We present a classical algorithm that, for any 3D geometrically-local, polylogarithmic-depth quantum circuit $C$ acting on $n$ qubits, and any bit string $x\in\{0,1\}^n$, can compute the quantity $|< x |C|0^{\otimes n}>|^2$ to within any inverse-polynomial additive error in quasi-polynomial time. It is known that it is $\#P$-hard to compute this same quantity to within $2^{-n^2}$ additive error [Mov20, KMM21]. The previous best known algorithm for this problem used $O(2^{n^{1/3}}\text{poly}(1/ε))$ time to compute probabilities to within additive error $ε$ [BGM20]. Notably, the [BGM20] paper included an elegant polynomial time algorithm for this estimation task restricted to 2D circuits, which makes a novel use of 1D Matrix Product States (MPS) carefully tailored to the 2D geometry of the circuit in question. Surprisingly, it is not clear that it is possible to extend this use of MPS to address the case of 3D circuits in polynomial time. This raises a natural question as to whether the computational complexity of the 3D problem might be drastically higher than that of the 2D problem. In this work we address this question by exhibiting a quasi-polynomial time algorithm for the 3D case. In order to surpass the technical barriers encountered by previously known techniques we are forced to pursue a novel approach.
Our algorithm has a Divide-and-Conquer structure, demonstrating how to approximate the desired quantity via several instantiations of the same problem type, each involving 3D-local circuits on about half the number of qubits as the original. This division step is then applied recursively, expressing the original quantity as a weighted combination of smaller and smaller 3D-local quantum circuits. A central technical challenge is to control correlations arising from entanglement that may exist between the different circuit ``pieces" produced this way.
We present a classical algorithm that, for any $D$-dimensional geometrically-local, quantum circuit $C$ of polylogarithmic-depth, and any bit string $x \in {0,1}^n$, can compute the quantity $|<x|C|0^{\otimes n}>|^2$ to within any inverse-polynomial additive error in quasi-polynomial time, for any fixed dimension $D$. This is an extension of the result [CC21], which originally proved this result for $D = 3$. To see why this is interesting, note that, while the $D = 1$ case of this result follows from standard use of Matrix Product States, known for decades, the $D = 2$ case required novel and …
Prior work has shown that there exists a relation problem which can be solved with certainty by a constant-depth quantum circuit composed of geometrically local gates in two dimensions, but cannot be solved with high probability by any classical constant depth circuit composed of bounded fan-in gates. Here we provide two extensions of this result. Firstly, we show that a separation in computational power persists even when the constant-depth quantum circuit is restricted to geometrically local gates in one dimension. The corresponding quantum algorithm is the simplest we know of which achieves…
Shallow Quantum Circuits for Geometrically Local Lindblad Dynamics
Center on Frontiers of Computing Studies, School of Computer Science, Peking University（北京大学计算机科学学院前沿计算研究中心）；School of Physics, Peking University（北京大学物理学院）；School of Artificial Intelligence, Beijing Normal University（北京师范大学人工智能学院）；Mathematical Institute, University of Oxford（牛津大学数学研究所）
Measurement-Guided State Refinement for Shallow Feedback-Based Quantum Optimization Algorithm
Instituto de Física Gleb Wataghin, Universidade de Campinas（坎皮纳斯大学格莱布·瓦塔金物理研究所）；São Paulo State University (UNESP), School of Sciences（圣保罗州立大学理学院）；QuaTI – Quantum Technology & Information（QuaTI量子技术与信息研究所）；Hospital Israelita Albert Einstein（阿尔伯特·爱因斯坦以色列医院）
Limitations of Noisy Geometrically Local Quantum Circuits
University of Maryland（马里兰大学）；Joint Center for Quantum Information & Computer Science, University of Maryland and NIST（马里兰大学与美国国家标准与技术研究院联合量子信息与计算中心）；IBM T. J. Watson Research Center（IBM托马斯·J·沃森研究中心）
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）
Conditional dependence and Scrooge ensembles in shallow random quantum circuits
University of Waterloo（滑铁卢大学）；TCM Group, Cavendish Laboratory, University of Cambridge（剑桥大学卡文迪许实验室 TCM 组）；Walter Burke Institute for Theoretical Physics and Institute for Quantum Information and Matter, California Institute of Technology（加州理工学院理论物理沃尔特·伯克研究所与量子信息物质研究所）；Google Quantum AI（谷歌量子人工智能）；Perimeter Institute for Theoretical Physics（理论物理珀米特研究所）
Hybrid quantum-classical algorithms provide ways to use noisy intermediate-scale quantum computers for practical applications. Expanding the portfolio of such techniques, we propose a quantum circuit learning algorithm that can be used to assist the characterization of quantum devices and to train shallow circuits for generative tasks. The procedure leverages quantum hardware capabilities to its fullest extent by using native gates and their qubit connectivity. We demonstrate that our approach can learn an optimal preparation of the Greenberger-Horne-Zeilinger states, also known as "cat states…
Probing the Ground State of the Antiferromagnetic Heisenberg Model on the Kagome Lattice using Geometrically Informed Variational Quantum Eigensolver
Constantine Quantum Technologies（君士坦丁量子技术）；Frères Mentouri University Constantine 1（费雷斯门图里大学君士坦丁1）；Laboratoire de Physique Mathématique et Subatomique（数学与亚原子物理实验室）；Theoretical Physics Laboratory（理论物理实验室）
On the distinguishability of geometrically uniform quantum states
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Illinois Quantum Information Science and Technology (IQUIST) Center, University of Illinois Urbana-Champaign（伊利诺伊量子信息与科学技术中心）
Toward Optimal Circuit Depth for Geometrically Local Hamiltonian Simulation
Center on Frontiers of Computing Studies, School of Computer Science, Peking University（北京大学计算机科学学院前沿计算研究中心）；School of Physics, Peking University（北京大学物理学院）；School of Artificial Intelligence, Beijing Normal University（北京师范大学人工智能学院）；State Key Laboratory for Novel Software Technology, Nanjing University（南京大学软件新技术国家重点实验室）；Hefei National Laboratory（合肥国家实验室）；Mathematical Institute, University of Oxford（牛津大学数学研究所）
A generalized Bloch sphere, in which the states of a quantum entity of arbitrary dimension are geometrically represented, is investigated and further extended, to also incorporate the measurements. This extended representation constitutes a general solution to the measurement problem, inasmuch it allows to derive the Born rule as an average over hidden-variables, describing not the state of the quantum entity, but its interaction with the measuring system. According to this modelization, a quantum measurement is to be understood, in general, as a tripartite process, formed by an initial determ…
We consider the task of estimating the expectation value of an $n$-qubit tensor product observable $O_1\otimes O_2\otimes \cdots \otimes O_n$ in the output state of a shallow quantum circuit. This task is a cornerstone of variational quantum algorithms for optimization, machine learning, and the simulation of quantum many-body systems. Here we study its computational complexity for constant-depth quantum circuits and three types of single-qubit observables $O_j$ which are (a) close to the identity, (b) positive semidefinite, (c) arbitrary. It is shown that the mean value problem admits a class…
A polynomial-time classical sampler for noisy quantum circuits from statistical mechanics
University of Maryland, College Park（马里兰大学帕克分校）；Electrical & Computer Engineering Department, UCLA（加州大学洛杉矶分校电气与计算机工程系）；Department of Electrical and Computer Engineering, Princeton University（普林斯顿大学电气与计算机工程系）；QuEra Computing Inc.（QuEra计算公司）；Department of Physics, Stanford University（斯坦福大学物理系）
Due to the unreliability and limited capacity of existing quantum computer prototypes, quantum circuit simulation continues to be a vital tool for validating next generation quantum computers and for studying variational quantum algorithms, which are among the leading candidates for useful quantum computation. Existing quantum circuit simulators do not address the common traits of variational algorithms, namely: 1) their ability to work with noisy qubits and operations, 2) their repeated execution of the same circuits but with different parameters, and 3) the fact that they sample from circuit…
