In [Phys. Rev. A 69, 022316 (2004)] we presented a description of the action of local Clifford operations on graph states in terms of a graph transformation rule, known in graph theory as \emph{local complementation}. It was shown that two graph states are equivalent under the local Clifford group if and only if there exists a sequence of local complementations which relates their associated graphs. In this short note we report the existence of a polynomial time algorithm, published in [Combinatorica 11 (4), 315 (1991)], which decides whether two given graphs are related by a sequence of local complementations. Hence an efficient algorithm to detect local Clifford equivalence of graph states is obtained.
Local unitary versus local Clifford equivalence of stabilizer and graph states
局域变换下稳定态的等价性对于理解纠缠的性质和应用具有重要意义。两个稳定器状态在通常的随机局部操作和经典通信准则下是等价的当且仅当它们在局部酉（LU）操作下是等价的。更令人惊讶的是，在某些条件下，两个LU等效稳定子状态在局部Clifford（LC）操作下也是等效的，如Van den Nest等人[\textbf{A71}，062323]所示。在这里，我们扩大了一类的稳定状态的LU等价意味着LC等价（$LU\Leftrightarrow LC$），包括所有的稳定状态表示的图既没有周期的长度为3或4。与Van den Nest等的结果进行比较。的条件下，我们证明了任何距离为$δ=2$的稳定态都超出了他们的判据。然后，我们进一步证明了$LU\Leftrightarrow LC$适用于$δ=2$的更一般类别的稳定态。我们还明确地构建图形表示$δ>2$稳定剂状态，超出了他们的标准：我们确定所有58个图形与多达11个顶点，并构建图形与$2^m-1$（$m\geq 4$）顶点使用量子纠错码具有非Clifford横向门。
Classical representation of local Clifford operators
Wuxi University（无锡学院）；Shanghai Jiao Tong University（上海交通大学）；Shanghai Seres Information Technology Co., Ltd.（上海赛力斯信息技术有限公司）；Southern University of Science and Technology（南方科技大学）；Capital Normal University（首都师范大学）；University of Tsinghua（清华大学）
We consider the problem of $\textit{subgroup testing}$ for a quantum circuit $C$: given access to $C$, determine whether it implements a unitary that is $a$-close or $b$-far from a subgroup $\mathcal{G}$ of the unitary group. It encompasses the problem of exact testing, property testing and tolerant testing. In this work, we study these problems with the group $\mathcal{G}$ as the trivial subgroup (i.e. identity testing) or the Pauli or Clifford group and their $q$-ary extension, and a $\textit{promise}$ version of these problems where $C$ is promised to be in some subgroup of the unitaries th…
Equivalence of non-local computation tasks beyond Clifford operations
Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，格勒诺布尔理工学院，信息学实验室）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；Institute for Quantum Computing, Waterloo, Ontario（量子计算研究所）；QuSoft, University of Amsterdam（QuSoft，阿姆斯特丹大学）；Sorbonne Université, CNRS, LIP6, Paris（索邦大学，法国国家科学研究中心，LIP6实验室）
We provide a purely quantum version of polar codes, achieving the symmetric coherent information of any qubit-input quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where a two-qubit gate randomly chosen from the Clifford group is used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by preshared EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that quantum polarization can be achieved by choosing the channel combining Clifford operator randomly, from …
Classifying locally equivalent graph states, and stabilizer states more broadly, is a significant problem in the theories of quantum information and multipartite entanglement. A special focus is given to those graph states for which equivalence through local unitaries implies equivalence through local Clifford unitaries (LU $\Leftrightarrow$ LC). Identification of locally equivalent states in this class is facilitated by a convenient transformation rule on the underlying graphs and an efficient algorithm. Here we investigate the question of local equivalence of the graph states behind the all-…
The Clifford group is the set of gates generated by the CZ gate, and the two local gates: the Hadamard and the Pi/2 phase shift gate. It is known that, for a two qubit system, the Clifford group C2 is a subgroup of order 92160 of the group of 4 by 4 unitary matrices. It is also known that the local Clifford gates LC2 is a subgroup of order 4608 of the group C2. In order to better understand the set C2, we make two matrices U1 and U2 in C2 equivalent if U_1=VU_2 for some V in LC2. We show that this equivalence relation splits C2 into 20 orbits, O1, ..., O20, each with 4608 elements. Moreover, f…
A five-qubit 1-resistant graph state and stabilizer marginal certificates
University of Science and Technology of China（中国科学技术大学）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）
Université de Lorraine（洛林大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Laboratoire Lorrain de Recherche en Informatique et ses Applications — UMR 7503（洛林信息与应用研究实验室）
Clifford Circuits Augmented Grassmann Matrix Product States
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Graduate School of Science and Technology, Keio University（庆应义塾大学理工学府）；Keio University Sustainable Quantum Artificial Intelligence Center (KSQAIC), Keio University（庆应义塾大学可持续量子人工智能中心）；Department of Physics, Niigata University（新潟大学物理学系）；Department of Physics, Graduate School of Science, Osaka Metropolitan University（大阪公立大学理学研究科物理学系）；Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University（大阪公立大学南部阳一郎理论与实验物理研究所）；Analytical Quantum Complexity RIKEN Hakubi Research Team, RIKEN Center for Quantum Computing (RQC)（理研量子计算中心分析量子复杂性理研白羽研究团队）
We introduce a framework for the formal specification and verification of quantum circuits based on the Feynman path integral. Our formalism, built around exponential sums of polynomial functions, provides a structured and natural way of specifying quantum operations, particularly for quantum implementations of classical functions. Verification of circuits over all levels of the Clifford hierarchy with respect to either a specification or reference circuit is enabled by a novel rewrite system for exponential sums with free variables. Our algorithm is further shown to give a polynomial-time dec…
Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describ…
Quantum information science strives to leverage the quantum-mechanical nature of our universe in order to achieve large improvements in certain information processing tasks. In deep-space optical communications, current receivers for the pure-state classical-quantum channel first measure each qubit channel output and then classically post-process the measurements. This approach is sub-optimal. In this dissertation we investigate a recently proposed quantum algorithm for this task, which is inspired by classical belief-propagation algorithms, and analyze its performance on a simple $5$-bit code…
We argue that all locality-preserving mappings between fermionic observables and Pauli matrices on a two-dimensional lattice can be generated from the exact bosonization in Ref. [1], whose gauge constraints project onto the subspace of the toric code with emergent fermions. Starting from the exact bosonization and applying Clifford finite-depth generalized local unitary (gLU) transformation, we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of Majorana fermions). In particular, we discover a new super-compact encoding using 1.25 qubits per fermion on the square lattic…
