Finite automata for caching in matrix product algorithms

Crosswhite, Gregory M.; Bacon, Dave

doi:10.1103/physreva.78.012356

Cited by 113 publications

(139 citation statements)

References 21 publications

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“…The numerical method used here is well-documented and compared to other methods in wide use (such as the density matrix renormalization group method) in the papers by Verstreate et al 68 and by Crosswhite and Bacon 83 . We will therefore not enter a discussion of the full numerical details but only sketch a few of the essential ideas.…”

Section: Appendix A: Numerical Detailsmentioning

confidence: 99%

“…The long-range Hamiltonian is represented by a single matrix product operator (MPO) in our calculations. A simple way of constructing the matrices of the MPO is to utilize the connection with finite automata or hidden Markov models 68,83 , where the matrices represents the action of a finite automaton or the transition rules of a hidden Markov model where the emitted symbols are exactly the strings of operators whose tensor product occurs in the Hamiltonian Eq. (3).…”

Section: Appendix A: Numerical Detailsmentioning

confidence: 99%

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Dipoles on a two-leg ladder

Gammelmark¹,

Zinner²

2013

Phys. Rev. B

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We study polar molecules with long-range dipole-dipole interactions confined to move on a two-leg ladder for different orientations of the molecular dipole moments with respect to the ladder. Matrix product states are employed to calculate the many-body ground state of the system as function of lattice filling fractions, perpendicular hopping between the legs, and dipole interaction strength. We show that the system exhibits zig-zag ordering when the dipolar interactions are predominantly repulsive. As a function of dipole moment orientation with respect to the ladder, we find that there is a critical angle at which ordering disappears. This angle is slightly larger than the angle at which the dipoles are non-interacting along a single leg. This behavior should be observable using current experimental techniques.

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Section: Appendix A: Numerical Detailsmentioning

confidence: 99%

Section: Appendix A: Numerical Detailsmentioning

confidence: 99%

Dipoles on a two-leg ladder

Gammelmark¹,

Zinner²

2013

Phys. Rev. B

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“…That is, a measurable statement such as "the d 2 × d 2 partial determinant is equal to the identity" is equivalent to the statement "the data can be modeled by uncorrelated states and measurements over a ddimensional Hilbert space." Similar forms of analysis have come up in the context of matrix product states [11][12][13], a way of representing various kinds of many-body quantum states that is particularly elegant for calculating correlation functions. Similar analyses can also be found in the more abstract context of (generalized) Baysian networks [14][15][16] where the presence of hidden variables with in a model or causal structure result in a rich set of testable constraints on the probabilities associated with the observed variables.…”

Section: Figmentioning

confidence: 99%

Nonholonomic tomography. II. Detecting correlations in multiqudit systems

Jackson¹,

Enk²

2017

Phys. Rev. A

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In the context of quantum tomography, quantities called partial determinants [1] were recently introduced. PDs (partial determinants) are explicit functions of the collected data which are sensitive to the presence of state-preparation-and-measurement (SPAM) correlations. In this paper, we demonstrate further applications of the PD and its generalizations. In particular we construct methods for detecting various types of SPAM correlation in multiqudit systems -e.g. measurementmeasurement correlations. The relationship between the PDs of each method and the correlations they are sensitive to is topological. We give a complete classification scheme for all such methods but focus on the explicit details of only the most scalable methods, for which the number of settings scales as O(d 4 ). This paper is the second of a two part series where the first paper[2] is about a theoretical perspective for the PD, particularly its interpretation as a holonomy.

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“…To see how operators of matrix-product form [9][10][11] arise naturally within such a prescription, we express each ancilla-qubit unitary in terms of an orthonormal basis…”

Section: Optimal Implementation Of Nonlocal Unitaries Within the mentioning

confidence: 99%

“…In other words, we demonstrate it is always possible to satisfy both conditions of the sequentiality and unitarity of the ancilla-qubit operations at the same time, however, at the price of ending up with a sequentially implemented unitary whose action is not perfectly equivalent to the original global unitary but is rather closest to that in some sense. This will be realized by exploiting the tools from matrix-product operator (MPO) theory [9][10][11]. We also study the role of initial correlations between the ancilla and qubits upon sequential preparation of unitaries and the way they could ever affect the evolution of the joint ancilla-qubit system.…”

Section: Introductionmentioning

confidence: 99%

Ancilla-assisted sequential approximation of nonlocal unitary operations

Saberi

2011

Phys. Rev. A

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We consider the recently proposed "no-go" theorem of Lamata et al. [Phys. Rev. Lett. 101, 180506 (2008)] on the impossibility of sequential implementation of global unitary operations with the aid of an itinerant ancillary system and view the claim within the language of Kraus representation. By virtue of an extremely useful tool for analyzing entanglement properties of quantum operations, namely, operator-Schmidt decomposition, we provide alternative proof to the no-go theorem and also study the role of initial correlations between the qubits and ancilla in sequential preparation of unitary entanglers. Despite the negative response from the no-go theorem, we demonstrate explicitly how the matrix-product operator (MPO) formalism provides a flexible structure to develop protocols for sequential implementation of such entanglers with an optimal fidelity. The proposed numerical technique, which we call variational matrix-product operator (VMPO), offers a computationally efficient tool for characterizing the "globalness" and entangling capabilities of nonlocal unitary operations.

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Finite automata for caching in matrix product algorithms

Cited by 113 publications

References 21 publications

Dipoles on a two-leg ladder

Dipoles on a two-leg ladder

Nonholonomic tomography. II. Detecting correlations in multiqudit systems

Ancilla-assisted sequential approximation of nonlocal unitary operations

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