我们构造了简单李群上的随机游走，对于直到 $t$ 阶的所有矩，它能快速收敛到 Haar 测度。具体而言，维度为 $2^{\mathsf n}$ 的酉群或正交群上的一步游走是随机 Pauli 旋转 $e^{\mathrm i θP /2}$。该随机游走的谱隙被证明为 $Ω(1/t)$，这与 $\{0,1\}^{\mathsf n}$ 上置换群随机游走先前已知的最佳界一致。这意味着该游走在深度 $O(\mathsf n t^2 + t \log 1/\varepsilon)d$ 内给出 $\varepsilon$-近似酉 $t$-设计，其中 $d=O(\log \mathsf n)$ 是实现 $e^{\mathrm i θP /2}$ 的电路深度。我们的简单证明使用了李代数的二次 Casimir 算子。
英文摘要：
We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to order $t$. Specifically, a step of the walk on the unitary or orthognoal group of dimension $2^{\mathsf n}$ is a random Pauli rotation $e^{\mathrm i θP /2}$. The spectral gap of this random walk is shown to be $Ω(1/t)$, which coincides with the best previously known bound for a random walk on the permutation group on $\{0,1\}^{\mathsf n}$. This implies that the walk gives an $\varepsilon$-approximate unitary $t$-design in depth $O(\mathsf n t^2 + t \log 1/\varepsilon)d$ where $d=O(\log \mathsf n)$ is the circuit depth to implement $e^{\mathrm i θP /2}$. Our simple proof uses quadratic Casimir operators of Lie algebras.
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…
A key task in quantum computation is the application of a sequence of gates implementing a specific unitary operation. However, the decomposition of an arbitrary unitary operation into simpler quantum gates is a nontrivial problem. Here we propose a general and robust protocol to decompose any target unitary into a sequence of Pauli rotations. The procedure involves identifying a commuting subset of Pauli operators having a high trace overlap with the target unitary, followed by a numerical optimization of their corresponding rotation angles. The protocol is demonstrated by decomposing several…
TARE: Block Encoding Linear Combinations of Pauli Strings Without Ancilla State Preparation
Fraunhofer-Institut für Arbeitswirtschaft und Organisation IAO（弗劳恩霍夫工作经济与组织研究所）；Universität Stuttgart, Institut für Arbeitswissenschaft und Technologiemanagement IAT（斯图加特大学劳动科学与技术管理研究所）；Fraunhofer-Institut für Angewandte Festkörperphysik IAF（弗劳恩霍夫应用固体物理研究所）；Albert-Ludwigs-Universität Freiburg（弗赖堡阿尔伯特·路德维希大学）
Unitary $k$-designs are probabilistic ensembles of unitary matrices whose first $k$ statistical moments match that of the full unitary group endowed with the Haar measure. In prior work, we showed that the automorphism group of classical $\mathbb{Z}_4$-linear Kerdock codes maps to a unitary $2$-design, which established a new classical-quantum connection via graph states. In this paper, we construct a Markov process that mixes this Kerdock $2$-design with symplectic transvections, and show that this process produces an $ε$-approximate unitary $3$-design. We construct a graph whose vertices are…
Unitary fault-tolerant encoding of Pauli states in surface codes
Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich（于利希研究中心）；Institute for Quantum Information, RWTH Aachen University（亚琛工业大学）；Max Planck Institute for the Science of Light（马克斯·普朗克光科学研究所）；School of Physics and Astronomy, Monash University（蒙纳士大学）；NVIDIA Corporation（英伟达公司）；Department of Physics, Friedrich-Alexander Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）
The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being …
Designs without disorder: unitary $k$-designs from a single-site bulk defect
University of Geneva（日内瓦大学）；Vrije Universiteit Brussel (VUB)（布鲁塞尔自由大学）；RIKEN Centre for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心）
Unitary $2$-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols, and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary $2$-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-$Z$ basis and that in the Pauli-$X$ basis. We also provide a converse about the number of repetitions …
In this work we explain the necessity for consistently labeled rotation maps for efficiently computing coined discrete time quantum walks on regular graphs.
Standard randomized benchmarking protocols entail sampling from a unitary 2 design, which is not always practical. In this article we examine randomized benchmarking protocols based on subgroups of the Clifford group that are not unitary 2 designs. We introduce a general method for analyzing such protocols and subsequently apply it to two subgroups, the group generated by controlled-NOT, Hadamard, and Pauli gates and that generated by only controlled-NOT and Pauli gates. In both cases the error probability can be estimated to within a factor of two or less where the factor can be arranged to b…
University of California San Diego（加利福尼亚大学圣地亚哥分校）；Department of Physics, University of California San Diego（加利福尼亚大学圣地亚哥分校物理系）
Classical shadow tomography provides an efficient method for predicting functions of an unknown quantum state from a few measurements of the state. It relies on a unitary channel that efficiently scrambles the quantum information of the state to the measurement basis. Facing the challenge of realizing deep unitary circuits on near-term quantum devices, we explore the scenario in which the unitary channel can be shallow and is generated by a quantum chaotic Hamiltonian via time evolution. We provide an unbiased estimator of the density matrix for all ranges of the evolution time. We analyze the…
The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organi…
Identifying the Sign of Coherent Over-Rotations with Logarithmically Many Pauli Settings
University of Georgia（佐治亚大学）；GyriQAI, Inc.（GyriQAI公司）；University of Tennessee at Chattanooga（田纳西大学查塔努加分校）；Augusta University（奥古斯塔大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
Unitary $k$-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices. Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design. We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected. Our proof strategy works for any distribution of unitaries satisfying a property we call Pauli 2-mixing and proceeds without the use of heavy mathematical machiner…
The Clifford group is a finite subgroup of the unitary group generated by the Hadamard, the CNOT, and the Phase gates. This group plays a prominent role in quantum error correction, randomized benchmarking protocols, and the study of entanglement. Here we consider the problem of finding a short quantum circuit implementing a given Clifford group element. Our methods aim to minimize the entangling gate count assuming all-to-all qubit connectivity. First, we consider circuit optimization based on template matching and design Clifford-specific templates that leverage the ability to factor out Pau…
Stochastic channels are ubiquitous in the field of quantum information because they are simple and easy to analyze. In particular, Pauli channels and depolarizing channels are widely studied because they can be efficiently simulated in many relevant quantum circuits. Despite their wide use, the properties of general stochastic channels have received little attention. In this paper, we prove that the diamond distance of a general stochastic channel from the identity coincides with its process infidelity to the identity. We demonstrate with an explicit example that there exist multi-qubit stocha…
Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, we show that $2(N-1)$ generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the $y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types which are defined in this paper are sufficient to simulate a controlled-unitary gate $\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling on $\mathbb{C}^2\otimes \mathbb{C}^N$. In the scenario of the unitary gate $\mathcal{U}_{cd(M\otimes N)}$ with $M\geq3$ that is locally equivalent to a diagonal uni…
We obtain the accuracy limit for estimating the expectation value of the position of a relativistic particle for an observer moving along one direction at a constant velocity. We use a specific model of a relativistic spin-1/2 particle described by a gaussian wave function with a spin down in the rest frame. To derive the state vector of the particle for the moving observer, we use the Wigner rotation that entangles the spin and the momentum of the particle. Based on this wave function for the moving frame, we obtain the symmetric logarithmic derivative (SLD) Cramer-Rao bound that sets the est…
Orientation-dependent Pauli noise in one-dimensional discrete-time quantum walks
Departamento de Física, CUCEI, Universidad de Guadalajara（瓜达拉哈拉大学CUCEI物理系）；Centro Universitario de Guadalajara, Universidad de Guadalajara（瓜达拉哈拉大学瓜达拉哈拉大学区）
Architectures and random properties of symplectic quantum circuits
Information Sciences, Los Alamos National Laboratory, 87545 NM, USA（信息科学，洛斯阿拉莫斯国家实验室）；Theoretical Division, Los Alamos National Laboratory, 87545 NM, USA（理论部门，洛斯阿拉莫斯国家实验室）
We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the "frame potential," which is minimized by unitary $k$-designs and measures the $2$-norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is out-of-time-order (OTO) four-point correlation functions. We show that the norm squared of a generalization of out-of-time-order $2k$-point correlators is proportional to the $k$th frame potential, providing a quantitative connection between chaos and pseudor…
