Dynamics of discrete-time quantum walk with time-correlated unitary noise
Y. F. Peng, X. X. Yi
备注：
The matlab code for Fig. 2 is wrong, and therefore, there is no doubt that the conclusion of the article needs to be revised. So, Please allow me to withdraw the current article
We investigate the dynamics of discrete-time quantum walk subject to time correlated noise. Noise is described as an unitary coin-type operator before each step, and attention is focused on the noise generated by a Gaussian Ornstein Uhlenbeck process, going beyond the usual telegraph noise, where the random variables are consist of only -1 and 1. Under the first-order approximation of BCH formula, the master equation of noisy discrete-time quantum walk is derived. The dynamics given by the master equation are in good agreement with those given by numerical simulations within a certain period of steps, which is controlled by noise parameters. Two remarker behaviors of long time noisy dynamics are observed in numerical simulations, corresponding to two opposite noise regimes: in slow noise regime, with the increase of the noise amplitude, the quantum coherence is suppressed, and the dynamics of noisy discrete-time quantum walk gradually transits to that of classical random walk. In fast noise regime, the walker is confined into few lattice sites, and the width of wave packet is much narrower compared with that in slow noise regime.
We present an idea to convert to a unitary quantum walk any open quantum walk which is defined on lattices as well as on finite graphs. This approach generalizes to the domain of open quantum walks (or quantum Markov chains) the framework introduced by Szegedy for quantizing Markov chains. For the unitary quantum walks formulated in this article, we define the probability and the mean probability of finding the walk at a node, then derive the asymptotic mean probability.
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The discrete-time quantum walk (QW) has been extensively and intensively investigated for the last decade, whose coin operator is defined by a unitary matrix. We extend the QW to a walk determined by a unitary matrix whose component is quaternion. We call this model quaternionic quantum walk (QQW) and present some properties. This paper is the first step for the study on QQWs.
Recent work on recurrence in quantum walks has provided a representation of Schur functions in terms of unitary operators. We propose a generalization of Schur functions by extending this operator representation to arbitrary operators on Banach spaces. Such generalized Schur functions meet the formal structure of first return generating functions, thus we call them FR-functions. We derive general properties of FR-functions, among them a generalization of the renewal equation already known for random and quantum walks, as well as splitting properties which extend useful factorizations of Schur …
The variational quantum eigensolver has been proposed as a low-depth quantum circuit that can be employed to examine strongly correlated systems on today's noisy intermediate-scale quantum computers. We examine details associated with the factorized form of the unitary coupled-cluster variant of this algorithm. We apply it to a simple strongly correlated condensed-matter system with nontrivial behavior---the four-site Hubbard model at half filling. This work show some of the subtle issues one needs to take into account when applying this algorithm in practice, especially to condensed-matter sy…
The two major discrete time formulations for quantum walks, coined and scattering, are unitarily equivalent for arbitrary position dependent transition amplitudes and any topology (PRA {\bf 80}, 052301 (2009)). Although the proof explicit describes the mapping obtention, its high technicality may hinder relevant physical aspects involved in the equivalence. Discussing concrete examples -- the most general constructions for the line, square and honeycomb lattices -- here we unveil the similarities and differences of these two versions of quantum walks. We moreover show how to derive the dynamic…
Quantum walk research has mainly focused on evolutions due to repeated applications of time-independent unitary coin operators. However, the idea of controlling the single particle evolution using time-dependent unitary coins has still been a subject of multiple studies as it not only hosts interesting possibilities for quantum information processing but also opens a much richer array of phenomena including static and dynamic localizations. So far, such studies have been performed only for single quantum walkers. In case of multi-walker systems, time-dependent coins may generate measurable phe…
We introduce a scheme for efficiently describing pure states of strongly correlated fermions in higher dimensions using unitary circuits featuring a causal cone. A local way of computing local expectation values is presented. We formulate a dynamical reordering scheme, corresponding to time-adaptive Jordan-Wigner transformation, that avoids nonlocal string operators. Primitives of such a reordering scheme are highlighted. Fermionic unitary circuits can be contracted with the same complexity as in the spin case. The scheme gives rise to a variational description of fermionic models not sufferin…
Topological phases and the associated multiple edge states are studied for parity and time-reversal $(\mathcal{PT})$ symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining $\mathcal{PT}$ symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the $\mathbb{Z}$ topological phase and numerically confirm that multiple edge states appear as expected from the bulk-edge correspondence. Therefore, the bulk-edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge s…
The mixing process of discrete-time quantum walks on one-dimensional lattices is revisited in a setting where the walker is coupled to an environment, and the time evolution of the walker and the environment is unitary. The mixing process is found to be incomplete, in the sense that the walker does not approach the maximally mixed state indefinitely, but the distance to the maximally mixed state saturates to some finite value depending on the size of the environment. The quantum speedup of mixing time is investigated numerically as the size of the environment decreases from infinity to a finit…
Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. To be analogous to the hitting time for a classical walk, the quantum hitting time must involve repeated measurements as well as unitary evolution. We derive an expression for hitting time using superoperators, and numerically evaluate it for the discrete walk on the hypercube. The values found are compared to other analogues of hitting time suggested in earlier work. The dependence of hitting times on the type of unitary ``coin'' is examined, and we give an example of an initia…
Quantum walks represent paradigmatic quantum evolutions, enabling powerful applications in the context of topological physics and quantum computation. They have been implemented in diverse photonic architectures, but the realization of a two-particle dynamics on a multi-dimensional lattice has hitherto been limited to continuous-time evolutions. To fully exploit the computational capabilities of quantum interference it is crucial to develop platforms handling multiple photons that propagate across multi-dimensional lattices. Here, we report a discrete-time quantum walk of two correlated photon…
The realism-based nonlocality (RBN) is a recently introduced measure that differs from the well-known Bell's nonlocality. For bipartite states, the RBN concerns how much an element of reality associated with a given observable is affected upon local measurements on a subsystem. Here, we present an analytical proof for the unitary invariance of the RBN and that it presents a monotonous behavior upon the action of unital and non-unital local quantum noise. We illustrate our results by employing the two-qubits Werner state and thermal quantum correlated states. We show how the RBN is limited by t…
We study the large-time behaviour of a sample $\mathcal{S}$ consisting of an ensemble of fermionic walkers on a graph interacting with a structured infinite reservoir of fermions $\mathcal{E}$ through an exchange of particles in preferred states. We describe the asymptotic state of $\mathcal{S}$ in terms the initial state of $\mathcal{E}$, with especially simple formulae in the limit of small coupling strength. We also study the particle fluxes into the different parts of the reservoir.
A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in $(1+1)$ curved spacetime. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of a test particle in curved spacetime, in spite of the fixed background lattice.…
We review some results on the equivalence of quantum states under local unitary transformations (LUT). In particular, the classification of two-qubit Schmidt correlated (SC) states under LUT is investigated. By presenting the standard form of quantum states under LUT, the sufficient and necessary conditions of whether two different SC states are local unitary equivalent are provided. The correlations of SC states are also discussed.
We investigated discrete-time quantum walks with an arbitary unitary coin. Here we discover that the average position $<x> =\max(<x> \sin(α+γ)$, while the initial state is $1/\sqrt{2}(\mid0L>+i\mid0R>)$. We prove the result and get some symmetry properties of quantum walks with a U(2) coin with $\mid0L>$ and $\mid0R>$ as the initial state.
We propose an optical cavity implementation of the two-dimensional coined quantum walk on the line. The implementation makes use of only classical resources, and is tunable in the sense that a large number of different unitary transformations can be implemented by tuning some parameters of the device.
