The name graph state is used to describe a certain class of pure quantum state which models a physical structure on which one can perform measurement-based quantum computing, and which has a natural graphical description. We present the two-graph state, this being a generalisation of the graph state and a two-graph representation of a stabilizer state. Mathematically, the two-graph state can be viewed as a simultaneous generalisation of a binary linear code and quadratic Boolean function. It describes precisely the coefficients of the pure quantum state vector resulting from the action of a member of the local Clifford group on a graph state, and comprises a graph which encodes the magnitude properties of the state, and a graph encoding its phase properties. This description facilitates a computationally efficient spectral analysis of the graph state with respect to operations from the local Clifford group on the state, as all operations can be realised graphically. By focusing on the so-called local transform group, which is a size 3 cyclic subgroup of the local Clifford group over one qubit, and over $n$ qubits is of size $3^n$, we can efficiently compute spectral properties of the graph state.
Graph states are quantum states that can be described by a stabilizer formalism and play an important role in quantum information processing. We consider the action of local unitary operations on graph states and hypergraph states. We focus on non-Clifford operations and find for certain transformations a graphical description in terms of weighted hypergraphs. This leads to the indentification of hypergraph states that are locally equivalent to graph states. Moreover, we present a systematic way to construct pairs of graph states which are equivalent under local unitary operations, but not equ…
We introduce an open-source software library Graphix, which optimizes and simulates measurement-based quantum computation (MBQC). By combining the measurement calculus with an efficient graph state simulator, Graphix allows the classical preprocessing of Pauli measurements in the measurement patterns, significantly reducing the number of operations required to perform the quantum computation while maintaining determinism. For a measurement pattern translated from a quantum circuit, this corresponds to the preprocessing of all Clifford gates, and this improvement in the one-way model is importa…
Clifford Manipulations of Stabilizer States: A graphical rule book for Clifford unitaries and measurements on cluster states, and application to photonic quantum computing
NSF-ERC Center for Quantum Networks, Wyant College of Optical Sciences, The University of Arizona（亚利桑那大学怀特光学科学学院量子网络 NSF-ERC 中心）
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。
Linear-depth quantum oracles for clique problems from edge colorings and graph states, with linear non-Clifford cost and a provably bounded-error k-clique search
School of Electrical and Computer Engineering, College of Engineering, University of Tehran（德黑兰大学工程学院电气与计算机工程学院）
In this article, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Strengths of our algebraic framework include the minimality of its assumptions, and the readiness by which one may give an explicit construction satisfying these assumptions. In terms of applications, this algebraic framework provides a solid foundation which opens the way for developing a graphical calculus for multi-qudit representations of ge…
Graph states, which include for example Bell states, GHZ states and cluster states, form a well-known class of quantum states with applications ranging from quantum networks to error-correction. Deciding whether two graph states are equivalent up to single-qubit Clifford operations is known to be decidable in polynomial time and have been studied both in the context of producing certain required states in a quantum network but also in relation to stabilizer codes. The reason for the latter this is that single-qubit Clifford equivalent graph states exactly corresponds to equivalent stabilizer c…
The ZX-Calculus is a graphical language for quantum mechanics. An axiomatisation has recently been proven to be complete for an approximatively universal fragment of quantum mechanics, the so-called Clifford+T fragment. We focus here on the expressive power of this axiomatisation beyond Clifford+T Quantum mechanics. We consider the full pure qubit quantum mechanics, and mainly prove two results: (i) First, the axiomatisation for Clifford+T quantum mechanics is also complete for all equations involving some kind of linear diagrams. The linearity of the diagrams reflects the phase group structur…
Simulating Clifford Circuits with Gaussian Elimination
Siebel School of Computing and Data Science, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校西贝尔计算与数据科学学院）；Computer Network Information Center, Chinese Academy of Sciences（中国科学院计算机网络信息中心）
We show that the formalism of "Sum-Over-Path" (SOP), used for symbolically representing linear maps or quantum operators, together with a proper rewrite system, has a structure of dagger-compact PROP. Several consequences arise from this observation: 1) Morphisms of SOP are very close to the diagrams of the graphical calculus called ZH-Calculus, so we give a system of interpretation between the two; 2) A construction, called the discard construction, can be applied to enrich the formalism so that, in particular, it can represent the quantum measurement. We also enrich the rewrite system so as …
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）
We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form $\exp(-i\fracπ{2^{n}} Z\otimes Z)$. When $n = 2$, these are equivalent, up to local Clifford unitaries, to graph states. However, when $n > 2$, their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an importan…
Quantum Accreditation with Non-Clifford Two-qubit Gates
School of Informatics, University of Edinburgh（爱丁堡大学信息学院）；Department of Physics, University of Warwick（华威大学物理系）；National Quantum Computing Centre, Rutherford Appleton Laboratory（拉瑟福德阿普尔顿实验室国家量子计算中心）
Constant-rate quantum codes with low-weight stabilizers and full logical Clifford actions via transversal and fold-transversal gates
OptQC Corp.（OptQC公司）；RIKEN Center for Quantum Computing (RQC)（理化学研究所量子计算中心）；Department of Applied Physics, Graduate School of Engineering, The University of Tokyo（东京大学工学研究科应用物理系）
ZX-calculus is a high-level graphical formalism for qubit computation. In this paper we give the ZX-rules that enable one to derive all equations between 2-qubit Clifford+T quantum circuits. Our rule set is only a small extension of the rules of stabilizer ZX-calculus, and substantially less than those needed for the recently achieved universal completeness. One of our rules is new, and we expect it to also have other utilities. These ZX-rules are much simpler than the complete of set Clifford+T circuit equations due to Selinger and Bian, which indicates that ZX-calculus provides a more conven…
Surface and color codes are two forms of topological quantum error correction in two spatial dimensions with complementary properties. Surface codes have lower-depth error detection circuits and well-developed decoders to interpret and correct errors, while color codes have transversal Clifford gates and better code efficiency in the number of physical qubits needed to achieve a given code distance. A formal equivalence exists between color codes and folded surface codes, but it does not guarantee the transferability of any of these favorable properties. However, the equivalence does imply the…
We study two-qubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlled-phase gate CS=diag(1,1,1,i). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes through magic state distillation. Since non-Clifford gates are typically more expensive to perform in a fault-tolerant manner, it is often desirable to construct circuits that use few CS gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for two-qubit Clifford+C…
Complexity separations for optimal matchgate-Clifford synthesis
Barcelona Supercomputing Center（巴塞罗那超级计算中心）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Johannes Kepler University（约翰内斯·开普勒大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Princeton University（普林斯顿大学）
The Clifford group is the quotient of the normalizer of the Weyl-Heisenberg group in dimension $d$ by its centre. We prove that when $d$ is not prime the Clifford group is not a group unitary $2$-design. Furthermore, we prove that the multipartite Clifford group is not a group unitary 2-design except for the known cases wherein the local Hilbert space dimensions are a constant prime number. We also clarify the structure of projective group unitary $2$-designs. We show that the adjoint action induced by a group unitary $2$-design decomposes into exactly two irreducible components; moreover, a g…
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 …
Hardware efficient transpilation of quantum circuits to a quantum devices native gateset is essential for the execution of quantum algorithms on noisy quantum computers. Typical quantum devices utilize a gateset with a single two-qubit Clifford entangling gate per pair of coupled qubits, however, in some applications access to a non-Clifford two-qubit gate can result in more optimal circuit decompositions and also allows more flexibility in optimizing over noise. We demonstrate calibration of a low error non-Clifford Controlled-$\fracπ{2}$ phase (CS) gate on a cloud based IBM Quantum computing…
Quantum computing has potential to provide exponential speedups over classical computing for many important applications. However, today's quantum computers are in their early stages, and hardware quality issues hinder the scale of program execution. Benchmarking and simulation of quantum circuits on classical computers is therefore essential to advance the understanding of how quantum computers and programs operate, enabling both algorithm discovery that leads to high-impact quantum computation and engineering improvements that deliver to more powerful quantum systems. Unfortunately, the natu…
We present a completely new approach to quantum circuit optimisation, based on the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, which provide a flexible, lower-level language for describing quantum computations graphically. Then, using the rules of the ZX-calculus, we give a simplification strategy for ZX-diagrams based on the two graph transformations of local complementation and pivoting and show that the resulting reduced diagram can be transformed back into a quantum circuit. While little is known about extracting circuits from arbitrary ZX-diagrams, we show that the un…
The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e. the ability to derive any true equation -- is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -- called cyclotomic supplementarity -- which consists in merging n subdiagrams at o…
