.
The design of time-independent local Hamiltonians that realise quantum algorithms is derived from the study of perfect state transfer. The novel features of this evolution are the perfect realisation of the computation, and the ability to implement operations in parallel rather than sequentially. This is applied to a quantum circuit for concatenated error correcting codes, and the properties of the eigenstates are compared to those of topological memories. An erroneous implementation of the algorithm (one where a single error could lead to a fault in the output) is found not to be topological,…
Simpler encoding and decoding networks are necessary for more reliable quantum error correcting codes (QECCs). The simplification of the encoder-decoder circuit for a perfect five-qubit QECC can be derived analytically if the QECC is converted from its equivalent one-way entanglement purification protocol (1-EPP). In this work, the analytical method to simplify the encoder-decoder circuit is introduced and a circuit that is as simple as the existent simplest circuits is presented as an example. The encoder-decoder circuit presented here involves nine single- and two-qubit unitary operations, o…
Topological quantum error-correcting codes are a promising candidate for building fault-tolerant quantum computers. Decoding topological codes optimally, however, is known to be a computationally hard problem. Various decoders have been proposed that achieve approximately optimal error thresholds. Due to practical constraints, it is not known if there exists an obvious choice for a decoder. In this paper, we introduce a framework which can combine arbitrary decoders for any given code to significantly reduce the logical error rates. We rely on the crucial observation that two different decodin…
We present and analyze a protocol for quantum steganography where the sender (Alice) encodes her steganographic information into the error syndromes of the perfect (five-qubit) quantum error-correcting code, and sends it to the receiver (Bob) over a depolarizing channel. Alice and Bob share a classical secret key, and hide quantum information in such a way that to an eavesdropper (Eve) without access to the secret key, the quantum message looks like an innocent codeword with a typical sequence of quantum errors. We calculate the average rate of key consumption, and show how the protocol improv…
We consider the problem of correcting the errors incurred from sending quantum information through a noisy quantum environment by using classical information obtained from a measurement on the environment. For discrete time Markovian evolutions, in the case of fixed measurement on the environment, we give criteria for quantum information to be perfectly corrigible and characterize the related feedback. Then we analyze the case when perfect correction is not possible and, in the qubit case, we find optimal feedback maximizing the channel fidelity.
The stabilizing properties of one-error correcting jump codes are explored under realistic non-ideal conditions. For this purpose the quantum algorithm of the tent-map is decomposed into a universal set of Hamiltonian quantum gates which ensure perfect correction of spontaneous decay processes under ideal circumstances even if they occur during a gate operation. An entanglement gate is presented which is capable of entangling any two logical qubits of different one-error correcting code spaces. With the help of this gate simultaneous spontaneous decay processes affecting physical qubits of dif…
We consider the problem of correcting the errors incurred from sending classical or quantum information through a noisy quantum environment by schemes using classical information obtained from a measurement on the environment. We give a conditions for quantum or classical information (prepared in a specified input basis B) to be corrigible based on a measurement M. Based on these criteria we give examples of noisy channels such that (1) no information can be corrected by such a scheme (2) for some basis B there is a correcting measurement M (3) for all bases B there is an M (4) there is a meas…
I describe a method for pasting together certain quantum error-correcting codes that correct one error to make a single larger one-error quantum code. I show how to construct codes encoding 7 qubits in 13 qubits using the method, as well as 15 qubits in 21 qubits and all the other "perfect" codes.
This paper introduces PyMatching, a fast open-source Python package for decoding quantum error-correcting codes with the minimum-weight perfect matching (MWPM) algorithm. PyMatching includes the standard MWPM decoder as well as a variant, which we call local matching, that restricts each syndrome defect to be matched to another defect within a local neighbourhood. The decoding performance of local matching is almost identical to that of the standard MWPM decoder in practice, while reducing the computational complexity approximately quadratically. We benchmark the performance of PyMatching, sho…
Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state, and state decoding. To address this challenge, we experimentally realise the $[\![5,1,3]\!]$ code, the so-called smallest perfect code that permits c…
We report two analytical bounds for quantum error-correcting codes that do not have preexisting classical counterparts. Firstly the quantum Hamming and Singleton bounds are combined into a single tighter bound, and then the combined bound is further strengthened via the well-known Lloyd's theorem in classical coding theory, which claims that perfect codes, codes attaining the Hamming bound, do not exist if the Lloyd's polynomial has some non-integer zeros. Our bound characterizes quantitatively the improvement over the Hamming bound via the non-integerness of the zeros of the Lloyd's polynomia…
Aix-Marseille Université（艾克斯-马赛大学）；École Normale Superieure de Lyon（里昂高等师范学院）；CNRS（法国国家科学研究中心）
Quantum machine learning algorithms could provide significant speed-ups over their classical counterparts; however, whether they could also achieve good generalization remains unclear. Recently, two quantum perceptron models which give a quadratic improvement over the classical perceptron algorithm using Grover's search have been proposed by Wiebe et al. arXiv:1602.04799 . While the first model reduces the complexity with respect to the size of the training set, the second one improves the bound on the number of mistakes made by the perceptron. In this paper, we introduce a hybrid quantum-clas…
Linear-Time Encodable and Decodable Quantum Error-Correcting Codes
Massachusetts Institute of Technology（麻省理工学院）；Hon Hai (Foxconn) Research Institute（鸿海（富士康）研究院）；University of California at San Diego（加州大学圣地亚哥分校）；CSAIL, Massachusetts Institute of Technology（麻省理工学院计算机科学与人工智能实验室）
We derive the effective channel for a logical qubit protected by an arbitrary quantum error-correcting code, and derive the map between channels induced by concatenation. For certain codes in the presence of single-bit Pauli errors, we calculate the exact threshold error probability for perfect fidelity in the infinite concatenation limit. We then use the control theory technique of balanced truncation to find low-order non-asymptotic approximations for the effective channel dynamics.
AI-Enabled Decoding of Qubit Loss for Quantum Error-Correcting Codes
Tsinghua University（清华大学）；Intelligent Quantum Inception Co., Ltd.（智量量子创新有限公司）；Institute for Advanced Study, Tsinghua University（清华大学高等研究院）；Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；iFLYTEK Research（科大讯飞研究院）
Quantum error correction and symmetries play central roles in quantum information science and physics. It is known that quantum error-correcting codes covariant with respect to continuous symmetries cannot correct erasure errors perfectly (an important case being the Eastin-Knill theorem), in contrast to the case without symmetry constraints. Furthermore, there are fundamental limits on the accuracy of such covariant codes for approximate quantum error correction. Here, we consider the quantum error correction capability of random covariant codes. In particular, we show that $U(1)$-covariant c…
The strongly correlated systems we use to realise quantum error-correcting codes may give rise to high-weight, problematic errors. Encouragingly, we can expect local quantum error-correcting codes with no string-like logical operators $-$ such as the cubic code $-$ to be robust to highly correlated, one-dimensional errors that span their lattice. The challenge remains to design decoding algorithms that utilise the high distance of these codes. Here, we begin the development of such algorithms by proposing an efficient decoder for the `Fibonacci code'; a two-dimensional classical code that mimi…
The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the $O(3)$ non-linear sigma model in $1+1$ dimensions, one of the simplest quantum field theories encompassing cr…
Following the introduction of the task of reference frame error correction, we show how, by using reference frame alignment with clocks, one can add a continuous Abelian group of transversal logical gates to any error-correcting code. With this we further explore a way of circumventing the no-go theorem of Eastin and Knill, which states that if local errors are correctable, the group of transversal gates must be of finite order. We are able to do this by introducing a small error on the decoding procedure that decreases with the dimension of the frames used. Furthermore, we show that there is …
A usual code for quantum wiretap channel requires an auxiliary random variable subject to the perfect uniform distribution. However, it is difficult to prepare such an auxiliary random variable. We propose a code that requires only an auxiliary random variable subject to a non-uniform distribution instead of the perfect uniform distribution. Further, we evaluate the exponential decreasing rate of leaked information and derive its equivocation rate. For practical constructions, we also discuss the security when our code consists of a linear error correcting code.
Mapping quantum error correcting codes to classical disordered statistical mechanics models and studying the phase diagram of the latter has proven a powerful tool to study the fundamental error robustness and associated critical error thresholds of leading quantum error correcting codes under phenomenological noise models. In this work, we extend this mapping to admit realistic, multi-parameter faulty quantum circuits in the description of quantum error correcting codes. Based on the underlying microscopic circuit noise model, we first systematically derive the associated strongly correlated …
In the current thesis we first talk about the six-qubit quantum error-correcting code and show its connections to entanglement-assisted error-correcting coding theory and then to subsystem codes. This code bridges the gap between the five-qubit (perfect) and Steane codes. We discuss two methods to encode one qubit into six physical qubits. Each of the two examples corrects an arbitrary single-qubit error. The first example is a degenerate six-qubit quantum error-correcting code. We prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Stean…
Quantum error correction and symmetries play central roles in quantum information science and physics. It is known that quantum error-correcting codes that obey (are covariant with respect to) continuous symmetries in a certain sense cannot correct erasure errors perfectly (a well-known result in this regard being the Eastin-Knill theorem in the context of fault-tolerant quantum computing), in contrast to the case without symmetry constraints. Furthermore, several quantitative fundamental limits on the accuracy of such covariant codes for approximate quantum error correction are known. Here, w…
qecGPT: decoding Quantum Error-correcting Codes with Generative Pre-trained Transformers
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（中国科学院大学杭州高等研究院基础物理与数学科学学院）；International Centre for Theoretical Physics Asia-Pacific（国际理论物理中心亚太地区）
We present a family of algorithms, combining real-space renormalization methods and belief propagation, to estimate the free energy of a topologically ordered system in the presence of defects. Such an algorithm is needed to preserve the quantum information stored in the ground space of a topologically ordered system and to decode topological error-correcting codes. For a system of linear size L, our algorithm runs in time log L compared to L^6 needed for the minimum-weight perfect matching algorithm previously used in this context and achieves a higher depolarizing error threshold.
