Noise is one of the main obstacles to realizing quantum devices that achieve a quantum computational advantage. A possible approach to minimize the noise effect is to employ shallow-depth quantum circuits since noise typically accumulates as circuit depth grows. In this work, we investigate the complexity of shallow-depth linear-optical circuits under the effects of photon loss and partial distinguishability. By establishing a correspondence between a linear-optical circuit and a bipartite graph, we show that the effects of photon loss and partial distinguishability are equivalent to removing the corresponding vertices. Using this correspondence and percolation theory, we prove that for constant-depth linear-optical circuits with single photons, there is a threshold of loss (noise) rate above which the linear-optical systems can be decomposed into smaller systems with high probability, which enables us to simulate the systems efficiently. Consequently, our result implies that even in shallow-depth circuits where noise is not accumulated enough, its effect may be sufficiently significant to make them efficiently simulable using classical algorithms due to its entanglement structure constituted by shallow-depth circuits.
Construction and simulability of quantum circuits with free fermions in disguise
Università di Bologna and INFN, Sezione di Bologna（博洛尼亚大学及意大利国家核物理研究所博洛尼亚分部）；Laboratoire de Probabilités, Statistique et Modélisation & CNRS, Université Paris Cité, Sorbonne Université Paris（巴黎概率、统计与建模实验室及法国国家科学研究中心、巴黎西岱大学、索邦巴黎大学）
In the task of quantum state learning, one receives some data about measurements performed on a state, and using that, must make predictions on the outcomes of unseen measurements. Computing a prediction is generally hard but it has been shown that learning can be performed efficiently for states that are generated by Clifford circuits, which are known to be efficiently classically simulable. This naturally leads to the question, how does efficient state learnability compare with efficient classical simulation? In this work we introduce an extra condition on top of classical simulablity that g…
Classical simulability of quantum circuits followed by sparse classical post-processing
Institute of Systems and Information Engineering, University of Tsukuba（筑波大学系统信息工程学院）；Tsukuba Institute for Advanced Research, University of Tsukuba（筑波大学筑波先进研究所）
As a promising candidate for exhibiting quantum computational supremacy, Gaussian Boson Sampling (GBS) is designed to exploit the ease of experimental preparation of Gaussian states. However, sufficiently large and inevitable experimental noise might render GBS classically simulable. In this work, we formalize this intuition by establishing a sufficient condition for approximate polynomial-time classical simulation of noisy GBS --- in the form of an inequality between the input squeezing parameter, the overall transmission rate and the quality of photon detectors. Our result serves as a non-cl…
Investigating the classical simulability of quantum circuits provides a promising avenue towards understanding the computational power of quantum systems. Whether a class of quantum circuits can be efficiently simulated with a probabilistic classical computer, or is provably hard to simulate, depends quite critically on the precise notion of "classical simulation" and in particular on the required accuracy. We argue that a notion of classical simulation, which we call epsilon-simulation, captures the essence of possessing "equivalent computational power" as the quantum system it simulates: It …
We consider Gaussian quantum circuits supplemented with non-Gaussian input states and derive sufficient conditions for efficient classical strong simulation of these circuits. In particular, we generalise the stellar representation of continuous-variable quantum states to the multimode setting and relate the stellar rank of the input non-Gaussian states, a recently introduced measure of non-Gaussianity, to the cost of evaluating classically the output probability densities of these circuits. Our results have consequences for the strong simulability of a large class of near-term continuous-vari…
London Centre for Nanotechnology, University College London（伦敦纳米技术中心，伦敦大学学院）；School of Physics and Astronomy, University of Leeds（利兹大学物理与天文学学院）
We study the evolution of an open quantum system using a Langevin unravelling of the density matrix evolution over matrix product states. As the strength of coupling to and temperature of the environment is increased, we find a transition where the entanglement of the individual trajectories saturates, permitting a classical simulation of the system for all times. This is the Hamiltonian open system counterpart of the saturation in entanglement found in random circuits with projective or weak measurements. If a system is open, there is a limit to the advantage in simulating its behaviour on a …
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）
The cost of simulating classically tractable quantum circuits and dynamics
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；European Organization for Nuclear Research (CERN)（欧洲核子研究组织（CERN））；Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院（EPFL））；Chulalongkorn University（朱拉隆功大学）
Sampling from probability distributions of quantum circuits is a fundamentally and practically important task which can be used to demonstrate quantum supremacy using noisy intermediate-scale quantum devices. In the present work, we examine classical simulability of sampling from the output photon-number distribution of linear-optical circuits composed of random beam splitters with equally distributed squeezed vacuum states and single-photon states input. We provide efficient classical algorithms to simulate linear-optical random circuits and show that the algorithms' error is exponentially sm…
Classical simulation of a quantum circuit with noisy magic inputs
Graduate School of Quantum Science and Technology, Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院量子科学与技术研究生院）；Department of Physics, Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院物理系）
Complexity of tensor network simulation for noisy quantum circuits
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Department of Mathematics, Tsinghua University（清华大学数学系）
Shallow Quantum Circuits for Deep Chemistry via Valence Bond Embeddings
Institute for Computer Science, University of Augsburg（奥格斯堡大学计算机科学研究所）；Center for Advanced Analytics and Predictive Sciences, University of Augsburg（奥格斯堡大学高级分析与预测科学中心）
We investigate the effect of quantum noise on the measurement-induced quantum phase transition in monitored random quantum circuits. Using the efficient simulability of random Clifford circuits, we find that the transition is broadened into a crossover and that the phase diagram as a function of projective measurements and noise exhibits several distinct regimes. We show that a mapping to a classical statistical mechanics problem accounts for the main features of the random circuit phase diagram. The bulk noise maps to an explicit permutation symmetry breaking coupling; this symmetry is sponta…
One of the crucial steps in building a scalable quantum computer is to identify the noise sources which lead to errors in the process of quantum evolution. Different implementations come with multiple hardware-dependent sources of noise and decoherence making the problem of their detection manyfoldly more complex. We develop a randomized benchmarking algorithm which uses Weyl unitaries to efficiently identify and learn a mixture of error models which occur during the computation. We provide an efficiently computable estimate of the overhead required to compute expectation values on outputs of …
We introduce a new family of quantum circuits for which the scrambling of a subspace of non-local operators is classically simulable. We call these circuits `super-Clifford circuits', since the Heisenberg time evolution of these operators corresponds to a Clifford evolution in operator space. By simulating the Clifford evolution in operator space we are able to simulate the time evolution of certain single Pauli strings into operators with an operator entanglement that grows linearly with the number of qubits. These circuits provide a new technique for studying scrambling in systems with a lar…
Does provable absence of barren plateaus imply classical simulability?
Information Sciences, Los Alamos National Laboratory（信息科学，洛斯阿拉莫斯国家实验室）；Quantum Science Center, Oak Ridge（量子科学中心，奥克兰岭）；Theoretical Division, Los Alamos National Laboratory（理论部，洛斯阿拉莫斯国家实验室）；Center for Nonlinear Studies, Los Alamos National Laboratory（非线性研究中心，洛斯阿拉莫斯国家实验室）；University of Strathclyde（斯特拉斯克莱德大学）；Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL)（物理研究所，洛桑联邦理工学院（EPFL））；Donostia International Physics Center（多斯通国际物理中心）；Department of Physics and Astronomy, University of Waterloo（物理与天文学系，滑铁卢大学）；Vector Institute, MaRS Centre（向量研究所，马尔伯里中心）；Complex Systems, Department of Physics, Faculty of Science, Chulalongkorn University（复杂系统，物理系，科学学院，朱拉隆梭大学）；Walter Burke Institute for Theoretical Physics, Caltech（沃尔特·伯克理论物理研究所，加州理工学院）
We show that low-depth random quantum circuits can be efficiently simulated by a quantum teleportation-inspired algorithm. By using logical qubits to redirect and teleport the quantum information in quantum circuits, the original circuits can be renormalized to new circuits with a smaller number of logical qubits. We demonstrate the algorithm to simulate several random quantum circuits, including 1D-chain 1000-qubit 42-depth, 2D-grid 125*8-qubit 42-depth and 2D-Bristlecone 72-qubit 32-depth circuits. Our results present a memory-efficient method with a clear physical picture to simulate low-de…
Practical Log-Depth Quantum State Preparation and Circuit Verification via Tree Tensor Network Compilation
University College London（伦敦大学学院）；Centre for Computational Science, Department of Chemistry, University College London（伦敦大学学院化学系计算科学中心）；Advanced Research Computing Centre, University College London（伦敦大学学院高级计算研究中心）；Informatics Institute, University of Amsterdam（阿姆斯特丹大学信息学研究所）
Efficient fidelity simulation of high-rate magic distillation circuits
QuEra Computing Inc.（QuEra 计算公司）；Joint Center for Quantum Information and Computer Science, University of Maryland（马里兰大学量子信息与计算机科学联合中心）；Simons Institute for the Theory of Computing, University of California at Berkeley（加州大学伯克利分校西蒙斯计算理论研究所）；Institute for Theoretical Physics, ETH Zürich（苏黎世联邦理工学院理论物理研究所）；Department of Physics, Harvard University（哈佛大学物理学系）
In breakthrough work, Bravyi, Gosset, and König (BGK) [Science, 2018] unconditionally proved that constant depth quantum circuits are more powerful than their classical counterparts. Their result is equivalent to saying that a particular family of constant depth quantum circuits takes classical circuits at least $Ω(\log n)$ depth to "simulate", in a certain sense. In our paper, we formalise their sense of simulation, which we call "possibilistic simulation" or "p-simulation", and construct explicit classical circuits that can p-simulate any depth-$d$ quantum circuit with Clifford and $t$ $T$-g…
We consider Kitaev's quantum double model based on a finite group $G$ and describe quantum circuits for (a) preparation of the ground state, (b) creation of anyon pairs separated by an arbitrary distance, and (c) non-destructive topological charge measurement. We show that for any solvable group $G$ all above tasks can be realized by constant-depth adaptive circuits with geometrically local unitary gates and mid-circuit measurements. Each gate may be chosen adaptively depending on previous measurement outcomes. Constant-depth circuits are well suited for implementation on a noisy hardware sinc…
We develop a global variable substitution method that reduces $n$-variable monomials in combinatorial optimization problems to equivalent instances with monomials in fewer variables. We apply this technique to $3$-SAT and analyze the optimal quantum circuit depth needed to solve the reduced problem using the quantum approximate optimization algorithm. For benchmark $3$-SAT problems, we find that the upper bound of the circuit depth is smaller when the problem is formulated as a product and uses the substitution method to decompose gates than when the problem is written in the linear formulatio…
Near term quantum computers with a high quantity (around 50) and quality (around 0.995 fidelity for two-qubit gates) of qubits will approximately sample from certain probability distributions beyond the capabilities of known classical algorithms on state-of-the-art computers, achieving the first milestone of so-called quantum supremacy. This has stimulated recent progress in classical algorithms to simulate quantum circuits. Classical simulations are also necessary to approximate the fidelity of multiqubit quantum computers using cross entropy benchmarking. Here we present numerical results of…
We present a scalable set of universal gates and multiply controlled gates in a qudit basis through a bijective mapping from N qubits to qudits with D = 2^N levels via rotations in U(2). For each of the universal gates (H, CNOT, and T), as well as the NOT gate and multiply-controlled-Z gates, we describe a systematic approach to identifying the set of U(2) rotations required to implement each gate for any qudit of size N and with minimal use of an ancilla level. The qudit gates are analyzed in terms of the total rotation count and gate depth as the system scales with D. We apply the qudit-basi…
