A Discrete Transmission Line Model for Discrete-time Quantum Walks

Fukushima, Yusuke; Wada, Tatsuaki

doi:10.4036/iis.2017.a.11

Cited by 3 publications

(4 citation statements)

References 12 publications

(9 reference statements)

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“…(66b) 20 The mathematical connection, via analytical continuation, between the standard, unitary, i.e., non-noisy DTQW, and the telegraph equation, is well-known [68][69][70]. In the present work, the connection is not merely mathematical, but physical, via the introduction of noise in the unitary dynamics.…”

Section: Flip Noise On Massless Dirac Fermions Yields the Telegraph E...mentioning

confidence: 88%

“…These two coin error channels 13 are purely and fully decohering, i.e. (this is our terminology), they make the coherences decrease, as time increases, monotonically and down to zero, respectively, and in any basis of the internal space; that the populations' difference go to zero is also independent 12 The mathematical connection, via analytical continuation, between the standard, unitary, i.e., non-noisy DTQW, and the telegraph equation, is well-known [66][67][68]. In the present work, the connection is not merely mathematical, but physical, via the introduction of noise in the unitary dynamics.…”

Section: System Of Equations In Position Space and Remarksmentioning

confidence: 99%

“…Regarding the role played by a spacetime-independent χ for the above Dirac Hamiltonian, equation(17), the reader may be interested in reference[57]. Regarding the role played by a spacetime-dependent χ in another class of continuum limits, the reader may be interested in reference[69]. Regarding the role played by the four spacetime-dependent angles in the original, spacetime-lattice model, equation (1), the reader may consult, for ξ 0 and ξ 1 , references[33,39,[47][48][49]58],-which show, among other results and in various, related settings, that these two angles correspond to lattice versions of the electromagnetic potential, having lattice U(1) gauge invariance -, and, for θ and χ, reference[58], which shows that these two angles encode the curvature of a discrete spacetime 14.…”

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confidence: 99%

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Quantum simulation of quantum relativistic diffusion via quantum walks

Arnault¹,

Macquet²,

Anglés-Castillo³

et al. 2020

J. Phys. A: Math. Theor.

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Two models are first presented, of one-dimensional discrete-time quantum walk (DTQW) with temporal noise on the internal degree of freedom (i.e., the coin): (i) a model with both a coin-flip and a phase-flip channel, and (ii) a model with random coin unitaries. It is then shown that both these models admit a common limit in the spacetime continuum, namely, a Lindblad equation with Dirac-fermion Hamiltonian part and, as Lindblad jumps, a chirality flip and a chirality-dependent phase flip, which are two of the three standard error channels for a two-level quantum system. This, as one may call it, Dirac Lindblad equation, provides a model of quantum relativistic spatial diffusion, which is evidenced both analytically and numerically. This model of spatial diffusion has the intriguing specificity of making sense only with original unitary models which are relativistic in the sense that they have chirality, on which the noise is introduced: The diffusion arises via the byconstruction (quantum) coupling of chirality to the position. For a particle with vanishing mass, the model of quantum relativistic diffusion introduced in the present work, reduces to the well-known telegraph equation, which yields propagation at short times, diffusion at long times, and exhibits no quantumness. Finally, the results are extended to temporal noises which depend smoothly on position.

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Section: Flip Noise On Massless Dirac Fermions Yields the Telegraph E...mentioning

confidence: 88%

Section: System Of Equations In Position Space and Remarksmentioning

confidence: 99%

mentioning

confidence: 99%

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Quantum simulation of quantum relativistic diffusion via quantum walks

Arnault¹,

Macquet²,

Anglés-Castillo³

et al. 2020

J. Phys. A: Math. Theor.

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show abstract

“…Through a simple example, we can observe our model behaves like an LC circuit, in the sense that, as the underlying non-reversible random walk is closer to the reversible one, the total energy of the internal graph in the stationary state increases to infinity [5]. See Section 5 for more detail.…”

Section: Introductionmentioning

confidence: 92%

Electric circuit induced by quantum walk

Higuchi,

Sabri,

Segawa

2020

Preprint

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We consider the Szegedy walk on graphs adding infinite length tails to a finite internal graph. We assume that on these tails, the dynamics is given by the free quantum walk. We set the ℓ ∞ -category initial state so that the internal graph receives time independent input from the tails, say α in , at every time step. We show that the response of the Szegedy walk to the input, which is the output, say β out , from the internal graph to the tails in the long time limit, is drastically changed depending on the reversibility of the underlying random walk. If the underlying random walk is reversible, we have β out = Sz(m δE )α in , where the unitary matrix Sz(m δE ) is the reflection matrix to the unit vector m δE which is determined by the boundary of the internal graph δE. Then the global dynamics so that the internal graph is regarded as one vertex recovers the local dynamics of the Szegedy walk in the long time limit. Moreover if the underlying random walk of the Szegedy walk is reversible, then we obtain that the stationary state is expressed by a linear combination of the reversible measure and the electric current on the electric circuit determined by the internal graph and the random walk's reversible measure. On the other hand, if the underlying random walk is not reversible, then the unitary matrix is just a phase flip; that is, β out = −α in , and the stationary state is similar to the current flow but satisfies a different type of the Kirchhoff laws.

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Development of a Discrete Mathematical Model for Medium-Length Transmission Lines

Al-Jufout

2020

2020 IEEE International Conference on Power and Energy (PECon)

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A Discrete Transmission Line Model for Discrete-time Quantum Walks

Cited by 3 publications

References 12 publications

Quantum simulation of quantum relativistic diffusion via quantum walks

Quantum simulation of quantum relativistic diffusion via quantum walks

Electric circuit induced by quantum walk

Development of a Discrete Mathematical Model for Medium-Length Transmission Lines

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