Richard Cleve, Daniel Gottesman
We show how, given any set of generators of the stabilizer of a quantum code, an efficient gate array that computes the codewords can be constructed. For an n-qubit code whose stabilizer has d generators, the resulting gate array consists of O(n d) operations, and converts k-qubit data (where k = n - d) into n-qubit codewords.
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Fault-tolerant correction-ready encoding of the [[7,1,3]] Steane code on a 2D grid
本研究提出在二维网格上利用少量 Flag-Bridge 量子比特的奇偶校验编码方法，容错地准备 [[7,1,3]] Steane 码零逻辑态，实现低错误率与紧凑混合协议，为噪声近期量子硬件提供稳健解决方案。
Quantum error correction beyond the toric code: dynamical systems meet encoding
本文在量子纠错背景下，受不可积经典台球不变积分曲面拓扑启发，构建了亏格大于一的表面码，显著提升了编码率与码距并具备抗噪免疫能力。
We introduce the notion of trace-norm isometric encoding and explore its implications for passive and active methods to protect quantum information against errors. Beside providing an operational foundations to the "subsystems principle" [E. Knill, Phys. Rev. A 74, 042301 (2006)] for faithfully realizing quantum information in physical systems, our approach allows additional explicit connections between noiseless, protectable, and correctable quantum codes to be identified. Robustness properties of isometric encodings against imperfect initialization and/or deviations from the intended error m…
We calculate the fidelity with which an arbitrary state can be encoded into a [7,1,3] CSS quantum error correction code in a non-equiprobable Pauli operator error environment with the goal of determining whether this encoding can be used for practical implementations of quantum computation. This determination is accomplished by applying ideal error correction to the encoded state which demonstrates the correctability of errors that occurred during the encoding process. We then apply single-qubit Clifford gates to the encoded state and determine the accuracy with which these gates can be applie…
We show that the problem of designing a quantum information error correcting procedure can be cast as a bi-convex optimization problem, iterating between encoding and recovery, each being a semidefinite program. For a given encoding operator the problem is convex in the recovery operator. For a given method of recovery, the problem is convex in the encoding scheme. This allows us to derive new codes that are locally optimal. We present examples of such codes that can handle errors which are too strong for codes derived by analogy to classical error correction techniques.
Structural encoding with classical codes for computational-basis bit-flip correction in the early fault-tolerant regime
本文提出了一种基于经典码的结构编码方法，用于在早期容错 regime 中通过计算基位翻正误校正提升量子计算性能。
We show how to perform error correction of single qubit dephasing by encoding a single qubit into a minimum of three. This may be performed in a manner closely analogous to classical error correction schemes. Further, the resulting quantum error correction schemes are trivially generalized to the minimal encoding of arbitrarily many qubits so as to allow for multiqubit dephasing correction under the sole condition that the environment acts independently on each qubit.
Approximate Quantum Error Correction with 1D Log-Depth Circuits
本研究证明一维对数深度随机电路在量子纠错中具有高性能，通过信息论分析证明其编码率满足哈希界限和信道容量，并展示纠错不准确度随深度增加而衰减。
Approximate quantum error correction, covariance symmetry, and their relation
本研究从信息论视角探究近似量子纠错与协方差对称性的关系，定义失真度量化近似纠错性能、量化编码信道非协变性，推导出等距编码下二者的权衡关系，并计算了一类随机码的相关平均测度，为量子纠错研究提供新的理论参考。
Generalized Number-Phase Lattice Encoding of a Bosonic Mode for Quantum Error Correction
本文针对玻色子系统量子纠错的对称性利用局限，提出基于数相变量相空间对称性的统一编码框架，构建数相格点码（含矩形、斜形、菱形），其斜形和菱形码通过数相涡旋效应高效纠错，在抗退相干噪声上优于传统正交码，为容错计算和通信扩展提供新方案。
Error Correction Properties of Covariant Bosonic Encodings
本文基于表示论框架形式化构建多模对称玻色子编码，利用舒尔引理分析纠错性质，并基于SU(2)有限子群构建三种新编码，优化种子态以提升保真度。
To implement fault-tolerant quantum computation with continuous variables, Gottesman-Kitaev-Preskill (GKP) qubits have been recognized as an important technological element. However, the analog outcome of GKP qubits, which includes beneficial information to improve the error tolerance, has been wasted, because the GKP qubits have been treated as only discrete variables. In this paper, we propose a hybrid quantum error correction approach that combines digital information with the analog information of the GKP qubits using the maximum-likelihood method. As an example, we demonstrate that the th…
Realistic Simulation of Quantum Repeater with Encoding and Classical Error Correction
本文通过SeQUeNCe模拟器实现并验证了带有编码和经典错误校正的量子中继协议，展示了其在2000公里距离内以0.91保真度分发逻辑贝尔对的能力，并揭示了实际模拟中的挑战。
It is often assumed that the ancilla qubits required for encoding a qubit in quantum error correction (QEC) have to be in pure states, $|00...0>$ for example. In this letter, we seek an encoding scheme, in which the ancillae may be in a uniformly mixed state. We demonstrate our scheme experimentally by making use of a three-qubit NMR quantum computer. Moreover, the encoded state has an interesting nature in terms of Quantum Discord, or purely quantum correlations between the data-qubit and ancillae.
Logical qubit encoding and quantum error correction (QEC) have been experimentally demonstrated in various physical systems with multiple physical qubits, however, logical operations are challenging due to the necessary nonlocal operations. Alternatively, logical qubits with bosonic-mode-encoding are of particular interest because their QEC protection is hardware efficient, but gate operations on QEC protected logical qubits remain elusive. Here, we experimentally demonstrate full control on a single logical qubit with a binomial bosonic code, including encoding, decoding, repetitive QEC, and …
The first generation of multi-qubit quantum technologies will consist of noisy, intermediate-scale devices for which active error correction remains out of reach. To exploit such devices, it is thus imperative to use passive error protection that meets a careful trade-off between noise protection and resource overhead. Here, we experimentally demonstrate that single-qubit encoding can significantly enhance the robustness of entanglement and coherence of four-qubit graph states against local noise with a preferred direction. In particular, we explicitly show that local encoding provides a signi…
Two-qubit operations for finite-energy Gottesman-Kitaev-Preskill encodings
针对有限能量 GKP 码，发现理想无限能量码的双量子比特门会产生不希望出现的纠缠，提出使用局部纠错协议缓解该问题并评估性能，同时设计能量守恒门实现以避免进一步纠错。
We show how to convert an arbitrary stabilizer code into a bipartite quantum code. A bipartite quantum code is one that involves two senders and one receiver. The two senders exploit both nonlocal and local quantum resources to encode quantum information with local encoding circuits. They transmit their encoded quantum data to a single receiver who then decodes the transmitted quantum information. The nonlocal resources in a bipartite code are ebits and nonlocal information qubits and the local resources are ancillas and local information qubits. The technique of bipartite quantum error correc…
We prove the existence of topological quantum error correcting codes with encoding rates $k/n$ asymptotically approaching the maximum possible value. Explicit constructions of these topological codes are presented using surfaces of arbitrary genus. We find a class of regular toric codes that are optimal. For physical implementations, we present planar topological codes.
We investigate an efficient quantum error correction of a fully correlated noise. Suppose the noise is characterized by a quantum channel whose error operators take fully correlated forms given by $σ_x^{\otimes n}$, $σ_y^{\otimes n}$ and $σ_z^{\otimes n}$, where $n>2$ is the number of qubits encoding the codeword. It is proved that (i) $n$ qubits codeword encodes $(n-1)$ data qubits when $n$ is odd and (ii) $n$ qubits codeword implements a noiseless subsystem encoding $(n-2)$ data qubits when $n$ is even. Quantum circuits implementing these schemes are constructed.
We present a quantum error correction code which protects three quantum bits (qubits) of quantum information against one erasure, i.e., a single-qubit arbitrary error at a known position. To accomplish this, we encode the original state by distributing quantum information over six qubits which is the minimal number for the present task (see reference [1]). The encoding and error recovery operations for such a code are presented. It is noted that the present code is also a three-qubit quantum hidded information code over each qubit. In addition, an encoding scheme for hiding $n$-qubit quantum i…
Error-Correction Transitions in Finite-Depth Quantum Channels
研究量子通道编码逻辑信息进入扩展希尔伯特空间的纠错协议，分析噪声在编码后及编码电路本身的作用，发现无限深度极限下随机矩阵理论预测临界噪声率相变，区分纠错相与信息丢失相。
Molecular Nanomagnets may enable the implementation of qudit-based quantum error-correction codes which exploit the many spin levels naturally embedded in a single molecule, a promising step towards scalable quantum processors. To fully realize the potential of this approach, a microscopic understanding of the errors corrupting the quantum information encoded in a molecular qudit is essential, together with the development of tailor-made quantum error correction strategies. We address these central points by first studying dephasing effects on the molecular spin qudit produced by the interacti…
We consider error correction procedures designed specifically for the amplitude damping channel. We analyze amplitude damping errors in the stabilizer formalism. This analysis allows a generalization of the [4,1] `approximate' amplitude damping code of quant-ph/9704002. We present this generalization as a class of [2(M+1),M] codes and present quantum circuits for encoding and recovery operations. We also present a [7,3] amplitude damping code based on the classical Hamming code. All of these are stabilizer codes whose encoding and recovery operations can be completely described with Clifford g…
Learning Encodings by Maximizing State Distinguishability: Variational Quantum Error Correction
本文提出通过最大化噪声通道后量子态可区分性来学习高效的纠错编码，采用变分量子纠错方法，在不同噪声环境下优于传统编码。
Surface-Code Quantum Error Correction for Molecular Tweezer Arrays: Encoding, Layout, and Correlated Noise
本文针对分子光镊阵列的表面码量子纠错，构建了编码映射，对比不同旋转量子位维度的编码，揭示关联损耗空间嵌入的重要性及单扇区基准的局限性。
High-Rank Encoding Can Improve Approximate Quantum Error Correction
该研究针对传统量子码构造的纯态编码限制，提出高秩编码方法，通过联合优化提升近似量子纠错的纠缠保真度，在小噪声下仍保持优势，对特定噪声需用高秩编码器实现近完美恢复。
Petz Recovery through a Single Polar Transformation for Approximate Quantum Error Correction
该研究针对Petz恢复映射电路实现复杂的问题，利用其极结构通过QSVT实现无后选择的近似Petz恢复，给出误差界并验证了电路深度与恢复精度的权衡。
Approximate quantum error correction theory of non-isometric codes
针对非等距编码在量子纠错中的局限性，基于近似量子纠错框架定量研究非等距编码对纠错精度和逻辑操作实现的基本限制，并应用于GKP和tiger码的能量约束分析及全息术讨论。
We present a semidefinite program optimization approach to quantum error correction that yields codes and recovery procedures that are robust against significant variations in the noise channel. Our approach allows us to optimize the encoding, recovery, or both, and is amenable to approximations that significantly improve computational cost while retaining fidelity. We illustrate our theory numerically for optimized 5-qubit codes, using the standard [5,1,3] code as a benchmark. Our optimized encoding and recovery yields fidelities that are uniformly higher by 1-2 orders of magnitude against ra…
Quantum error correction is vital for implementing universal quantum computing. A key component is the encoding circuit that maps a product state of physical qubits into the encoded multipartite entangled logical state. Known methods are typically not 'optimal' either in terms of the circuit depth (and therefore the error burden) or the specifics of the target platform, i.e. the native gates and topology of a system. This work introduces a variational compiler for efficiently finding the encoding circuit of general quantum error correcting codes with given quantum hardware. Focusing on the noi…
Fault-tolerant qubit encoding using a spin-7/2 qudit
针对量子信息技术的纠错难题，提出基于自旋7/2原子核与电子自旋1/2量子比特耦合的量子存储器编码方案，以更少资源实现一阶X/Y/Z纠错，可在现有分子自旋系统中实现并能扩展高阶保护。
Current approaches to fault-tolerant quantum computation will not enable useful quantum computation on near-term devices of 50 to 100 qubits. Leading proposals, such as the color code and surface code schemes, must devote a large fraction of their physical quantum bits to quantum error correction. Building from recent quantum machine learning techniques, we propose an alternative approach to quantum error correction aimed at reducing this overhead, which can be implemented in existing quantum hardware and on a myriad of quantum computing architectures. This method aims to optimize the average …
We present an encoding technique that reduces the effects of noise on quantum spin systems whose operation is driven by Hamiltonian evolution. This technique is widely applicable, being most relevant to the scenarios where there are insufficient qubits to permit full scale error correction. Instead, our technique can be implemented over small numbers of qubits and still leads to noticeable improvements in the fidelity of operations. The encoding scheme is easy to implement, flexible with respect to choice of Hamiltonian, and close to optimal.
A permutation-invariant code on m qubits is a subspace of the symmetric subspace of the m qubits. We derive permutation-invariant codes that can encode an increasing amount of quantum information while suppressing leading order spontaneous decay errors. To prove the result, we use elementary number theory with prior theory on permutation invariant codes and quantum error correction.
Error correction of parity-encoding-based annealing through post-readout decoding
本研究针对奇偶编码量子退火中冗余自旋导致的效率与成本问题，提出在读出后加入经典解码纠错，并通过Monte Carlo模拟验证其可恢复逻辑信息、提升方案可行性。
We show that multidimensional Zeno effect combined with non-holonomic control allows to efficiently protect quantum systems from decoherence by a method similar to classical coding. Contrary to the conventional approach, our method is applicable to arbitrary error-inducing Hamiltonians and general quantum systems. We also propose algorithms of finding encoding that approaches the Hamming upper bound along with methods of practical realizations of the encodings. Two new codes protecting 2 information qubits out of 7 and 4 information qubits out of 9 against a single error with arbitrarily small…
We present a general error-correcting scheme for quantum annealing that allows for the encoding of a logical qubit into an arbitrarily large number of physical qubits. Given any Ising model optimization problem, the encoding replaces each logical qubit by a complete graph of degree $C$, representing the distance of the error-correcting code. A subsequent minor-embedding step then implements the encoding on the underlying hardware graph of the quantum annealer. We demonstrate experimentally that the performance of a D-Wave Two quantum annealing device improves as $C$ grows. We show that the per…
Encoding lattice structures in Quantum Computational Basis States
本文提出了一种将晶格结构编码到量子比特计算基态的通用方法，并展示了其在蛋白质结构预测中的具体应用。
A new approach to efficient quantum computation with probabilistic gates is proposed and analyzed in both a local and non-local setting. It combines heralded gates previously studied for atom or atom-like qubits with logical encoding from linear optical quantum computation in order to perform high fidelity quantum gates across a quantum network. The error-detecting properties of the heralded operations ensure high fidelity while the encoding makes it possible to correct for failed attempts such that deterministic and high-quality gates can be achieved. Importantly, this is robust to photon los…
From the set of operators for errors and its correction code, we introduce the so-called complete unitary transformation. It can be used for encoding while the inverse of it can be applied for correcting the errors of the encoded qubit. We show that this unitary protocol can be applied for any code which satisfies the quantum error correction condition.
Optimal Particle-Conserved Linear Encoding for Practical Fermionic Simulation
本文通过纠错复杂度视角建立费米子门与测量复杂度框架，提出随机奇偶校验码最优编码和费米子期望解码器，实现粒子数守恒子空间编码的可扩展量子模拟，并在 LiH 和 H₂ 的变分量子本征求解实验中验证了协议有效性。
I calculate the fidelity of a [7,1,3] CSS quantum error correction code logical zero state constructed in a non-equiprobable Pauli operator error environment for two methods of encoding. The first method is to apply fault tolerant error correction to an arbitrary state of 7 qubits utilizing Shor states for syndrome measurement. The Shor states are themselves constructed in the non-equiprobable Pauli operator error environment and their fidelity depends on the number of verifications done to ensure multiple errors will not propagate into the encoded quantum information. Surprisingly, performing…
We propose a new approach to implement quantum repeaters for long distance quantum communication. Our protocol generates a backbone of encoded Bell pairs and uses the procedure of classical error correction during simultaneous entanglement connection. We illustrate that the repeater protocol with simple Calderbank-Shor-Steane (CSS) encoding can significantly extend the communication distance, while still maintaining a fast key generation rate.
Error correction phase transition in noisy random quantum circuits
本文研究噪声随机量子电路中的信息编码，发现存在临界深度相变，并证明随机电路在深度与噪声率权衡上达到最优。
Quantum Error Correction from Complexity in Brownian SYK
本文利用 Brownian SYK 码系综证明编码复杂性会使相互纯度指数下降并带来鲁棒量子纠错，为 AdS/CFT 中高复杂性体自由度的不可因果触及提供解释。
We propose a post-selection technique, based on quantum error detection, for quantum key distribution (QKD) systems that run over quantum repeaters with encoding. In such repeaters, quantum error correction techniques are used for entanglement distillation. By developing an analytical approach to study such quantum repeaters, we show that, in the context of QKD, it is often more efficient to use the error detection, rather than the error correction, capability of the underlying code to sift out cases where an error has been detected. We implement our technique for three-qubit repetition codes …
This paper investigates quantum error correction schemes for fully-correlated noise channels on an $n$-qubit system, where error operators take the form $W^{\otimes n}$, with $W$ being an arbitrary $2\times 2$ unitary operator. In previous literature, a recursive quantum error correction scheme can be used to protect $k$ qubits using $(k+1)$-qubit ancilla. We implement this scheme on 3-qubit and 5-qubit channels using the IBM quantum computers, where we uncover an error in the previous paper related to the decomposition of the encoding/decoding operator into elementary quantum gates. Here, we …
Quantum error correction protects the quantum state against noise and decoherence in quantum communication and quantum computation, which enables one to perform fault-torrent quantum information processing. We experimentally demonstrate a quantum error correction scheme with a five-wave-packet code against a single stochastic error, the original theoretical model of which is firstly proposed by S. L. Braunstein and T. A. Walker. Five submodes of a continuous variable cluster entangled state of light are used for five encoding channels. Especially, in our encoding scheme the information of the …
Simultaneous Discovery of Quantum Error Correction Codes and Encoders with a Noise-Aware Reinforcement Learning Agent
本研究针对量子纠错码与编码器的硬件定制化发现问题，提出可扩展的噪声感知强化学习方法，能从零生成适配特定硬件与错误模型的编码方案，还通过元智能体实现跨噪声模型的见解迁移，提升硬件适配QEC发现效率。
In the paper titled "Encoding A Qubit In An Oscillator" Gottesman, Kitaev, and Preskill [quant-ph/0008040] described a method to encode a qubit in the continuous Hilbert space of an oscillator's position and momentum variables. This encoding provides a natural error correction scheme that can correct errors due to small shifts of the position or momentum wave functions (i.e., use of the displacement operator). We present bounds on the size of correctable shift errors when both qubit and ancilla states may contain errors. We then use these bounds to constrain the quality of input qubit and anci…
Compact quantum data representations are essential to the emerging field of quantum algorithms for data analysis. We introduce two new data encoding schemes, QCrank and QBArt, which have a high degree of quantum parallelism through uniformly controlled rotation gates. QCrank encodes a sequence of real-valued data as rotations of the data qubits, allowing for high storage density. QBArt directly embeds a binary representation of the data in the computational basis, requiring fewer quantum measurements and lending itself to well-understood arithmetic operations on binary data. We present several…
Stabilization of encoded logical qubits using quantum error correction is key to the realization of reliable quantum computers. While qubit codes require many physical systems to be controlled, oscillator codes offer the possibility to perform error correction on a single physical entity. One powerful encoding for oscillators is the grid state or GKP encoding, which allows small displacement errors to be corrected. Here we introduce and implement a dissipative map designed for physically realistic finite GKP codes which performs quantum error correction of a logical qubit implemented in the mo…
An outstanding challenge for quantum information processing using bosonic systems is Gaussian errors such as excitation loss and added thermal noise errors. Thus, bosonic quantum error correction (QEC) is essential. Most bosonic QEC schemes encode a finite-dimensional logical qubit or qudit into noisy bosonic oscillator modes. In this case, however, the infinite-dimensional bosonic nature of the physical system is lost at the error-corrected logical level. On the other hand, there are several proposals for encoding an oscillator mode into many noisy oscillator modes. However, these oscillator-…
We explore the effect of Shor state construction methods on logical state encoding and quantum error correction for the [[7,1,3]] Calderbank-Shor-Steane quantum error correction code in a nonequiprobable error environment. We determine the optimum number of verification steps to be used in Shor state construction and whether Shor states without verification are usable for practical quantum computation. These results are compared to the same processes of encoding and error correction where Shor states are not used. We demonstrate that the construction of logical zero states with no first order …
We construct a theory of continuous-variable entanglement-assisted quantum error correction. We present an example of a continuous-variable entanglement-assisted code that corrects for an arbitrary single-mode error. We also show how to implement encoding circuits using passive optical devices, homodyne measurements, feedforward classical communication, conditional displacements, and off-line squeezers.
In classical case there is simplest method of error correction with using three equal bits instead of one. In the paper is shown, how the scheme fails for quantum error correction with complex vector spaces of usual quantum mechanics, but works in real and quaternionic cases. It is discussed also, how to implement the three qubits scheme with using encoding of quaternionic qubit by Majorana spinor. Necessary concepts and formulae from area of quantum error corrections are closely introduced and proved.
It has recently been shown that there are efficient algorithms for quantum computers to solve certain problems, such as prime factorization, which are intractable to date on classical computers. The chances for practical implementation, however, are limited by decoherence, in which the effect of an external environment causes random errors in the quantum calculation. To combat this problem, quantum error correction schemes have been proposed, in which a single quantum bit (qubit) is ``encoded'' as a state of some larger number of qubits, chosen to resist particular types of errors. Most such s…
We propose to use a collective excitation blockade mechanism to identify errors that occur due to disturbances of single atoms in ensemble quantum registers where qubits are stored in the collective population of different internal atomic states. A simple error correction procedure and a simple decoherence-free encoding of ensemble qubits in the hyperfine states of alkali atoms are presented.
The smallest quantum code that can correct all one-qubit errors is based on five qubits. We experimentally implemented the encoding, decoding and error-correction quantum networks using nuclear magnetic resonance on a five spin subsystem of labeled crotonic acid. The ability to correct each error was verified by tomography of the process. The use of error-correction for benchmarking quantum networks is discussed, and we infer that the fidelity achieved in our experiment is sufficient for preserving entanglement.
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