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arXiv:1606.06156 · InTech eBooks

Invariance in Quantum Walks

Miquel Montero

Published Aug 24, 2016• 1 citation
arXivHTMLPDF

Abstract

In this Chapter, we present some interesting properties of quantum walks on the line.We concentrate our attention in the emergence of invariance and provide some insights into the ultimate origin of the observed behavior.In the first part of the Chapter, we review the building blocks of the quantum-mechanical version of the standard random walk in one dimension.The most distinctive difference between random and quantum walks is the replacement of the random coin in the former by the action of a unitary operator upon some internal property of the later.We provide explicit expressions for the solution to the problem when the most general form for the homogeneous unitary operator is considered, and we analyze several key features of the system as the presence of symmetries or stationary limits.After that, we analyze the consequences of letting the properties of the coin operator change from site to site, and from time step to time step.In spite of this lack of homogeneity, the probabilistic properties of the motion of the walker can remain unaltered if the coin variability is chosen adequately.Finally, we show how this invariance can be connected to the gauge freedom of electromagnetism.

DOI
10.5772/62872

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In this Chapter, we present some interesting properties of quantum walks on the line.We concentrate our attention in the emergence of invariance and provide some insights into the ultimate origin of the observed behavior.In the first part of the Chapter, we review the building blocks of the quantum-mechanical version of the standard random walk in one dimension.The most distinctive difference between random and quantum walks is the replacement of the random coin in the former by the action of a unitary operator upon some internal property of the later.We provide explicit expressions for the solution to the problem when the most general form for the homogeneous unitary operator is considered, and we analyze several key features of the system as the presence of symmetries or stationary limits.After that, we analyze the consequences of letting the properties of the coin operator change from site to site, and from time step to time step.In spite of this lack of homogeneity, the probabilistic properties of the motion of the walker can remain unaltered if the coin variability is chosen adequately.Finally, we show how this invariance can be connected to the gauge freedom of electromagnetism.
This paper unveils some features of a discrete-time quantum walk on the line whose coin depends on the temporal variable. After considering the most general form of the unitary coin operator, it focuses on the role played by the two phase factors that one can incorporate there, and shows how both terms influence the evolution of the system. A closer analysis reveals that the probabilistic properties of the motion of the walker remain unaltered when the update rule of these phases is chosen adequately. This invariance is based on a symmetry with consequences not yet fully explored.
In this paper, a study on discrete-time coined quantum walks on the line is presented. Clear mathematical foundations are still lacking for this quantum walk model. As a step toward this objective, the following question is being addressed: Given a graph, what is the probability that a quantum walk arrives at a given vertex after some number of steps? This is a very natural question, and for random walks it can be answered by several different combinatorial arguments. For quantum walks this is a highly non-trivial task. Furthermore, this was only achieved before for one specific coin operator (Hadamard operator) for walks on the line. Even considering only walks on lines, generalizing these computations to a general SU(2) coin operator is a complex task. The main contribution is a closed-form formula for the amplitudes of the state of the walk (which includes the question above) for a general symmetric SU(2) operator for walks on the line. To this end, a coin operator with parameters that alters the phase of the state of the walk is defined. Then, closed-form solutions are computed by means of Fourier analysis and asymptotic approximation methods. We also present some basic properties of the walk which can be deducted using weak convergence theorems for quantum walks. In particular, the support of the induced probability distribution of the walk is calculated. Then, it is shown how changing the parameters in the coin operator affects the resulting probability distribution.
Quantum walks on graphs are ubiquitous in quantum computing finding a myriad of applications. Likewise, random walks on graphs are a fundamental building block for a large number of algorithms with diverse applications. While the relationship between quantum and random walks has been recently discussed in specific scenarios, this work establishes a formal equivalence between the two processes on arbitrary finite graphs and general conditions for shift and coin operators. It requires empowering random walks with time heterogeneity, where the transition probability of the walker is non-uniform and time dependent. The equivalence is obtained by equating the probability of measuring the quantum walk on a given node of the graph and the probability that the random walk is at that same node, for all nodes and time steps. The first result establishes procedure for a stochastic matrix sequence to induce a random walk that yields the exact same vertex probability distribution sequence of any given quantum walk, including the scenario with multiple interfering walkers. The second result establishes a similar procedure in the opposite direction. Given any random walk, a time-dependent quantum walk with the exact same vertex probability distribution is constructed. Interestingly, the matrices constructed by the first procedure allows for a different simulation approach for quantum walks where node samples respect neighbor locality and convergence is guaranteed by the law of large numbers, enabling efficient (polynomial-time) sampling of quantum graph trajectories. Furthermore, the complexity of constructing this sequence of matrices is discussed in the general case.
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.
The quantum random walk has been much studied recently, largely due to its highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum walk on the line: the presence of decoherence in the quantum ``coin'' which drives the walk. We find exact analytical expressions for the time dependence of the first two moments of position, and show that in the long-time limit the variance grows linearly with time, unlike the unitary walk. We compare this to the results of direct numerical simulation, and see how the form of the position distribution changes from the unitary to the usual classical result as we increase the strength of the decoherence.
Based on studies of four specific networks, we conjecture a general relation between the walk dimensions dw{d}_{w}dw​ of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that dw{d}_{w}dw​ of the quantum walk takes on exactly half the value found for the classical random walk on the same geometry. Since walks on homogeneous lattices satisfy this relation trivially, our results for heterogeneous networks suggest that such a relation holds irrespective of whether translational invariance is maintained or not. To develop our results, we extend the renormalization-group analysis (RG) of the stochastic master equation to one with a unitary propagator. As in the classical case, the solution $\ensuremath{\rho}(x,t)$ in space and time of this quantum-walk equation exhibits a scaling collapse for a variable xdw/t{x}^{{d}_{w}}/txdw​/t in the weak limit, which defines dw{d}_{w}dw​ and illuminates fundamental aspects of the walk dynamics, e.g., its mean-square displacement. We confirm the collapse for $\ensuremath{\rho}(x,t)$ in each case with extensive numerical simulation. The exact values for dw{d}_{w}dw​ themselves demonstrate that RG is a powerful complementary approach to study the asymptotics of quantum walks that weak-limit theorems have not been able to access, such as for systems lacking translational symmetries beyond simple trees.