Having protected quantum information is essential to perform quantum computations. One possibility is to reduce the number of particles needing to be protected from noise and instead use systems with more states, so called qudit quantum computers. In this paper we show that codes for these systems can be derived from already known codes, and in particular this procedure removes the need for shared entanglement in entanglement-assisted quantum error-correcting codes, which is a result which could prove to be useful for fault-tolerant qudit, and even qubit, quantum computers.
In this paper, we show how to construct non-binary entanglement-assisted stabilizer quantum codes by using pre-shared entanglement between the sender and receiver. We also give an algorithm to determine the circuit for non-binary entanglement-assisted stabilizer quantum codes and some illustrated examples. The codes we constructed do not require the dual-containing constraint, and many non-binary classical codes, like non-binary LDPC codes, which do not satisfy the condition, can be used to construct non-binary entanglement-assisted stabilizer quantum codes.
An Entanglement-Assisted Stabilizer Framework for Distributed Sensing of Local Phases
School of Electrical Engineering, Korea University（韩国大学电气工程学院）；Quantum Network Research Center, Korea Institute of Science and Technology Information（韩国科学技术信息研究院量子网络研究中心）
The scheme of entanglement-assisted quantum error-correcting (EAQEC) codes assumes that the ebits of the receiver are error-free. In practical situations, errors on these ebits are unavoidable, which diminishes the error-correcting ability of these codes. We consider two different versions of this problem. We first show that any (nondegenerate) standard stabilizer code can be transformed into an EAQEC code that can correct errors on the qubits of both sender and receiver. These EAQEC codes are equivalent to standard stabilizer codes, and hence the decoding techniques of standard stabilizer cod…
Entangled qubit can increase the capacity of quantum error correcting codes based on stabilizer codes. In addition, by using entanglement quantum stabilizer codes can be construct from classical linear codes that do not satisfy the dual-containing constraint. We show that it is possible to construct both additive and non-additive quantum codes using the codeword stabilized quantum code framework. Nonadditive codes may offer improved performance over the more common sta- bilizer codes. Like other entanglement-assisted codes, the encoding procedure acts only the qubits on Alice's side, and only …
Quantum error-correcting codes are used to protect quantum information from decoherence. A raw state is mapped, by an encoding circuit, to a codeword so that the most likely quantum errors from a noisy quantum channel can be removed after a decoding process. A good encoding circuit should have some desired features, such as low depth, few gates, and so on. In this paper, we show how to practically implement an encoding circuit of gate complexity $O(n(n-k+c)/\log n)$ for an $[[n,k;c]]$ quantum stabilizer code with the help of $c$ pairs of maximally-entangled states. For the special case of an $…
In the previous research by Grassl, Huber and Winter, they proved a theorem which can make entanglement-assisted quantum error-correcting codes (EAQECC) from general quantum error-correcting codes (QECC). In this paper, we prove that the shortened EAQECC is a stabilizer code if the original EAQECC is a stabilizer code.
Operator quantum error correction is a recently developed theory that provides a generalized framework for active error correction and passive error avoiding schemes. In this paper, we describe these codes in the stabilizer formalism of standard quantum error correction theory. This is achieved by adding a "gauge" group to the standard stabilizer definition of a code that defines an equivalence class between encoded states. Gauge transformations leave the encoded information unchanged; their effect is absorbed by virtual gauge qubits that do not carry useful information. We illustrate the cons…
We develop the theory of entanglement-assisted quantum error correcting (EAQEC) codes, a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to pre-shared entanglement. Conventional stabilizer codes are equivalent to dual-containing symplectic codes. In contrast, EAQEC codes do not require the dual-containing condition, which greatly simplifies their construction. We show how any quaternary classical code can be made into a EAQEC code. In particular, efficient modern codes, like LDPC codes, which attain the Shannon capacity, can be made into E…
We construct a CCCZ gate using six T gates, assisted by stabilizer operations and classical feedback. More generally, we reduce the T cost of a $C^{n}Z$ gate from $4n-4$ to $4n-6$, for $n > 2$.
We present a local hidden-variable model supplemented by classical communication that reproduces the quantum-mechanical predictions for measurements of all products of Pauli operators on an n-qubit GHZ state (or "cat state"). The simulation is efficient since the required amount of communication scales linearly with the number of qubits, even though there are Bell-type inequalities for these states for which the amount of violation grows exponentially with n. The structure of our model yields insight into the Gottesman-Knill theorem by demonstrating that, at least in this limited case, the cor…
We present a model, motivated by the criterion of reality put forward by Einstein, Podolsky, and Rosen and supplemented by classical communication, which correctly reproduces the quantum-mechanical predictions for measurements of all products of Pauli operators on an n-qubit GHZ state (or ``cat state''). The n-2 bits employed by our model are shown to be optimal for the allowed set of measurements, demonstrating that the required communication overhead scales linearly with n. We formulate a connection between the generation of the local values utilized by our model and the stabilizer formalism…
Zeno-Enhanced Probabilistic Error Cancellation with Quantum Error Detection Codes
Tsinghua University（清华大学）；Frontier Science Center for Quantum Information（量子信息前沿科学中心）；Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；Hefei National Laboratory（合肥国家实验室）
Non-local Magic: closed-form solution and equivalence with magic of purification
Università degli Studi di Salerno（萨莱诺大学）；Istituto Nazionale di Fisica Nucleare (INFN)（意大利国家核物理研究所）；Scuola Superiore Meridionale（南方高等学院）；Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）
Unified and Generalized Approach to Entanglement-Assisted Quantum Error Correction
Xanadu；Department of Mathematics & Statistics, University of Guelph（圭尔夫大学数学与统计系）；Perimeter Institute for Theoretical Physics（理论物理珀米特研究所）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is defined from the orthogonal group of a simplified stabilizer group. From the Poisson summation formula, this duality leads to the MacWilliams identities and linear programming bounds for EAQEC codes. We establish a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits.
Recently, there has been an emergence of useful applications for noisy intermediate-scale quantum (NISQ) devices notably, though not exclusively, in the fields of quantum machine learning and variational quantum algorithms. In such applications, circuits of various depths and composed of different sets of gates are run on NISQ devices. Therefore, it is crucial to find practical ways to capture the general performance of circuits on these devices. Motivated by this pressing need, we modified the standard Clifford randomized benchmarking (RB) and interleaved RB schemes targeting them to hardware…
Contrary to the assumption that most quantum error-correcting codes (QECC) make, it is expected that phase errors are much more likely than bit errors in physical devices. By employing the entanglement-assisted stabilizer formalism, we develop a new kind of error-correcting protocol which can flexibly trade error correction abilities between the two types of errors, such that high error correction performance is achieved both in symmetric and in asymmetric situations. The characteristics of the QECCs can be optimized in an adaptive manner during information transmission. The proposed entanglem…
Entanglement-assisted quantum (QUENTA) codes are a subclass of quantum error-correcting codes which use entanglement as a resource. These codes can provide error correction capability higher than the codes derived from the traditional stabilizer formalism. In this paper, it is shown a general method to construct QUENTA codes from cyclic codes. Afterwards, the method is applied to Reed-Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of QUENTA codes. Two families of QUENTA codes are maximal distance separable (MDS), and one is almost MDS or alm…
The entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum…
Binary Gauss Stabilizers for Abelian Lattice Gauge Theories
Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院）；PQI – Portuguese Quantum Institute（葡萄牙量子研究所）；Centro de Física e Engenharia de Materiais Avançados (CeFEMA)（先进材料与工程物理中心）；University of Trento（特伦托大学）；INFN-TIFPA Trento Institute of Fundamental Physics and Applications（特伦托基础物理与应用研究所）
Environmental records unlock universal quantum computation from thermal decoherence
HK Institute of Quantum Science & Technology, The University of Hong Kong（香港量子科学研究所，香港大学）；QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong（QICI量子信息与计算倡议，计算与数据科学学院，香港大学）
We present a general formalism for quantum error-correcting codes that encode both classical and quantum information (the EACQ formalism). This formalism unifies the entanglement-assisted formalism and classical error correction, and includes encoding, error correction, and decoding steps such that the encoded quantum and classical information can be correctly recovered by the receiver. We formally define this kind of quantum code using both stabilizer and symplectic language, and derive the appropriate error-correcting conditions. We give several examples to demonstrate the construction of su…
As in classical coding theory, quantum analogues of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane (CSS) construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the …
Anti-Flatness and Non-Local Non-stabilizerness in Two-Particle Scattering Processes
Fakultät für Physik, Universität Bielefeld（比勒费尔德大学物理学院）；GSI Helmholtzzentrum für Schwerionenforschung（德国重离子研究中心）；Institute for Nuclear Theory（核理论研究所）；InQubator for Quantum Simulation (IQuS), Department of Physics, University of Washington（华盛顿大学物理系量子模拟创新中心）
We incorporate active and passive quantum error-correcting techniques to protect a set of optical information modes of a continuous-variable quantum information system. Our method uses ancilla modes, entangled modes, and gauge modes (modes in a mixed state) to help correct errors on a set of information modes. A linear-optical encoding circuit consisting of offline squeezers, passive optical devices, feedforward control, conditional modulation, and homodyne measurements performs the encoding. The result is that we extend the entanglement-assisted operator stabilizer formalism for discrete vari…
Fault-tolerant quantum computation (FTQC) schemes that use multi-qubit large block codes can potentially reduce the resource overhead to a great extent. A major obstacle is the requirement of a large number of clean ancilla states of different types without correlated errors inside each block. These ancilla states are usually logical stabilizer states of the data code blocks, which are generally difficult to prepare if the code size is large. Previously we have proposed an ancilla distillation protocol for Calderbank-Shor-Steane (CSS) codes by classical error-correcting codes. It was assumed t…
Many-body chirality of topological stabilizer states
Purdue University（普渡大学）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；University of Waterloo（滑铁卢大学）；IBM Quantum, T.J. Watson Research Center（IBM量子，T.J.沃森研究中心）；University of Toronto（多伦多大学）
Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct several families of QUENTA codes via the Euclidean and the Hermitian construction. Two of the families crea…
Power and Limits of Collective Local Measurements in Multicopy State Discrimination
School of Mathematics, South China University of Technology（华南理工大学数学学院）；School of Computer Science and Technology, Dongguan University of Technology（东莞理工学院计算机科学与技术学院）
It is commonly expected that quantum theory is universal, in that it describes the world at all scales. Yet, quantum effects at the macroscopic scale continue to elude our experimental observation. This fact is commonly attributed to decoherence processes affecting systems of sufficiently large number of constituent subsystems leading to an effective macro-scale beyond which a quantum description of the whole system becomes superfluous. Here, we show both theoretically and experimentally that the existence of such a scale is unjustifiable from an information-theoretic perspective. We introduce…
As a quantum-informative window into quantum many-body physics, the concept and application of entanglement renormalization group (ERG) have been playing a vital role in the study of novel quantum phases of matter, especially long-range entangled (LRE) states in topologically ordered systems. For instance, by recursively applying local unitaries as well as adding/removing qubits that form product states, the 2D toric code ground states, i.e., fixed point of Z_2 topological order, are efficiently coarse-grained with respect to the system size. As a further improvement, the addition/removal of 2…
Guiding many-body systems to desired states is a central challenge of modern quantum science, with applications from quantum computation to many-body physics and quantum-enhanced metrology. Approaches to solving this problem include step-by-step assembly, reservoir engineering to irreversibly pump towards a target state, and adiabatic evolution from a known initial state. Here we construct low-entropy quantum fluids of light in a Bose Hubbard circuit by combining particle-by-particle assembly and adiabatic preparation. We inject individual photons into a disordered lattice where the eigenstate…
