Matthew B. Elliott, Bryan Eastin, Carlton M. Caves
We use a graphical representation of stabilizer states to describe, simply and efficiently, the effect of measurements of Pauli products on stabilizer states. This work complements our earlier work [Phys. Rev. A 77, 042307 (2008)], which described in graphical terms the action of Clifford operations on stabilizer states.
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Hypergraph states of many quantum bits share the rich interplay between simple combinatorial description and nontrivial entanglement properties enjoyed by the graph states that they generalize. In this paper, we consider hypergraph states that are also permutationally invariant. We characterize the states in this class that have nontrivial local Pauli stabilizers and give applications to nonlocality and error correction.
We introduce a graphical representation of stabilizer states and translate the action of Clifford operators on stabilizer states into graph operations on the corresponding stabilizer-state graphs. Our stabilizer graphs are constructed of solid and hollow nodes, with (undirected) edges between nodes and with loops and signs attached to individual nodes. We find that local Clifford transformations are completely described in terms of local complementation on nodes and along edges, loop complementation, and change of node type or sign. Additionally, we show that a small set of equivalence rules g…
Error-mitigated initialization of surface codes with non-Pauli stabilizers
通过引入非泡利稳定子提出两种创新初始化协议，提升了非克利福德逻辑态初始化保真度，降低了态蒸馏资源开销，并利用晶格手术技术实现逻辑量子比特纠缠，兼容传统泡利表面码计算。
Exact efficient simulation of noisy logical magic states using Clifford stabilizers
本文提出Clifford稳定子模拟算法，通过将含噪非Clifford电路映射为等价Clifford电路，实现逻辑魔态制备协议的精确高效经典模拟，并成功模拟至故障距离7，为FTQC基准测试提供新途径。
Clifford Manipulations of Stabilizer States: A graphical rule book for Clifford unitaries and measurements on cluster states, and application to photonic quantum computing
本文提出了一种基于克莱因群框架的图示规则手册，用于高效操作集群态，并展示了其在光子量子计算中的应用。
SymFT: Universal Fault-Tolerant Quantum Circuit Simulation via Symbolic Clifford--Pauli Frames and Stabilizer Coordinates
SymFT是一款高吞吐量容错量子电路模拟器，通过符号Clifford–Pauli框架分解和自适应稳定子坐标规划优化采样，在多种量子电路上实现了比Stim、Clifft、SOFT更优的采样性能。
Noisy Stabilizer Formalism
针对多量子比特态描述的指数开销问题，提出含噪稳定子形式体系，可高效追踪稳定子态在Clifford操作、Pauli测量及各类Pauli噪声下的演化，适用于局域测量生成少数量子比特多体纠缠态的场景。
We introduce a purely graph-theoretical object, namely the coding clique, to construct quantum errorcorrecting codes. Almost all quantum codes constructed so far are stabilizer (additive) codes and the construction of nonadditive codes, which are potentially more efficient, is not as well understood as that of stabilizer codes. Our graphical approach provides a unified and classical way to construct both stabilizer and nonadditive codes. In particular we have explicitly constructed the optimal ((10,24,3)) code and a family of 1-error detecting nonadditive codes with the highest encoding rate s…
In 2008 Coecke and Duncan proposed the graphical ZX-calculus rewrite system which came to formalize reasoning with quantum circuits, measurements and quantum states. The ZX-calculus is sound for qubit quantum mechanics. Hence, equality of diagrams under ZX-equivalent transformations lifts to an equality of corresponding equations over matrices. Conversely, in 2014 Backens proved completeness, establishing that any derivation done in stabilizer quantum mechanics with matrices can be derived graphically using the ZX-calculus. A graphical rewrite system that is both confluent and also terminates …
Qudit Noisy Stabilizer Formalism
本文提出量子比特噪声稳定子形式体系，高效描述噪声下稳定子态的演化，复杂度随初始量子比特数线性、最终数指数增长，并验证了生成广义贝尔对的效率。
We describe generalizations of the Pauli group, the Clifford group and stabilizer states for qudits in a Hilbert space of arbitrary dimension d. We examine a link with modular arithmetic, which yields an efficient way of representing the Pauli group and the Clifford group with matrices over the integers modulo d. We further show how a Clifford operation can be efficiently decomposed into one and two-qudit operations. We also focus in detail on standard basis expansions of stabilizer states.
Improving the simulation of quantum circuits on classical computers is important for understanding quantum advantage and increasing development speed. In this paper, we explore a new way to express stabilizer states and further improve the speed of simulating stabilizer circuits with a current existing approach. First, we discover a unique and elegant canonical form for stabilizer states based on graph states to better represent stabilizer states and show how to efficiently simplify stabilizer states to canonical form. Second, we develop an improved algorithm for graph state stabilizer simulat…
Entanglement on Two Bloch Spheres: Exploring Two-Qubit Stabilizer Group Structure
本研究借助稳定子形式体系，建立两量子比特态的密度矩阵与双布洛赫球图形表示的关联（二者均源于稳定子群结构），并以此探究最大纠缠态的对称性。
In this paper we investigate stabilizer quantum error correction codes using controlled phase rotations of strong coherent probe states. We explicitly describe two methods to measure the Pauli operators which generate the stabilizer group of a quantum code. First, we show how to measure a Pauli operator acting on physical qubits using a single coherent state with large average photon number, displacement operations, and photon detection. Second, we show how to measure the stabilizer operators fault-tolerantly by the deterministic preparation of coherent cat states along with one-bit teleportat…
Verification of NISQ era quantum devices demands fast classical simulation of large noisy quantum circuits. We present an algorithm based on the stabilizer formalism that can efficiently simulate noisy stabilizer circuits. Additionally, the protocol can efficiently simulate a large set of multi-qubit mixed states that are not mixtures of stabilizer states. The existence of these 'bound states' was previously only known for odd-dimensional systems like qutrits. The algorithm also has the favorable property that circuits with depolarizing noise are simulated much faster than unitary circuits. Th…
We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group $\langle αI, X,S\rangle$, where $α=e^{iπ/4}$ and $S=\operatorname{diag}(1,i)$. We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians etc. We also highligh…
Many-body quantum states, as a matter of fact, are extremely essential to solve certain mathematical problems or simulate quantum systems in measurement-based quantum computation. However, how to verify large scale quantum states, such as hypergraph states, is an exceedingly hard task for multi-body quantum systems. Here, we propose a novel fault-tolerant solution for verification of colorable hypergraph states by using stabilizer test. Compared with the adaptive stabilizer test, our protocol is dramatically facilitating by making only Pauli-X and Pauli-Z measurement. As to appliance, it will …
Clifford circuits play an important role in quantum computation. Gottesman and Chuang proposed a gate teleportation protocol so that a quantum circuit can be implemented by the teleportation circuit with specific ancillary qubits. In particular, an $n$-qubit Clifford circuit $U$ can be implemented by preparing an ancillary stabilizer state $(I\otimes U)|Φ^+\rangle^{\otimes n}$ for teleportation and doing a Pauli correction conditioned on the measurement. In this paper, we provide an alternative procedure to implement a Clifford circuit through Pauli measurements, by preparing $O(1)$ ancillas t…
Existence of Pauli-like stabilizers for every quantum error-correcting code
本文证明所有量子纠错码均可由交换的“Paulian”算子构成Paulian稳定子群来稳定，并通过受控门测量获取错误伴随式，为实现和发现新纠错码提供了可能途径。
We construct a Pauli stabilizer model for every two-dimensional Abelian topological order that admits a gapped boundary. Our primary example is a Pauli stabilizer model on four-dimensional qudits that belongs to the double semion (DS) phase of matter. The DS stabilizer Hamiltonian is constructed by condensing an emergent boson in a $\mathbb{Z}_4$ toric code, where the condensation is implemented by making certain two-body measurements. We rigorously verify the topological order of the DS stabilizer model by identifying an explicit finite-depth quantum circuit (with ancillary qubits) that maps …
Complete Flow-Preserving Rewrite Rules for MBQC Patterns with Pauli Measurements
本文提出并证明了带泡利测量的MBQC模式中，引入新Z测量量子比特可保持泡利流，并给出稳定子ZX图的唯一规范形式及完整保流重写规则。
Graphical Stabilizer Decompositions for Multi-Control Toffoli Gate Dense Quantum Circuits
本文基于ZX-calculus研究量子计算的图形化稳定子分解，提出了非稳定子态的新型分解与针对多控制Toffoli门密集量子电路的加权算法，并在模拟量子查询和随机基准中较传统方法偶有改进。
Mapping an error syndrome to the error operator is the core of quantum decoding network and is also the key step of recovery. The definitions of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix were presented, and then the error syndromes of quantum errors were expressed in terms of the columns of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix. It also showed that the error syndrome matrices of a stabilizer code are determined by its check matrix, which is similar to the classical case. So, the error-detection and recovery techniques of c…
In this paper we study the quantum graphs of mixed-unitary channels generated by tensor products of Pauli operators, which we call Pauli channels. We show that most quantum graphs arising from Pauli channels have non-trivial quantum cliques or quantum anticliques which are stabilizer codes. However, a reformulation of Nik Weaver's quantum Ramsey theorem in terms of stabilizer codes and Pauli channels fails. Specifically, for every positive integer $n$, there exists an $n$-qubit Pauli channel for which any non-trivial quantum clique or quantum anticlique fails to be a stabilizer code.
A strong converse for stabilizer codes over Pauli channels via the blowing-up lemma
研究泡利信道上稳定子码的量子通信强逆定理，通过爆破引理证明，当码传输速率高于输入态相干信息时纠缠保真度指数衰减，确定了无记忆信道容量，给出反退化信道情况，还限制了近确定性解码并分离出编码器端陈述。
Learning Noise-Robust Stabilizer Structure via Bell Sampling
针对噪声态稳定子结构学习，提出Bell采样查找-验证算法，在弱Pauli噪声下实现多项式样本与运行时间，缩小了不可知学习与纯态算法的差距。
Graphical quantum Clifford-encoder compilers from the ZX calculus
本文提出一种基于ZX演算的量子编译算法，将Clifford编码器映射为唯一规范图形表示，并证明其规范性与高效可归约性，为通过图论分析设计新型稳定子量子码提供新途径。
We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with code distance being the linear system size, is decomposed by a local Clifford circuit of constant depth into a finite number of copies of the toric code (abelian discrete gauge theory) stabilizer group. This means that under local Clifford circuits the number of toric code copies is the complete invariant of topological Pauli stabilizer codes. Previously, the same conclusion was obtained under the assumption of nonchirality for qubit cod…
The Gottesman-Knill theorem allows for the efficient simulation of stabilizer-based quantum error-correction circuits. Errors in these circuits are commonly modeled as depolarizing channels by using Monte Carlo methods to insert Pauli gates randomly throughout the circuit. Although convenient, these channels are poor approximations of common, realistic channels like amplitude damping. Here we analyze a larger set of efficiently simulable error channels by allowing the random insertion of any one-qubit gate or measurement that can be efficiently simulated within the stabilizer formalism. Our ne…
The code distance of Floquet codes
本文证明Floquet码中可纠正且不可检测的时空错误可分解为良性错误，并据此定义码距为最小非良性不可检测错误权重，结果推广至动态码。
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes ar…
We present novel algorithms to estimate outcomes for qubit quantum circuits. Notably, these methods can simulate a Clifford circuit in linear time without ever writing down stabilizer states explicitly. These algorithms outperform previous noisy near-Clifford techniques for most circuits. We identify a large class of input states that can be efficiently simulated despite not being stabilizer states. The algorithms leverage probability distributions constructed from Bloch vectors, paralleling previously known algorithms that use the discrete Wigner function for qutrits.
The Pauli groups are ubiquitous in quantum information theory because of their usefulness in describing quantum states and operations and their readily understood symmetry properties. In addition, the most well-understood quantum error correcting codes -- stabilizer codes -- are built using Pauli operators. The eigenstates of these operators -- stabilizer states -- display a structure (e.g., mutual orthogonality relationships) that has made them useful in examples of multi-qubit non-locality and contextuality. Here, we apply the graph-theoretical contextuality formalism of Cabello, Severini an…
Statistical verification of a quantum state aims to certify whether a given unknown state is close to the target state with confidence. So far, sample-optimal verification protocols based on local measurements have been found only for disparate groups of states: bipartite pure states, GHZ states, and antisymmetric basis states. In this work, we investigate systematically optimal verification of entangled stabilizer states using Pauli measurements. First, we provide a lower bound on the sample complexity of any verification protocol based on separable measurements, which is independent of the n…
The Classification of Pauli Stabilizer Codes: A Lattice and Continuum Treatise
本文基于代数L理论，通过间隙界面和粗粒化对Pauli稳定子码进行分类，并与框架TQFT进行比较，揭示两者结构关系，建立晶格理论的体-边对应关系。
Learning the stabilizer group of a Matrix Product State
该研究提出一种基于泡利基有偏采样的经典算法，可学习矩阵乘积态的稳定子群以获取非稳定子单调量，在高纠缠MPS上表现准确且标度良好，为非平衡量子多体物理研究提供了有效工具。
Simulating general noise nearly as cheaply as Pauli noise
本文通过分层重要性采样在稳定子形式化中展示如何以合理时间模拟通用噪声，非幺正噪声几乎与Pauli噪声成本相当，而幺正噪声需更多时间但仍有显著改进，使深入研究真实设备噪声性能成为可能。
Stabilizer-Code Channel Transforms Beyond Repetition Codes for Improved Hashing Bounds
本文提出利用稳定码作为通道变换，改进哈希界限，通过构造全辛表来计算逻辑Pauli误差和syndrome的联合分布，从而获得更高的可实现速率。
It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group coho…
We find a scaling reduction in the stabilizer rank of the twelve-qubit tensored $T$ gate magic state. This lowers its asymptotic bound to $2^{\sim 0.463 t}$ for multi-Pauli measurements on $t$ magic states, improving over the best previously found bound of $2^{\sim 0.468 t}$. We numerically demonstrate this reduction. This constructively produces the most efficient strong simulation algorithm of the Clifford+$T$ gateset to relative or multiplicative error. We then examine the cost of Pauli measurement in terms of its Gauss sum rank, which is a slight generalization of the stabilizer rank and i…
Analytical and Compressed Simulation of Noisy Stabilizer Circuits
本文提出分析与算法技术，高效模拟广泛噪声稳定子电路，通过推导张量积Pauli算符的期望值闭式表达式，实现高效强模拟方法，并引入电路压缩框架降低弱模拟的每样本成本，扩展分析框架以包含确定性测量、通用旋转和非对角噪声通道。
Learning logical Pauli noise in quantum error correction
本文研究稳定子量子纠错中量子器件的表征问题，证明对于任意稳定子码、子系统码和数据综合征码，只要码能纠正噪声，即可在最小条件下从综合征数据估计泡利噪声诱导的逻辑错误信道。
Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases
本文引入马约拉纳-泡利稳定子码，构造了费米子环面码的精确可解稳定子实现，并通过任意子凝聚和对称性规范对偶网络连接了多种拓扑相。
We propose a measurement scheme that validates the preparation of an $n$-qubit stabilizer state. The scheme involves a measurement of $n$ Pauli observables, a priori determined from the stabilizer state and which can be realized using single-qubit gates. Based on the proposed validation scheme, we derive an explicit expression for the worst-case fidelity, i.e., the minimum fidelity between the stabilizer state and any other state consistent with the measured data. We also show that the worst-case fidelity can be certified, with high probability, using ${\cal O}(n^2)$ copies of the state.
Measurement-Assisted Clifford Synthesis
本文提出了一种基于稳定器表和测量的量子Clifford单元合成方法，通过辅助量子比特和测量实现两量子比特门数量与稳定器表重量相等的高效合成。
We present a general protocol for stabilizer measurements and pumping in a system of N superconducting qubits. We assume always-on, equal dispersive couplings to a single mode of a high-Q microwave resonator in the ultra-strong dispersive limit where the dispersive shifts largely exceed the spectral linewidth. In this limit, we show how to map the two eigenvalues of an arbitrary weight M < N Pauli operator, onto two quasi-orthogonal coherent states of the cavity. Together with a fast cavity readout, this enables the efficient measurement of stabilizer operators.
Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
本文提出一种适用于$\u2124_d$量子比特（含非素数d）的算法，可提取二维平移不变广义Pauli稳定子码的拓扑数据，识别新拓扑序并为量子纠错码研究提供系统方法。
Clifford-hierarchy stabilizer formalism with applications to twisted quantum doubles
该文提出 $X\mathcal C^{(k)}_d$ 稳定子形式体系，推广 Pauli 稳定子以涵盖扭曲量子双重模型，证明其可规避 Bravyi-Koenig 界，并展示通过测量该稳定子可确定性制备逻辑魔法态。
Phased outcome-complete simulation
本文提出了一种能够精确跟踪全局相位的完整模拟方法，扩展了稳定器电路的多项式时间完整模拟算法，适用于包含单量子位旋转的非稳定器电路等重要家族。
Pauli topological subsystem codes from Abelian anyon theories
本文构造了由任意二维阿贝尔任意子理论刻画的Pauli拓扑子系统码，通过“规范掉”拓扑稳定子码中的任意子类型实现，扩展了分类至复合维度量子比特，并证明每个阿贝尔任意子理论都是环面码和扭曲量子双模的子理论。
Local equivalence of stabilizer states: a graphical characterisation
研究稳定器状态的 LU 等价性，引入局部补运算的推广以图形化刻画，基于最小局部集展示图态等价存在严格无限层次结构，为图态 LU 等价性研究提供新方法与结论。
The form of a local Clifford (LC, also called local Gaussian (LG)) operation for the continuous-variable (CV) weighted graph states is presented in this paper, which is the counterpart of the LC operation of local complementation for qubit graph states. The novel property of the CV weighted graph states is shown, which can be expressed by the stabilizer formalism. It is distinctively different from the qubit weighted graph states, which can not be expressed by the stabilizer formalism. The corresponding graph rule, stated in purely graph theoretical terms, is described, which completely charac…
Pauli spectrum and nonstabilizerness of random fermionic Gaussian states
本文刻画了随机费米子高斯态的魔法性，推导了其Pauli谱与稳定化熵的精确表达式，发现冻结现象，并证明典型态在丢弃一定比例模式后仍保持魔法性。
Exponentially Accelerated Sampling of Pauli Strings for Nonstabilizerness
本文提出一种高效经典框架，用于计算通用N-量子比特态的稳定子Rényi熵和稳定子空度，通过快速沃尔什-哈达玛变换和精确的Pauli算子划分，将每个采样的Pauli字符串的平均成本从O(2^N)降低到O(N)。
This paper presents ``Stim", a fast simulator for quantum stabilizer circuits. The paper explains how Stim works and compares it to existing tools. With no foreknowledge, Stim can analyze a distance 100 surface code circuit (20 thousand qubits, 8 million gates, 1 million measurements) in 15 seconds and then begin sampling full circuit shots at a rate of 1 kHz. Stim uses a stabilizer tableau representation, similar to Aaronson and Gottesman's CHP simulator, but with three main improvements. First, Stim improves the asymptotic complexity of deterministic measurement from quadratic to linear by t…
On Error Thresholds for Pauli Channels: Some answers with many more questions
本文研究Pauli通道的误差阈值，通过分析余集重量生成函数框架，探讨了稳定化码的非加性特性及拼接码的性能优化。
We introduce a novel measure for the quantum property of nonstabilizerness - commonly known as "magic" - by considering the Rényi entropy of the probability distribution associated to a pure quantum state given by the square of the expectation value of Pauli strings in that state. We show that this is a good measure of nonstabilizerness from the point of view of resource theory and show bounds with other known measures. The stabilizer Rényi entropy has the advantage of being easily computable because it does not need a minimization procedure. We present a protocol for an experimental measureme…
The performance of quantum error correction can be significantly improved if detailed information about the noise is available, allowing to optimize both codes and decoders. It has been proposed to estimate error rates from the syndrome measurements done anyway during quantum error correction. While these measurements preserve the encoded quantum state, it is currently not clear how much information about the noise can be extracted in this way. So far, apart from the limit of vanishing error rates, rigorous results have only been established for some specific codes. In this work, we rigorously…
We prove a general lower bound to the spectral gap of the Davies generator for Hamiltonians that can be written as the sum of commuting Pauli operators. These Hamiltonians, defined on the Hilbert space of $N$-qubits, serve as one of the most frequently considered candidates for a self-correcting quantum memory. A spectral gap bound on the Davies generator establishes an upper limit on the life time of such a quantum memory and can be used to estimate the time until the system relaxes to thermal equilibrium when brought into contact with a thermal heat bath. The bound can be shown to behave as …
Quantum error-correction codes (QECCs) are a vital ingredient of quantum computation and communication systems. In that context it is highly desirable to design QECCs that can be represented by graphical models which possess a structure that enables efficient and close-to-optimal iterative decoding. In this paper we focus on stabilizer QECCs, a class of QECCs whose construction is rendered non-trivial by the fact that the stabilizer label code, a code that is associated with a stabilizer QECC, has to satisfy a certain self-orthogonality condition. In order to design graphical models of stabili…
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