Problem of equilibration and the computation of correlation functions on a quantum computer

Terhal, Barbara M.; DiVincenzo, David P.

doi:10.1103/physreva.61.022301

Cited by 171 publications

(192 citation statements)

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“…Since 1 is a simple eigenvalue, Φ has an unique fixed point which is (by Lemma 3.1) a density matrix ρ ∞ ∈ M 1,+ d (C). Using the Jordan form of Φ, one can show the following result ( [25]). Proposition 4.1.…”

Section: Non-random Repeated Interactions and A New Model Of Random Dmentioning

confidence: 99%

Random repeated quantum interactions and random invariant states

Nechita

Pellegrini

2010

Probab. Theory Relat. Fields

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Abstract. We consider a generalized model of repeated quantum interactions, where a system H is interacting in a random way with a sequence of independent quantum systems Kn, n 1. Two types of randomness are studied in detail. One is provided by considering Haar-distributed unitaries to describe each interaction between H and Kn. The other involves random quantum states describing each copy Kn. In the limit of a large number of interactions, we present convergence results for the asymptotic state of H. This is achieved by studying spectral properties of (random) quantum channels which guarantee the existence of unique invariant states. Finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble. IntroductionInitially introduced in [2] as a discrete approximation of Langevin dynamics, the model of repeated quantum interactions has found since many applications (quantum trajectories, stochastic control, etc.). In this work we generalize this model by allowing random interactions at each time step. Our main focus is the long-time behavior of the reduced dynamics.Our viewpoint is that of Quantum Open Systems, where a "small" system is in interaction with an inaccessible environment (or an auxiliary system). We are interested in the reduced dynamics of the small system, which is described by the action of quantum channels. When repeating such interactions, under some mild conditions on the spectrum of the quantum channel, we show that the successive states of the small system converge to the invariant density matrix of the channel.These considerations motivated us to consider random invariant states, and we introduce a new probability measure on the set of density matrices. There exists extensive literature [8,28,19,3] on what is a "typical" density matrix. There are two general categories of such probability measures on M 1,+ d (C): measures that come from metrics with statistical significance and the so-called "induced measures", where density matrices are obtained as partial traces of larger, random pure states. Our construction from Section 4 falls into the second category, since our model involves an open system in interaction with a chain of "auxiliary" systems.Next, we introduce two models of random quantum channels. In the first model, we allow for the states of the auxiliary system to be random. In the second one, the unitary matrices acting on the coupled system are assumed random, distributed along the Haar invariant probability on the unitary group, and independent between different interactions. Since the (random) state of the system fluctuates, almost sure convergence does not hold, and we state results in the ergodic sense.2000 Mathematics Subject Classification. Primary 15A52; Secondary 94A40, 60F15.

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Section: Non-random Repeated Interactions and A New Model Of Random Dmentioning

confidence: 99%

Random repeated quantum interactions and random invariant states

Nechita

Pellegrini

2010

Probab. Theory Relat. Fields

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“…By general results on quantum channels the vast majority of CPT maps are known to be mixing (or relaxing) [16,17,18,19]. This means that for a generic choice of Φ, in the limit m → ∞ the transformation (3) will send all input states into a unique fix point identified as the unique eigen-operator of Φ associated with its largest eigenvalue.…”

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“…The speed of convergence, evaluated through the trace distance, is exponentially fast [18,19] inm and, a part from some constant prefactor, can be upper-bounded by the quantitym d 3 κm with κ < 1 being the modulus of the largest eigenvalue of Φ whose associated eigenvector contribute in the expansion of Eq. (6).…”

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“…The paper is organized as follows: after a brief review of the MERA, we show how the local expectation values and correlations functions can be casted in terms of concatenated quantum channels. Using general properties of mixing quantum channels [16,17,18,19] we then provide a way for expressing the thermodynamic limit of these quantities.The MERA tensor network:-Consider a many-body quantum system composed by N = 2 n sites of dimension d (qudits). Any pure state can be expressed as |ψ = T ℓ1,ℓ2,··· ,ℓN |ξ ℓ1 , ξ ℓ2 , · · · , ξ ℓN , where for j ∈ {1, · · · , N } and ℓ ∈ {1, · · · , d} the vectors |ξ ℓ j ∈ H d form the basis of the j-th d-dimensional system component.…”

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Quantum Multiscale Entanglement Renormalization Ansatz Channels

2008

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Tensor networks representations of many-body quantum systems can be described in terms of quantum channels. We focus on channels associated with the Multi-scale Entanglement Renormalization Ansatz (MERA) tensor network that has been recently introduced to efficiently describe critical systems. Our approach allows us to compute the MERA correspondent to the thermodynamic limit of a critical system introducing a transfer matrix formalism, and to relate the system critical exponents to the convergence rates of the associated channels.PACS numbers: 03.67.-a,05.30.-d,89.70.-a Understanding the properties of strongly interacting many-body quantum system is central in many areas of physics. Whenever it is hard to device reliable analytical approaches, as in many situations of experimental relevance, ingenious numerical methods are necessary to grasp the essential properties of these systems. Our ability of simulating them is based on the possibility to find an efficient description of their ground state. This is the case, for example, of White's Density Matrix Renormalization Group [1] which can be recasted in terms of Matrix Product States (MPS) [2,3,4,5,6]. Such representations are characterized by a simple tensor decomposition of the many-body wave-function which allows one i) to efficiently compute all the relevant observables of the system (e.g. energy, local observables, and correlation functions), and ii) to reduce the effective number of parameters over which the numerical optimization needs to be performed. MPS fulfill these requirements and can be used to describe faithfully the ground states of not critical, short range one-dimensional many-body Hamiltonians at zero-temperature. However MPS typically fail to provide an accurate description in other relevant situations, i.e. when the system is critical, in higher physical dimensions or if the model possesses longrange couplings. Several proposals have been put forward to overcome this problem. Projected Entangled Pair States (PEPS) [7] generalize MPS in dimensions higher than one. Weighted graph states [8] can deal with long-range correlations. In this Letter we focus on a solution recently proposed by Vidal [9] who introduced a tensor structure based on the so called Multiscale Entanglement Renormalization Ansatz (MERA). The MERA tensor network satisfies both the constraints i) and ii) and accommodates the scale invariance typical of critical systems [10,11]. The relevance of this approach might represent a major breakthrough in our simulation capabilities [12] and motivates an intensive study of the MERA [13,14].Here we point out a previously unnoticed connection between the MERA and the theory of completely positive quantum maps [15] establishing a link between two important areas of quantum information science. This allows us to introduce a transfer matrix formalism in the same spirit as it has been done for MPS [3,5], providing new tools to compute physical observables using MERA. The main outcomes of our work are i) a method for determining the prope...

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Introduction to Quantum Algorithms for Physics and Chemistry

Yung¹,

Whitfield²,

Boixo³

et al. 2014

Advances in Chemical Physics

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An enormous number of model chemistries are used in computational chemistry to solve or approximately solve the Schrödinger equation; each with their own drawbacks. One key limitation is that the hardware used in computational chemistry is based on classical physics, and is often not well suited for simulating models in quantum physics. In this review, we focus on applications of quantum computation to chemical physics problems. We describe the algorithms that have been proposed for the electronicstructure problem, the simulation of chemical dynamics, thermal state preparation, density functional theory and adiabatic quantum simulation.

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Problem of equilibration and the computation of correlation functions on a quantum computer

Cited by 171 publications

References 38 publications

Random repeated quantum interactions and random invariant states

Random repeated quantum interactions and random invariant states

Quantum Multiscale Entanglement Renormalization Ansatz Channels

Introduction to Quantum Algorithms for Physics and Chemistry

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