Discrete-time quantum walk on circular graph: Simulations and effect of gate depth and errors

2021 ◽  
pp. 2150008
Author(s):  
Iyed Ben Slimen ◽  
Amor Gueddana ◽  
Vasudevan Lakshminarayanan
Keyword(s):  
Discrete Time ◽  
Quantum Computer ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Quantum Circuits ◽  
Time Quantum ◽  
Universal Quantum ◽  
Circuit Decomposition ◽  
Circular Graph ◽  
The Impact

We investigate the counterparts of random walks in universal quantum computing and their implementation using standard quantum circuits. Quantum walks have been recently well investigated for traversing graphs with certain oracles. We focus our study on traversing a 1D graph, namely a circle, and show how to implement a discrete-time quantum walk in quantum circuits built with universal CNOT and single qubit gates. We review elementary quantum gates and circuit decomposition techniques and propose a generalized version of all CNOT-based circuits of the quantum walk. We simulated these circuits on five different qubits IBM-Q quantum devices. This quantum computer has nonperfect gates based on superconducting qubits, and, therefore, we analyzed the impact of the CNOT errors and CNOT-depth on the fidelity of the circuit.

scholarly journals A limit theorem for a splitting distribution of a quantum walk

2018 ◽  
Vol 16 (03) ◽  
pp. 1850023
Author(s):  
Takuya Machida
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Random Walks ◽  
Discrete Time ◽  
Limit Distribution ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Unitary Evolution ◽  
Time Quantum ◽  
Long Time

Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

The fog on: Generalized teleportation by means of discrete-time quantum walks on N-lines and N-cycles

2019 ◽  
Vol 33 (23) ◽  
pp. 1950270 ◽  
Author(s):  
Duc Manh Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Discrete Time ◽  
Quantum Teleportation ◽  
Quantum Walk ◽  
Quantum Walks ◽  
One Dimensional ◽  
Time Quantum ◽  
Infinite Line ◽  
The One

The recent paper entitled “Generalized teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles” by Yang et al. [Mod. Phys. Lett. B 33(6) (2019) 1950069] proposed the quantum teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles. However, further investigation shows that the quantum walk over the one-dimensional infinite line can be based over the [Formula: see text]-cycles and cannot be based on [Formula: see text]-lines. The proofs of our claims on quantum walks based on finite lines are also provided in detail.

scholarly journals Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 504
Author(s):  
Ce Wang ◽  
Caishi Wang
Keyword(s):  
Discrete Time ◽  
Quantum Information Processing ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Integer Lattice ◽  
Initial State ◽  
Higher Dimensional ◽  
Time Quantum ◽  
Gauss Type ◽  
Limit Probability

As a discrete-time quantum walk model on the one-dimensional integer lattice Z , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2 , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z d , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

scholarly journals Borromean states in discrete-time quantum walks

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 523
Author(s):  
Marcin Markiewicz ◽  
Marcin Karczewski ◽  
Pawel Kurzynski
Keyword(s):  
Discrete Time ◽  
Bound States ◽  
Quantum Walk ◽  
Bound State ◽  
Quantum Walks ◽  
Interacting Particles ◽  
Time Quantum ◽  
The Right ◽  
Efimov States ◽  
Peculiar Behavior

In the right conditions, removing one particle from a multipartite bound state can make it fall apart. This feature, known as the "Borromean property", has been recently demonstrated experimentally in Efimov states. One could expect that such peculiar behavior should be linked with the presence of strong inter-particle correlations. However, any exploration of this connection is hindered by the complexity of the physical systems exhibiting the Borromean property. To overcome this problem, we introduce a simple dynamical toy model based on a discrete-time quantum walk of many interacting particles. We show that the particles described by it need to exhibit the Greenberger-Horne-Zeillinger (GHZ) entanglement to form Borromean bound states. As this type of entanglement is very prone to particle losses, our work demonstrates an intuitive link between correlations and Borromean properties of the system. Moreover, we discuss our findings in the context of the formation of composite particles.

scholarly journals Unstructured search by random and quantum walk

2022 ◽  
Vol 22 (1&2) ◽  
pp. 53-85
Author(s):  
Thomas G. Wong
Keyword(s):  
Random Walks ◽  
Continuous Time ◽  
Quantum Computer ◽  
Success Probability ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Search Problem ◽  
Large N ◽  
Time Quantum ◽  
Classical Computer

The task of finding an entry in an unsorted list of $N$ elements famously takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$ queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large $N$, the random walks converge to the same evolution, both taking $N \ln(1/\epsilon)$ time to reach a success probability of $1-\epsilon$. In contrast, the discrete-time quantum walk asymptotically takes $\pi\sqrt{N}/2\sqrt{2}$ timesteps to reach a success probability of $1/2$, while the continuous-time quantum walk takes $\pi\sqrt{N}/2$ time to reach a success probability of $1$.

scholarly journals General methods and properties to evaluate continuum limits of the 1D discrete time quantum walk

2020 ◽  
Vol 19 (10) ◽  
Author(s):  
Michael Manighalam ◽  
Mark Kon
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Systems ◽  
Quantum Walks ◽  
Time Limit ◽  
Continuous Limit ◽  
Continuous Space ◽  
Time Quantum ◽  
Space Limit

Abstract Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Molfetta and Arrighi (A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk when the coin is allowed to transition through the continuous limit process.

scholarly journals Universal quantum computing using single-particle discrete-time quantum walk

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Shivani Singh ◽  
Prateek Chawla ◽  
Anupam Sarkar ◽  
C. M. Chandrashekar
Keyword(s):  
Hilbert Space ◽  
Discrete Time ◽  
Single Particle ◽  
Quantum Walk ◽  
Quantum Gates ◽  
Position Space ◽  
Single Qubit ◽  
Time Quantum ◽  
Universal Quantum ◽  
Qubit States

AbstractQuantum walk has been regarded as a primitive to universal quantum computation. In this paper, we demonstrate the realization of the universal set of quantum gates on two- and three-qubit systems by using the operations required to describe the single particle discrete-time quantum walk on a position space. The idea is to utilize the effective Hilbert space of the single qubit and the position space on which it evolves in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates and engineering arbitrary states is simpler in the proposed quantum walk model when compared to the circuit based model of computation. We will also discuss the scalability of the model and some propositions for using lesser number of qubits in realizing larger qubit systems.

scholarly journals Limit theorems for the discrete-time quantum walk on a graph with joined half lines

2012 ◽  
Vol 12 (3&4) ◽  
pp. 314-333
Author(s):  
Kota Chisaki ◽  
Norio Konno ◽  
Etsuo Segawa
Keyword(s):  
Asymptotic Behavior ◽  
Discrete Time ◽  
Limit Theorems ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Half Line ◽  
Initial State ◽  
Time Quantum ◽  
Weak Convergence Theorem ◽  
The One

We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.

Computational power of single qubit discrete-time quantum walk

2021 ◽  
Author(s):  
Shivani Singh ◽  
Prateek Chawla ◽  
Anupam Sarkar ◽  
C. M. Chandrashekar
Keyword(s):  
Discrete Time ◽  
Quantum Walk ◽  
Quantum Gates ◽  
Computational Power ◽  
Position Space ◽  
Single Qubit ◽  
Qubit System ◽  
Time Quantum ◽  
Universal Quantum ◽  
Qubit States

Abstract Quantum walk has been regarded as a primitive to universal quantum computation. By using the operations required to describe the single particle discrete-time quantum walk on a position space we demonstrate the realization of the universal set of gates on two-and three-qubit system. The idea is to reap the effective Hilbert space of the single qubit and the position space on which it evolves in superposition of position space in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates in the form of engineering arbitrary states is simpler in the proposed quantum walk model when compared to the circuit based model of computation. We will also discuss the scalability of the model and some propositions for using lesser number of qubits in realizing larger qubit systems.

Localization of quantum walks via the CGMV method

2011 ◽  
Vol 11 (5&6) ◽  
pp. 485-496
Author(s):  
Norio Konno ◽  
Etsuo Segawa
Keyword(s):  
Spectral Analysis ◽  
Discrete Time ◽  
Spectral Measure ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Laurent Polynomials ◽  
Cmv Matrix ◽  
Orthogonal Laurent Polynomials ◽  
Homogeneous Trees ◽  
Time Quantum

We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle, expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantum walk. As a corollary, we give another proof for localization of the quantum walk on homogeneous trees shown by Chisaki et al.

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