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

arXiv:1208.0424 · Physical Review A

Quantum random walk: Effect of quenching

Sanchari Goswami, Parongama Sen

Published Aug 13, 2012• 8 citations
arXivHTMLPDF

Abstract

We study the effect of quenching on a discrete quantum random walk by removing a detector placed at a position ${x}_{D}$ abruptly at time ${t}_{R}$ from its path. The results show that this may lead to an enhancement of the occurrence probability at ${x}_{D}$ provided the time of removal ${t}_{R}<{t}_{R}^{\mathrm{lim}}$, where ${t}_{R}^{\mathrm{lim}}$ scales as ${x}_{D}{}^{2}$. The ratio of the occurrence probabilities for a quenched walker (${t}_{R}\ensuremath{\ne}0$) and free walker (${t}_{R}=0$) shows that it scales as $1/{t}_{R}$ at large values of ${t}_{R}$ independent of ${x}_{D}$. On the other hand, if ${t}_{R}$ is fixed, this ratio varies as ${x}_{D}^{2}$ for small ${x}_{D}$. The results are compared to the classical case. We also calculate the correlations as functions of both time and position.

DOI
10.1103/physreva.86.022314

Related papers

Moving Detector Quantum Walk with Random Relocation

Md Aquib Molla, Sanchari Goswami

We study a discrete-time quantum walk in presence of a detector at $x_D$ initially. The detector here is repeatedly removed after a span of $t_R$, the removal time, and reinserted at random locations. Two relocation rules are considered here: In Model~1, the detector is reinserted at any site beyond $x_D$, while in Model~2, reinsertion is done within a restricted window around the position of the detector at that time. Both variants behave like Semi Infinite Walk (SIW) for large $t_R$, where the detector behaves effectively as a fixed boundary. However, in the rapid-relocation regime, i.e., when $t_R$ is small, the behaviours are different. Model~1 permits greater spreading due to unrestricted reinsertion, which is different from Model~2. The time evolution of occupation probability ratio of our walker to that of an infinite walker at $x_D$, i.e., $f(x_D,t)/f_\infty(x_D,t)$, initially show the feature of a SIW upto $t=t_R$, then show some oscillatory behaviour and finally reach a saturation value for both the models. The ratio enhancing under certain conditions of $x_D$ and $t_R$, is a purely quantum mechanical effect. The saturation ratio shows a crossover behavior below and above a removal time $t_R^*$. At sites $x \neq x_D$ the occupation probablity ratios at a certain time reveals that for small $t_R$, the behaviours of the two models are drastically different from each other, as well as from Semi Infinite Walk (SIW), Quenched Quantum Walk (QQW) and Moving Detector Quantum Walk (MDQW). The correlation ratios of the two models with that of Infinite Walk (IW) show interesting time dependence for sites to the left or right of the initial detector position $x_D$.

quant-pharXiv:2604.03593•Apr 7, 2026•0 citations

Persistence of unvisited sites in presence of a quantum random walker

Sanchari Goswami, Parongama Sen

A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at $x=0$ at time $t=0$, the persistence probability $P(x,t)$ that a site at $x$ has not been visited till time $t$ has been calculated. $P(x,t)$ behaves as $(t/|x|-1)^{-α}$ with $α\sim 0.3$ while the global fraction ${\cal{P}}(t) = \sum_xP(x,t)/2t$ of sites remaining unvisited at time $t$ attains a constant value. $F(x,t)$, the probability that the site at $x$ is visited for the first time at $t$ behaves as $(t/|x|-1)^{-β}/|x|$ where $β= 1+ α$ for $t/|x|>> 1$,and ${\cal{F}}(t) =\sum_xF(x,t)/2t \sim 1/t$. A few other properties related to the persistence and first passage times are studied and some fundamental differences between the classical and the quantum cases are observed.

arXiv:0908.1012•Aug 7, 2009•0 citations

One-dimensional quantum random walks with two entangled coins

Chaobin Liu, Nelson Petulante

We offer theoretical explanations for some recent observations in numerical simulations of quantum random walks (QRWs). Specifically, in the case of a QRW on the line with one particle (walker) and two entangled coins, we explain the phenomenon, called ``localization,'' whereby the probability distribution of the walker's position is seen to exhibit a persistent major ``spike'' (or ``peak'') at the initial position and two other minor spikes which drift to infinity in either direction. Another interesting finding in connection with QRWs of this sort pertains to the limiting behavior of the position probability distribution. It is seen that the probability of finding the walker at any given location eventually becomes stationary and nonvanishing. We explain these observations in terms of the degeneration of some eigenvalue of the time evolution operator $U(k)$. An explicit general formula is derived for the limiting probability, from which we deduce the limiting value of the height of the observed spike at the origin. We show that the limiting probability decreases quadratically for large values of the position $x$. We locate the two minor spikes and demonstrate that their positions are determined by the phases of nondegenerated eigenvalues of $U(k)$. Finally, for fixed time $t$ sufficiently large, we examine the dependence on $t$ of the probability of finding a particle at a given location $x$.

arXiv:0807.2263•Physical Review A•Mar 12, 2009•44 citations

Properties of distribution and entanglement in discrete-time quantum walk with percolation

An Zhiyun, Zhi-Jian Li

We investigate one-dimensional discrete-time quantum walk on the line where the links between neighboring sites are randomly broken. Two link-broken ways, static percolation and dynamical percolation, are considered. The former means that the broken links are fixed in position space at each time step, while the latter is that broken links are varied with time step. Our attention focuses on the effects of these disorders on two physical quantities, the probability distribution and the entanglement between the coin degree of freedom and position degree of freedom. Choosing Hadamard coin operator and assuming the walker to start from the position eigenstate|0〉and attach itself to a coherent coin state 1/√2 (|↑〉+ i|↓〉), we give the statistical average results after making numerical calculations many times. The choices of coin operator and initial state, resulting in a symmetric probability distribution about origin in the ideal case, is helpful in comparing with different cases in different disorder strengths. It is shown that the probability distribution of static percolation quantum walk can change from a coherent behavior at short time to Anderson localization at longer time, while the dynamical percolation quantum walk can change to a classical diffusive behavior. With the decrease of the percolation probability, these transitions become faster. The entanglement for ideal case without disorder reaches a constant value after a short time evolution. The static percolation makes the entanglement less than that of ideal case and fluctuate irregularly around a certain value. The situation is very different for the dynamical percolation:the entanglement increases smoothly with the time step and can exceed the constant value in the ideal case at some time. Both of entanglements for two types of percolations decrease with reducing percolation probability. As a striking characteristic, the entanglement in dynamical case can tend to maximum regardless of percolation probability in long time limit, while the static case cannot. In the model for our study, the randomized unitary operations, induced by the static and dynamical percolations, can lead to some noticeable effects on the transport and entanglement of discrete time quantum walk. The results about the interplay between disorder and entanglement not only assist quantum information processing, but also give more options to further explore and understand disorder physical processes in nature.

Acta Physica Sinica•Jan 1, 2017•0 citations

Randomizing Quantum Walk

Rashid Ahmad, Bibi, Safia, Uzma Sajjad

The evolution of a walker in standard "Discrete-time Quantum Walk (DTQW)" is determined by coin and shift unitary operators. The conditional shift operator shifts the position of the walker to right or left by unit step size while the direction of motion is specified by the coin operator. This scenario can be generalized by choosing the step size randomly at each step in some specific interval. For example, the value of the roll of a dice can be used to specify the step size after throwing the coin. Let us call such a quantum walk "Discrete-time Random Step Quantum Walk (DTRSQW)". A completely random probability distribution is obtained whenever the walker follows the DTRSQW. We have also analyzed two more types of quantum walks, the "Discrete-time Un-biased Quantum Walk (DTUBQW)" and the "Discrete-time Biased Quantum Walk (DTBQW)". In the first type, the step size is kept different than unit size but the same for left and right shifts, whereas in the second type left and right shifts can also be different. The probability distribution in DTUBQW is found to follow a certain rule. The standard deviation ($σ$) of DTRSQW is higher than DTQW and hence DTRSQW spreads faster. The $σ$ of DTUBQW shows sawtooth behavior with faster spread than DTQW for some specific values of rotation angles and steps.

arXiv:2003.00440•Mar 1, 2020•0 citations

Coined quantum walk on a quantum network

Jigyen Bhavsar, Shashank Shekhar, Siddhartha Santra

We explore a discrete-time, coined quantum walk on a quantum network where the coherent superposition of walker moves originates from the unitary interaction of the walker coin with the qubit degrees of freedom in the quantum network. The walk dynamics leads to a growth of entanglement between the walker and the network on one hand and, on the other, between the network qubits among themselves. The initial entanglement among the network qubits plays a crucial role in determining the asymptotic values of these entanglement measures and the quantum walk statistics. Specifically, the entanglement entropy of the walker-network state and the negativity of the quantum network-qubit state saturate to values increasing with the initial network entanglement. The asymptotic time-averaged walker-position probability distribution shows increasing localization around the initial walker position with higher initial network entanglement. A potential application of these results as a characterization tool for quantum network properties is suggested.

arXiv:2406.01558•Physical Review A•Nov 20, 2024•2 citations
We study the effect of quenching on a discrete quantum random walk by removing a detector placed at a position xD{x}_{D}xD​ abruptly at time tR{t}_{R}tR​ from its path. The results show that this may lead to an enhancement of the occurrence probability at xD{x}_{D}xD​ provided the time of removal tR<tRlim{t}_{R}<{t}_{R}^{\mathrm{lim}}tR​<tRlim​, where tRlim{t}_{R}^{\mathrm{lim}}tRlim​ scales as xD2{x}_{D}{}^{2}xD​2. The ratio of the occurrence probabilities for a quenched walker (tR\ensuremath≠0{t}_{R}\ensuremath{\ne}0tR​\ensuremath=0) and free walker (tR=0{t}_{R}=0tR​=0) shows that it scales as 1/tR1/{t}_{R}1/tR​ at large values of tR{t}_{R}tR​ independent of xD{x}_{D}xD​. On the other hand, if tR{t}_{R}tR​ is fixed, this ratio varies as xD2{x}_{D}^{2}xD2​ for small xD{x}_{D}xD​. The results are compared to the classical case. We also calculate the correlations as functions of both time and position.
We study a discrete-time quantum walk in presence of a detector at xDx_DxD​ initially. The detector here is repeatedly removed after a span of tRt_RtR​, the removal time, and reinserted at random locations. Two relocation rules are considered here: In Model~1, the detector is reinserted at any site beyond xDx_DxD​, while in Model~2, reinsertion is done within a restricted window around the position of the detector at that time. Both variants behave like Semi Infinite Walk (SIW) for large tRt_RtR​, where the detector behaves effectively as a fixed boundary. However, in the rapid-relocation regime, i.e., when tRt_RtR​ is small, the behaviours are different. Model~1 permits greater spreading due to unrestricted reinsertion, which is different from Model~2. The time evolution of occupation probability ratio of our walker to that of an infinite walker at xDx_DxD​, i.e., f(xD,t)/f∞(xD,t)f(x_D,t)/f_\infty(x_D,t)f(xD​,t)/f∞​(xD​,t), initially show the feature of a SIW upto t=tRt=t_Rt=tR​, then show some oscillatory behaviour and finally reach a saturation value for both the models. The ratio enhancing under certain conditions of xDx_DxD​ and tRt_RtR​, is a purely quantum mechanical effect. The saturation ratio shows a crossover behavior below and above a removal time tR∗t_R^*tR∗​. At sites x≠xDx \neq x_Dx=xD​ the occupation probablity ratios at a certain time reveals that for small tRt_RtR​, the behaviours of the two models are drastically different from each other, as well as from Semi Infinite Walk (SIW), Quenched Quantum Walk (QQW) and Moving Detector Quantum Walk (MDQW). The correlation ratios of the two models with that of Infinite Walk (IW) show interesting time dependence for sites to the left or right of the initial detector position xDx_DxD​.
A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at x=0x=0x=0 at time t=0t=0t=0, the persistence probability P(x,t)P(x,t)P(x,t) that a site at xxx has not been visited till time ttt has been calculated. P(x,t)P(x,t)P(x,t) behaves as (t/∣x∣−1)−α(t/|x|-1)^{-α}(t/∣x∣−1)−α with α∼0.3α\sim 0.3α∼0.3 while the global fraction P(t)=∑xP(x,t)/2t{\cal{P}}(t) = \sum_xP(x,t)/2tP(t)=∑x​P(x,t)/2t of sites remaining unvisited at time ttt attains a constant value. F(x,t)F(x,t)F(x,t), the probability that the site at xxx is visited for the first time at ttt behaves as (t/∣x∣−1)−β/∣x∣(t/|x|-1)^{-β}/|x|(t/∣x∣−1)−β/∣x∣ where β=1+αβ= 1+ αβ=1+α for t/∣x∣>>1t/|x|>> 1t/∣x∣>>1,and F(t)=∑xF(x,t)/2t∼1/t{\cal{F}}(t) =\sum_xF(x,t)/2t \sim 1/tF(t)=∑x​F(x,t)/2t∼1/t. A few other properties related to the persistence and first passage times are studied and some fundamental differences between the classical and the quantum cases are observed.
We offer theoretical explanations for some recent observations in numerical simulations of quantum random walks (QRWs). Specifically, in the case of a QRW on the line with one particle (walker) and two entangled coins, we explain the phenomenon, called ``localization,'' whereby the probability distribution of the walker's position is seen to exhibit a persistent major ``spike'' (or ``peak'') at the initial position and two other minor spikes which drift to infinity in either direction. Another interesting finding in connection with QRWs of this sort pertains to the limiting behavior of the position probability distribution. It is seen that the probability of finding the walker at any given location eventually becomes stationary and nonvanishing. We explain these observations in terms of the degeneration of some eigenvalue of the time evolution operator U(k)U(k)U(k). An explicit general formula is derived for the limiting probability, from which we deduce the limiting value of the height of the observed spike at the origin. We show that the limiting probability decreases quadratically for large values of the position xxx. We locate the two minor spikes and demonstrate that their positions are determined by the phases of nondegenerated eigenvalues of U(k)U(k)U(k). Finally, for fixed time ttt sufficiently large, we examine the dependence on ttt of the probability of finding a particle at a given location xxx.
We investigate one-dimensional discrete-time quantum walk on the line where the links between neighboring sites are randomly broken. Two link-broken ways, static percolation and dynamical percolation, are considered. The former means that the broken links are fixed in position space at each time step, while the latter is that broken links are varied with time step. Our attention focuses on the effects of these disorders on two physical quantities, the probability distribution and the entanglement between the coin degree of freedom and position degree of freedom. Choosing Hadamard coin operator and assuming the walker to start from the position eigenstate|0〉and attach itself to a coherent coin state 1/√2 (|↑〉+ i|↓〉), we give the statistical average results after making numerical calculations many times. The choices of coin operator and initial state, resulting in a symmetric probability distribution about origin in the ideal case, is helpful in comparing with different cases in different disorder strengths. It is shown that the probability distribution of static percolation quantum walk can change from a coherent behavior at short time to Anderson localization at longer time, while the dynamical percolation quantum walk can change to a classical diffusive behavior. With the decrease of the percolation probability, these transitions become faster. The entanglement for ideal case without disorder reaches a constant value after a short time evolution. The static percolation makes the entanglement less than that of ideal case and fluctuate irregularly around a certain value. The situation is very different for the dynamical percolation:the entanglement increases smoothly with the time step and can exceed the constant value in the ideal case at some time. Both of entanglements for two types of percolations decrease with reducing percolation probability. As a striking characteristic, the entanglement in dynamical case can tend to maximum regardless of percolation probability in long time limit, while the static case cannot. In the model for our study, the randomized unitary operations, induced by the static and dynamical percolations, can lead to some noticeable effects on the transport and entanglement of discrete time quantum walk. The results about the interplay between disorder and entanglement not only assist quantum information processing, but also give more options to further explore and understand disorder physical processes in nature.
The evolution of a walker in standard "Discrete-time Quantum Walk (DTQW)" is determined by coin and shift unitary operators. The conditional shift operator shifts the position of the walker to right or left by unit step size while the direction of motion is specified by the coin operator. This scenario can be generalized by choosing the step size randomly at each step in some specific interval. For example, the value of the roll of a dice can be used to specify the step size after throwing the coin. Let us call such a quantum walk "Discrete-time Random Step Quantum Walk (DTRSQW)". A completely random probability distribution is obtained whenever the walker follows the DTRSQW. We have also analyzed two more types of quantum walks, the "Discrete-time Un-biased Quantum Walk (DTUBQW)" and the "Discrete-time Biased Quantum Walk (DTBQW)". In the first type, the step size is kept different than unit size but the same for left and right shifts, whereas in the second type left and right shifts can also be different. The probability distribution in DTUBQW is found to follow a certain rule. The standard deviation (σσσ) of DTRSQW is higher than DTQW and hence DTRSQW spreads faster. The σσσ of DTUBQW shows sawtooth behavior with faster spread than DTQW for some specific values of rotation angles and steps.
We explore a discrete-time, coined quantum walk on a quantum network where the coherent superposition of walker moves originates from the unitary interaction of the walker coin with the qubit degrees of freedom in the quantum network. The walk dynamics leads to a growth of entanglement between the walker and the network on one hand and, on the other, between the network qubits among themselves. The initial entanglement among the network qubits plays a crucial role in determining the asymptotic values of these entanglement measures and the quantum walk statistics. Specifically, the entanglement entropy of the walker-network state and the negativity of the quantum network-qubit state saturate to values increasing with the initial network entanglement. The asymptotic time-averaged walker-position probability distribution shows increasing localization around the initial walker position with higher initial network entanglement. A potential application of these results as a characterization tool for quantum network properties is suggested.