Simulating Quantum Computation by Contracting Tensor Networks
01 Jun 2008-SIAM Journal on Computing (Society for Industrial and Applied Mathematics)-Vol. 38, Iss: 3, pp 963-981
TL;DR: It is proved that a quantum circuit with T gates whose underlying graph has a treewidth d can be simulated deterministically in T^{O(1)}\exp[O(d)]$ time, which, in particular, is polynomial in $T$ if d=O(\log T)$.
Abstract: The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with $T$ gates whose underlying graph has a treewidth $d$ can be simulated deterministically in $T^{O(1)}\exp[O(d)]$ time, which, in particular, is polynomial in $T$ if $d=O(\log T)$. Among many implications, we show efficient simulations for log-depth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that one-way quantum computation of Raussendorf and Briegel (Phys. Rev. Lett., 86 (2001), pp. 5188-5191), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small-treewidth graph with a constant maximum degree. (The requirement on the maximum degree was removed in [I. L. Markov and Y. Shi, preprint:quant-ph/0511069].)
Citations
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TL;DR: In this paper, an updated version of supplementary information to accompany "Quantum supremacy using a programmable superconducting processor", an article published in the October 24, 2019 issue of Nature, is presented.
Abstract: This is an updated version of supplementary information to accompany "Quantum supremacy using a programmable superconducting processor", an article published in the October 24, 2019 issue of Nature. The main article is freely available at this https URL. Summary of changes since arXiv:1910.11333v1 (submitted 23 Oct 2019): added URL for qFlex source code; added Erratum section; added Figure S41 comparing statistical and total uncertainty for log and linear XEB; new References [1,65]; miscellaneous updates for clarity and style consistency; miscellaneous typographical and formatting corrections.
4,873 citations
TL;DR: An overview of the field of Variational Quantum Algorithms is presented and strategies to overcome their challenges as well as the exciting prospects for using them as a means to obtain quantum advantage are discussed.
Abstract: Applications such as simulating complicated quantum systems or solving large-scale linear algebra problems are very challenging for classical computers due to the extremely high computational cost. Quantum computers promise a solution, although fault-tolerant quantum computers will likely not be available in the near future. Current quantum devices have serious constraints, including limited numbers of qubits and noise processes that limit circuit depth. Variational Quantum Algorithms (VQAs), which use a classical optimizer to train a parametrized quantum circuit, have emerged as a leading strategy to address these constraints. VQAs have now been proposed for essentially all applications that researchers have envisioned for quantum computers, and they appear to the best hope for obtaining quantum advantage. Nevertheless, challenges remain including the trainability, accuracy, and efficiency of VQAs. Here we overview the field of VQAs, discuss strategies to overcome their challenges, and highlight the exciting prospects for using them to obtain quantum advantage.
842 citations
Cites background from "Simulating Quantum Computation by C..."
...Moreover, as quantum circuits can be viewed as tensor networks [51], it is quite natural to combine the existing tensor network techniques with a quantum ansatz [52– 54]....
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TL;DR: In this article, the authors study the task of sampling from the output distributions of (pseudo-)random quantum circuits, a natural task for benchmarking quantum computers, and show that this sampling task must take exponential time in a classical computer.
Abstract: A critical question for the field of quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of state-of-the-art classical computers, achieving so-called quantum supremacy. We study the task of sampling from the output distributions of (pseudo-)random quantum circuits, a natural task for benchmarking quantum computers. Crucially, sampling this distribution classically requires a direct numerical simulation of the circuit, with computational cost exponential in the number of qubits. This requirement is typical of chaotic systems. We extend previous results in computational complexity to argue more formally that this sampling task must take exponential time in a classical computer. We study the convergence to the chaotic regime using extensive supercomputer simulations, modeling circuits with up to 42 qubits - the largest quantum circuits simulated to date for a computational task that approaches quantum supremacy. We argue that while chaotic states are extremely sensitive to errors, quantum supremacy can be achieved in the near-term with approximately fifty superconducting qubits. We introduce cross entropy as a useful benchmark of quantum circuits which approximates the circuit fidelity. We show that the cross entropy can be efficiently measured when circuit simulations are available. Beyond the classically tractable regime, the cross entropy can be extrapolated and compared with theoretical estimates of circuit fidelity to define a practical quantum supremacy test.
730 citations
TL;DR: This work presents the leading proposals to achieve quantum supremacy, and discusses how to reliably compare the power of a classical computer to thePower of a quantum computer.
Abstract: The field of quantum algorithms aims to find ways to speed up the solution of computational problems by using a quantum computer. A key milestone in this field will be when a universal quantum computer performs a computational task that is beyond the capability of any classical computer, an event known as quantum supremacy. This would be easier to achieve experimentally than full-scale quantum computing, but involves new theoretical challenges. Here we present the leading proposals to achieve quantum supremacy, and discuss how we can reliably compare the power of a classical computer to the power of a quantum computer. Proposals for demonstrating quantum supremacy, when a quantum computer supersedes any possible classical computer at a specific task, are reviewed.
717 citations
TL;DR: In this paper, the authors review recent developments in measurement-based quantum computation with a view to both fundamental and practical issues, in particular the power of quantum computation, the protection against noise (fault tolerance) and steps towards experimental realization.
Abstract: Quantum computation offers a promising new kind of information processing, where the non-classical features of quantum mechanics are harnessed and exploited. A number of models of quantum computation exist. These models have been shown to be formally equivalent, but their underlying elementary concepts and the requirements for their practical realization can differ significantly. A particularly exciting paradigm is that of measurement-based quantum computation, where the processing of quantum information takes place by rounds of simple measurements on qubits prepared in a highly entangled state. We review recent developments in measurement-based quantum computation with a view to both fundamental and practical issues, in particular the power of quantum computation, the protection against noise (fault tolerance) and steps towards experimental realization. Finally, we highlight a number of connections between this field and other branches of physics and mathematics. So-called one-way schemes have emerged as a powerful model to describe and implement quantum computation. This article reviews recent progress, highlights connections to other areas of physics and discusses future directions.
706 citations
Cites methods from "Simulating Quantum Computation by C..."
...Recently, several techniques have been developed to tackle classical simulatability of MQC (refs&nbs...
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References
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Book•
01 Jan 2000
TL;DR: In this article, the quantum Fourier transform and its application in quantum information theory is discussed, and distance measures for quantum information are defined. And quantum error-correction and entropy and information are discussed.
Abstract: Part I Fundamental Concepts: 1 Introduction and overview 2 Introduction to quantum mechanics 3 Introduction to computer science Part II Quantum Computation: 4 Quantum circuits 5 The quantum Fourier transform and its application 6 Quantum search algorithms 7 Quantum computers: physical realization Part III Quantum Information: 8 Quantum noise and quantum operations 9 Distance measures for quantum information 10 Quantum error-correction 11 Entropy and information 12 Quantum information theory Appendices References Index
25,929 citations
01 Dec 2010
TL;DR: This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.
Abstract: Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.
14,825 citations
"Simulating Quantum Computation by C..." refers methods in this paper
...ange the treewidth. Quantum circuits. We review some basic concepts of quantum mechanics and quantum computation. For a more detailed treatment, we refer the readers to the book by Nielsen and Chuang [24]. The state space of one qubit is denoted by Hdef= C2. We fix an orthonormal basis for Hand label the basis vectors with |0i and |1i. The space of operators on a vector space V is denoted by L(V). The ...
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TL;DR: A scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states, which are thus one-way quantum computers and the measurements form the program.
Abstract: We present a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states. The measurements are used to imprint a quantum logic circuit on the state, thereby destroying its entanglement at the same time. Cluster states are thus one-way quantum computers and the measurements form the program.
3,260 citations
TL;DR: The results imply that a necessary condition for an exponential computational speedup is that the amount of entanglement increases with the size n of the computation, and provide an explicit lower bound on the required growth.
Abstract: We present a classical protocol to efficiently simulate any pure-state quantum computation that involves only a restricted amount of entanglement. More generally, we show how to classically simulate pure-state quantum computations on n qubits by using computational resources that grow linearly in n and exponentially in the amount of entanglement in the quantum computer. Our results imply that a necessary condition for an exponential computational speedup (with respect to classical computations) is that the amount of entanglement increases with the size n of the computation, and provide an explicit lower bound on the required growth.
2,019 citations
"Simulating Quantum Computation by C..." refers background in this paper
..., [26, 27, 24, 30, 24, 25, 16], have been developed to simulate quantum evoluti on by contracting variances of tensor networks under the names of Matrix Product States (MPS) , Projected Entangled Pairs States (PEPS) , etc....
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...Jozsa and Linden [13], as well as Vidal [26] demonstrate efficient classical simulation of such circuits and conclude that achieving quant m speed-ups requires more than a bounded amount of entanglement....
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...Jozsa and Linden [13], as we ll as Vidal [26] demonstrate efficient classical simulation of such circuits and conclude that achieving qua nt m speed-ups requires more than a bounded amount of entanglement....
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...We are grateful to Guifré Vidal and Frank Verstraetefor pointing out relevant previous works....
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TL;DR: An invariant of graphs called the tree-width is introduced, and used to obtain a polynomially bounded algorithm to test if a graph has a subgraph contractible to H, where H is any fixed planar graph.
1,726 citations