Two-party one-way quantum communication has been extensively studied in the recent literature. We target the size of minimal information that is necessary for a feasible party to finish a given combinatorial task, such as distinction of instances, using one-way communication from another party. This type of complexity measure has been studied under various names: advice complexity, Kolmogorov complexity, distinguishing complexity, and instance complexity. We present a general framework focusing on underlying combinatorial takes to study these complexity measures using quantum information processing. We introduce the key notions of relative hardness and quantum advantage, which provide the foundations for task-based quantum minimal one-way information complexity theory.
On computational complexity and average-case hardness of shallow-depth boson sampling
NextQuantum Innovation Research Center, Department of Physics and Astronomy, Seoul National University（首尔大学物理与天文学系Next量子创新研究中心）；Department of Physics, Korea Advanced Institute of Science and Technology（韩国科学技术院物理系）
Recent work of Bravyi et al. and follow-up work by Bene Watts et al. demonstrates a quantum advantage for shallow circuits: constant-depth quantum circuits can perform a task which constant-depth classical (i.e., AC$^0$) circuits cannot. Their results have the advantage that the quantum circuit is fairly practical, and their proofs are free of hardness assumptions (e.g., factoring is classically hard, etc.). Unfortunately, constant-depth classical circuits are too weak to yield a convincing real-world demonstration of quantum advantage. We attempt to hold on to the advantages of the above resu…
We observe that fault-tolerant quantum computers have an optimal advantage over classical computers in approximating solutions to many NP optimization problems. This observation however gives nothing in practice.
Quantum Circuit Representation of Combinatorial Matrix Functions
SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University（成均馆大学纳米技术高级研究所）；IBM Quantum（IBM量子）；Department of Physics, Chungbuk National University（忠北国立大学物理系）；Yonsei University（延世大学）
Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians
Joint Center for Quantum Information & Computer Science, National Institute of Standards & Technology and University of Maryland, College Park（量子信息与计算联合中心，美国国家标准与技术研究院和马里兰大学帕克分校）；Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park（计算机科学系和高级计算机研究所，马里兰大学帕克分校）
Efficiently verifiable quantum advantage using error correction
IBM Research（IBM研究院）；University of Chicago（芝加哥大学）；Massachusetts Institute of Technology（麻省理工学院）；Simons Institute for the Theory of Computing, University of California at Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；QuEra Computing Inc.（QuEra计算公司）；Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland（马里兰大学帕克分校量子信息与计算机科学联合中心）；ETH Zürich（苏黎世联邦理工学院）
A combinatorial framework for clustering graph states: Algorithms and hardness for rank-integrity
Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800（波尔多大学）；University of Innsbruck（因斯布鲁克大学）；School of Computer Science, Georgia Institute of Technology（佐治亚理工学院计算机学院）；School of Computer Science and School of Mathematics, Georgia Institute of Technology（佐治亚理工学院计算机学院与数学学院）；Collegium de Lyon, ENS de Lyon, LIP（里昂学院，巴黎高等师范学院里昂分校，计算与语言研究所）；Kahlert School of Computing, University of Utah（犹他大学卡尔赫特计算机学院）；Univ. Lyon, ENS de Lyon, UCBL, CNRS, LIP（里昂大学，巴黎高等师范学院里昂分校，里昂第一大学，法国国家科学研究中心，计算与语言研究所）
Entanglement geometry separates circuit cutting, classical hardness, and trainability
Quantum Software Lab（量子软件实验室）；University of Edinburgh, UK（爱丁堡大学）；IBM Research（IBM研究院）；Centre for Quantum Technologies（量子技术中心）；National University of Singapore（新加坡国立大学）
Photonics is a promising platform for demonstrating a quantum computational advantage (QCA) by outperforming the most powerful classical supercomputers on a well-defined computational task. Despite this promise, existing proposals and demonstrations face challenges. Experimentally, current implementations of Gaussian boson sampling (GBS) lack programmability or have prohibitive loss rates. Theoretically, there is a comparative lack of rigorous evidence for the classical hardness of GBS. In this work, we make progress in improving both the theoretical evidence and experimental prospects. We pro…
Hardness and Complexity Transition of Noisy Random Circuit Sampling
NextQuantum Center, Department of Physics and Astronomy, Seoul National University（首尔大学物理与天文学系 NextQuantum 中心）；Department of Physics, Korea Advanced Institute of Science and Technology（韩国科学技术院物理系）
Fermionic Linear Optics (FLO) is a restricted model of quantum computation which in its original form is known to be efficiently classically simulable. We show that, when initialized with suitable input states, FLO circuits can be used to demonstrate quantum computational advantage with strong hardness guarantees. Based on this, we propose a quantum advantage scheme which is a fermionic analogue of Boson Sampling: Fermion Sampling with magic input states. We consider in parallel two classes of circuits: particle-number conserving (passive) FLO and active FLO that preserves only fermionic parit…
Quantum variational optimization has been posed as an alternative to solve optimization problems faster and at a larger scale than what classical methods allow. In this paper we study systematically the role of entanglement, the structure of the variational quantum circuit, and the structure of the optimization problem, in the success and efficiency of these algorithms. For this purpose, our study focuses on the variational quantum eigensolver (VQE) algorithm, as applied to quadratic unconstrained binary optimization (QUBO) problems on random graphs with tunable density. Our numerical results …
The Quest for Quantum Advantage in Combinatorial Optimization: End-to-end Benchmarking of Quantum Solvers vs. Multi-core Classical Solvers
Kipu Quantum GmbH（基普量子有限公司）；Department of Physical Chemistry, University of the Basque Country UPV/EHU（巴斯克大学物理化学系）；Zuse Institute Berlin（柏林祖斯研究所）；Technische Universität Berlin（柏林工业大学）；IBM Quantum, IBM Research Europe – Zurich（IBM量子，IBM欧洲苏黎世研究院）；IDAL, Electronic Engineering Department, ETSE-UV, University of Valencia（瓦伦西亚大学电子工程学院IDAL）
We study the computational power of unitary Clifford circuits with solely magic state inputs (CM circuits), supplemented by classical efficient computation. We show that CM circuits are hard to classically simulate up to multiplicative error (assuming PH non-collapse), and also up to additive error under plausible average-case hardness conjectures. Unlike other such known classes, a broad variety of possible conjectures apply. Along the way we give an extension of the Gottesman-Knill theorem that applies to universal computation, showing that for Clifford circuits with joint stabiliser and non…
An accurate calculation of the properties of quantum many-body systems is one of the most important yet intricate challenges of modern physics and computer science. In recent years, the tensor network ansatz has established itself as one of the most promising approaches enabling striking efficiency of simulating static properties of one-dimensional systems and abounding numerical applications in condensed matter theory. In higher dimensions, however, a connection to the field of computational complexity theory has shown that the accurate normalization of the two-dimensional tensor networks cal…
Improving Performance in Combinatorial Optimization Problems with Inequality Constraints: An Evaluation of the Unbalanced Penalization Method on D-Wave Advantage
One-parameter interpolations between any two unitary matrices (e.g., quantum gates) $U_1$ and $U_2$ along efficient paths contained in the unitary group are constructed. Motivated by applications, we propose the continuous unitary path $U(θ)$ obtained from the QR-factorization \[ U(θ)R(θ)=(1-θ)A+θB, \] where $U_1 R_1=A$ and $U_2 R_2=B$ are the QR-factorizations of $A$ and $B$, and $U(θ)$ is a unitary for all $θ$ with $U(0)=U_1$ and $U(1)=U_2$. The QR-algorithm is modified to, instead of $U(θ)$, output a matrix whose columns are proportional to the corresponding columns of $U(θ)$ and whose entr…
