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arXiv:quant-ph/0608060 · Physical Review A

Classical simulation versus universality in measurement-based quantum computation

M. Van den Nest, Wolfgang Dür, Guifré Vidal, Hans J. Briegel

Published Jan 31, 2007• 95 citations
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Abstract

We investigate for which resource states an efficient classical simulation of measurement-based quantum computation is possible. We show that the Schmidt-rank width, a measure recently introduced to assess universality of resource states, plays a crucial role in also this context. We relate Schmidt-rank width to the optimal description of states in terms of tree tensor networks and show that an efficient classical simulation of measurement-based quantum computation is possible for all states with logarithmically bounded Schmidt-rank width (with respect to the system size). For graph states where the Schmidt-rank width scales in this way, we efficiently construct the optimal tree tensor network descriptions, and provide several examples. We highlight parallels in the efficient description of complex systems in quantum information theory and graph theory.

DOI
10.1103/physreva.75.012337

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We investigate for which resource states an efficient classical simulation of measurement-based quantum computation is possible. We show that the Schmidt-rank width, a measure recently introduced to assess universality of resource states, plays a crucial role in also this context. We relate Schmidt-rank width to the optimal description of states in terms of tree tensor networks and show that an efficient classical simulation of measurement-based quantum computation is possible for all states with logarithmically bounded Schmidt-rank width (with respect to the system size). For graph states where the Schmidt-rank width scales in this way, we efficiently construct the optimal tree tensor network descriptions, and provide several examples. We highlight parallels in the efficient description of complex systems in quantum information theory and graph theory.
In this paper, we build a framework allowing for a systematic investigation of the fundamental issue: ' Which quantum states serve as universal resources for measurement-based (one-way) quantum computation? ' We start our study by re-examining what is exactly meant by 'universality' in quantum computation, and what the implications are for universal one-way quantum computation. Given the framework of a measurement-based quantum computer, where quantum information is processed by local operations only, we find that the most general universal one-way quantum computer is one which is capable of accepting arbitrary classical inputs and producing arbitrary quantum outputs—we refer to this property as CQ-universality. We then show that a systematic study of CQ-universality in one-way quantum computation is possible by identifying entanglement features that are required to be present in every universal resource. In particular, we find that a large class of entanglement measures must reach its supremum on every universal resource. These insights are used to identify several families of states as being not universal, such as one-dimensional (1D) cluster states, Greenberger–Horne–Zeilinger (GHZ) states, W states, and ground states of non-critical 1D spin systems. Our criteria are strengthened by considering the efficiency of a quantum computation, and we find that entanglement measures must obey a certain scaling law with the system size for all efficient universal resources. This again leads to examples of non-universal resources, such as, e.g. ground states of critical 1D spin systems. On the other hand, we provide several examples of efficient universal resources, namely graph states corresponding to hexagonal, triangular and Kagome lattices. Finally, we consider the more general notion of encoded CQ-universality, where quantum outputs are allowed to be produced in an encoded form. Again we provide entanglement-based criteria for encoded universality. Moreover, we present a general procedure to construct encoded universal resources.
Various algorithms have been developed to simulate quantum circuits on classical hardware. Among the most prominent are approaches based on \emph{stabilizer decompositions} and \emph{tensor network contraction}. In this work, we present a unified framework that bridges these two approaches, placing them under a common formalism. Using this, we present two new algorithms to simulate an nnn-qubit circuit CCC: one that runs in O~(Ttw(C))\tilde{O}(T^{\mathsf{tw}(C)})O~(Ttw(C)) time and the other in O~(Tγ⋅tw(C))\tilde{O}(T^{γ\cdot \mathsf{tw}(C)})O~(Tγ⋅tw(C)) time, where tw(C)\mathsf{tw}(C)tw(C) and rw(C)\mathsf{rw}(C)rw(C) refer to the the tree-width and rank-width, respectively, of a tensor network associated to CCC, TTT is the number of non-Clifford gates in CCC, and γ≈3.42γ\approx 3.42γ≈3.42. The proposed algorithms are simple, only require a linear amount of memory, are trivially parallelizable, and interact nicely with ZX-diagram simplification routines. Furthermore, we introduce the refined complexity measures \emph{focused tree-width} and \emph{focused rank-width}, which are always at least as efficient as their standard equivalent; these can be directly applied within our simulation algorithms, allowing for a more precise upper bound on the run time.
We propose an efficient algorithm to classically simulate quantum evolution on tree graphs with bounded entanglement for any bipartite partition along an edge of the tree. The algorithm combines the Schmidt decomposition with the tensor network state, where the latter is a natural extension of the matrix product states from chains to graphs. As an interesting application of this algorithm, we show that any one-way quantum computation on tree graphs can be efficiently simulated with a classical computer.
We describe and analyze algorithms for classically simulating measurement of an n-qubit quantum state in the standard basis, that is, sampling a bit string from the probability distribution determined by the Born rule. Our algorithms reduce the sampling task to computing poly(n) amplitudes of n-qubit states; unlike previously known techniques they do not require computation of marginal probabilities. Two classes of quantum states are considered: output states of polynomial-size quantum circuits, and ground states of local Hamiltonians with an inverse polynomial spectral gap. We show that our algorithms can significantly accelerate quantum circuit simulations based on tensor network contraction or low-rank stabilizer decompositions. As another striking consequence we obtain the first efficient classical simulation algorithm for measurement-based quantum computation with the surface code resource state on any planar graph and any schedule of measurements.