Quantum Computation and the Evaluation of Tensor Networks

Arad, Itai; Landau, Zeph

doi:10.1137/080739379

Cited by 66 publications

(84 citation statements)

References 42 publications

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“…We will now show how this worry can be circumvented; we can efficiently measure Tr(W x,z ρ) up to an additive approximation. The protocol that we will follow is largely inspired by studies of the evaluation of tensor networks on a quantum computer [30] which proved particularly useful for calculating partition functions in classical statistical mechanics. Related problems also arose in [31].…”

Section: A Implementing the Entanglement Witnessesmentioning

confidence: 99%

Bound entanglement in tree graphs

Kay¹

2010

J. Phys. A: Math. Theor.

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In this paper, we discuss the entanglement properties of graph-diagonal states, with particular emphasis on calculating the threshold for the transition between the presence and absence of entanglement (i.e. the separability point). Special consideration is made of the thermal states of trees, including the linear cluster state. We characterise the type of entanglement present, and describe the optimal entanglement witnesses and their implementation on a quantum computer, up to an additive approximation. In the case of general graphs, we invoke a relation with the partition function of the classical Ising model, thereby intimating a connection to computational complexity theoretic tasks. Finally, we show that the entanglement is robust to some classes of local perturbations.

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Section: A Implementing the Entanglement Witnessesmentioning

confidence: 99%

Bound entanglement in tree graphs

Kay¹

2010

J. Phys. A: Math. Theor.

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“…Whilst algorithms which can speed up tensor network contractions by optimising the bubbling used [3][4][5], as discusssed above, the underlying computational problem is NP-complete [6,7] Even ignoring the specific bubbling used, the complexity of the overall contraction procedure can also be shown to be prohibitive in general. Consider a network made from the binary tensors e and n. The value of e is 1 if and only if all indices are identical, and zero otherwise, whilst n has value 1 if and only if all legs differ and 0 otherwise.…”

Section: Computational Complexitymentioning

confidence: 99%

Hand-waving and interpretive dance: an introductory course on tensor networks

Bridgeman

Chubb

2017

J. Phys. A: Math. Theor.

321

300

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View the article online for updates and enhancements. AbstractThe curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this difficulty in both the numerical and analytic regimes.These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states.The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file.

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“…There it was found that the number Fourier components needed to be sampled scales polynomially with the lattice size, but in order to obtain a mulitplicative approximation of the partition function, the requisite accuracy of estimation of each coefficient scaled exponentially with the system size. An algorithm, based on using a quantum computer to contract tensor networks yields similar approximation scales [3]. Even preparing a quantum state which coherently encodes a classical thermal state of an Ising appears to be difficult, e.g.…”

Section: Partition Functionsmentioning

confidence: 99%

“…During the last ten years, significant efforts have been devoted to investigating whether quantum mechanics could help in this respect. Various methods, all involving the superposition principle, have been proposed to compute the Jones polynomial at particular values of its variable [1], partition functions of classical statistical models [2][3][4], the Tutte polynomial [5], or more generally to contract tensor networks [3].…”

Section: Introductionmentioning

confidence: 99%

Low depth quantum circuits for Ising models

et al. 2014

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A scheme for measuring complex temperature partition functions of Ising models is introduced. In the context of ordered qubit registers this scheme finds a natural translation in terms of global operations, and single particle measurements on the edge of the array. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimation error, valid with high confidence, are provided through a central-limit theorem, which validity extends beyond the present context. It holds for example for estimations of the Jones polynomial. Interestingly, the kind of state preparations and measurements involved in this application can in principle be made "instantaneous", i.e. independent of the system size or the parameters being simulated. Second, the scheme allows to accurately estimate some non-trivial invariants of links. A third result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models on a square lattice. We provide conditions under which estimating such partition functions allows one to reconstruct scattering amplitudes of quantum circuits making the problem BQP-hard. Using this mapping, we show that fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we show that the ability to accurately measure corner magnetizations on thermal states of two-dimensional Ising models with magnetic field leads to fully polynomial random approximation schemes (FPRAS) for the partition function. Each of these results corresponds to a section of the text that can be essentially read independently.

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Quantum Computation and the Evaluation of Tensor Networks

Cited by 66 publications

References 42 publications

Bound entanglement in tree graphs

Bound entanglement in tree graphs

Hand-waving and interpretive dance: an introductory course on tensor networks

Low depth quantum circuits for Ising models

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