Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Information and Computation

Limit distributions for different forms of four-state quantum walks on a two-dimensional lattice

Takuya Machida, C. M. Chandrashekar, Norio Konno, Thomas Busch

Published Oct 1, 2015• 5 citations
Publisher

Abstract

Long-time limit distributions are key quantities for understanding the asymptotic dynamics of quantum walks, and they are known for most forms of one-dimensional quantum walks using two-state coin systems. For two-dimensional quantum walks using a four-state coin system, however, the only known limit distribution is for a walk using a parameterized Grover coin operation and analytical complexities have been a major obstacle for obtaining long-time limit distributions for other coins. In this work however, we present two new types of long-time limit distributions for walks using different forms of coin-flip operations in a four-state coin system. This opens the road towards understanding the dynamics and asymptotic behaviour for higher state coin system from a mathematical view point.

DOI
10.26421/qic15.13-14-10

Related papers

Limit distributions for different forms of four-state quantum walks on a two-dimensional lattice

MachidaTakuya, M. ChandrashekarC., KonnoNorio +1 more

Long-time limit distributions are key quantities for understanding the asymptotic dynamics of quantum walks, and they are known for most forms of one-dimensional quantum walks using two-state coin ...

Quantum Information and Computation•Oct 1, 2015•0 citations

Two-state quantum walk on two- and three-dimensional lattices

C. M. Chandrashekar

We present a new scheme for a discrete-time quantum walk on two- and three-dimensional lattices using a two-state particle. We use different Pauli basis as translational eigestates for different axis and show that the coin operation, which is necessary for one-dimensional walk is not a necessary requirement for two- and three- dimensional walk but can serve as an additional resource. Using this scheme, the probability distribution from Grover walk using four-state particle and other equivalent schemes on a square lattice using coin operation is reproduced. We also present the Hamiltonian form of evolution which can serve as a general framework to simulate, control, and study the dynamics in different physical systems.

arXiv:1103.2704•Mar 14, 2011•4 citations

A limit theorem for a 3-period time-dependent quantum walk

F. Alberto Grünbaum, Takuya Machida

We consider a discrete-time 2-state quantum walk on the line. The state of the quantum walker evolves according to a rule which is determined by a coin-flip operator and a position-shift operator. In this paper we take a 3-periodic time evolution as the rule. For such a quantum walk, we get a limit distribution which expresses the asymptotic behavior of the walker after a long time. The limit distribution is different from that of a time-independent quantum walk or a 2-period time-dependent quantum walk. We give some analytical results and then consider a number of variants of our model and indicate the result of simulations for these ones.

arXiv:1404.1509•Quantum Information and Computation•Jan 1, 2015•3 citations

Limit distributions of two-dimensional quantum walks

恭平 渡部, Naoki Kobayashi, Makoto Katori +1 more

One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t\ensuremath{\rightarrow}\ensuremath{\infty}$ of all joint moments of two components of walker's pseudovelocity, ${X}_{t}/t$ and ${Y}_{t}/t$, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer simulations is also shown.

arXiv:0802.2749•Physical Review A•Jun 19, 2008•91 citations

Complete classification of trapping coins for quantum walks on the two-dimensional square lattice

Bálint Kollár, András Gilyén, Iva Tkáčová +3 more

One of the unique features of discrete-time quantum walks is called trapping, meaning the inability of the quantum walker to completely escape from its initial position, although the system is translationally invariant. The effect is dependent on the dimension and the explicit form of the local coin. A four-state discrete-time quantum walk on a square lattice is defined by its unitary coin operator, acting on the four-dimensional coin Hilbert space. The well-known example of the Grover coin leads to a partial trapping, i.e., there exists some escaping initial state for which the probability of staying at the initial position vanishes. On the other hand, some other coins are known to exhibit strong trapping, where such an escaping state does not exist. We present a systematic study of coins leading to trapping, explicitly construct all such coins for discrete-time quantum walks on the two-dimensional square lattice, and classify them according to the structure of the operator and the manifestation of the trapping effect. We distinguish three types of trapping coins exhibiting distinct dynamical properties, as exemplified by the existence or nonexistence of the escaping state and the area covered by the spreading wave packet.

arXiv:2002.08070•Physical Review A•Jul 7, 2020•7 citations

A QUANTUM WALK WITH A DELOCALIZED INITIAL STATE: CONTRIBUTION FROM A COIN-FLIP OPERATOR

Takuya Machida

A unit evolution step of discrete-time quantum walks (QWs) is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the QWs starting with a localized initial state have been derived. In this paper, we compute limit distributions of a 2-state QW with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the (α, β) delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.

arXiv:1203.5396•International Journal of Quantum Information•Aug 1, 2013•7 citations
Long-time limit distributions are key quantities for understanding the asymptotic dynamics of quantum walks, and they are known for most forms of one-dimensional quantum walks using two-state coin systems. For two-dimensional quantum walks using a four-state coin system, however, the only known limit distribution is for a walk using a parameterized Grover coin operation and analytical complexities have been a major obstacle for obtaining long-time limit distributions for other coins. In this work however, we present two new types of long-time limit distributions for walks using different forms of coin-flip operations in a four-state coin system. This opens the road towards understanding the dynamics and asymptotic behaviour for higher state coin system from a mathematical view point.
We present a new scheme for a discrete-time quantum walk on two- and three-dimensional lattices using a two-state particle. We use different Pauli basis as translational eigestates for different axis and show that the coin operation, which is necessary for one-dimensional walk is not a necessary requirement for two- and three- dimensional walk but can serve as an additional resource. Using this scheme, the probability distribution from Grover walk using four-state particle and other equivalent schemes on a square lattice using coin operation is reproduced. We also present the Hamiltonian form of evolution which can serve as a general framework to simulate, control, and study the dynamics in different physical systems.
We consider a discrete-time 2-state quantum walk on the line. The state of the quantum walker evolves according to a rule which is determined by a coin-flip operator and a position-shift operator. In this paper we take a 3-periodic time evolution as the rule. For such a quantum walk, we get a limit distribution which expresses the asymptotic behavior of the walker after a long time. The limit distribution is different from that of a time-independent quantum walk or a 2-period time-dependent quantum walk. We give some analytical results and then consider a number of variants of our model and indicate the result of simulations for these ones.
One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t\ensuremath{\rightarrow}\ensuremath{\infty}$ of all joint moments of two components of walker's pseudovelocity, Xt/t{X}_{t}/tXt​/t and Yt/t{Y}_{t}/tYt​/t, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer simulations is also shown.
One of the unique features of discrete-time quantum walks is called trapping, meaning the inability of the quantum walker to completely escape from its initial position, although the system is translationally invariant. The effect is dependent on the dimension and the explicit form of the local coin. A four-state discrete-time quantum walk on a square lattice is defined by its unitary coin operator, acting on the four-dimensional coin Hilbert space. The well-known example of the Grover coin leads to a partial trapping, i.e., there exists some escaping initial state for which the probability of staying at the initial position vanishes. On the other hand, some other coins are known to exhibit strong trapping, where such an escaping state does not exist. We present a systematic study of coins leading to trapping, explicitly construct all such coins for discrete-time quantum walks on the two-dimensional square lattice, and classify them according to the structure of the operator and the manifestation of the trapping effect. We distinguish three types of trapping coins exhibiting distinct dynamical properties, as exemplified by the existence or nonexistence of the escaping state and the area covered by the spreading wave packet.
A unit evolution step of discrete-time quantum walks (QWs) is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the QWs starting with a localized initial state have been derived. In this paper, we compute limit distributions of a 2-state QW with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the (α, β) delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.