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Recurrence in the classical random walk is well known and described by the
Polya number. For quantum walks, recurrence is similarly understood in terms of
the probability of a localized quantum walker to return to its origin. Under
certain circumstances the quantum walker may also return to an arbitrary
initial quantum state in a ?nite number of st...
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Citations
... The periods of M-type Hadamard walks are already known by Dukes [4] and Konno et al. [10] The result is as follows: In this paper, we point out that the same approach as for M type is effective for F type and give the proof below. Now, let T H,F N be the period of F-type Hadamard walk on C N (N ≥ 2). ...
... x 3 + 3x 2 + 3x + 1. F type: (2,4,4)) , ...
... x 3 + 3x 2 + 3x + 1. F type: (2,4,4)) , ...
The quantum walk is a quantum counterpart of the classical random walk. On the other hand, absolute zeta functions can be considered as zeta functions over F1. This study presents a connection between quantum walks and absolute zeta functions. In this paper, we focus on Hadamard walks and 3-state Grover walks on cycle graphs. The Hadamard walks and the Grover walks are typical models of the quantum walks. We consider the periods and zeta functions of such quantum walks. Moreover, we derive the explicit forms of the absolute zeta functions of corresponding zeta functions. Also, it is shown that our zeta functions of quantum walks are absolute automorphic forms.
... Another property that separates quantum walks to their classical counterpart is the fact that the walker in quantum random walk may return to the initial position periodically unlike classical random walks where the walker returns to its starting point at irregular and unpredictable times [17,45]. Such a phenomenon in quantum walks is defined as periodicity. ...
In this paper, we study discrete-time quantum walks on Cayley graphs corresponding to Dihedral groups, which are graphs with both directed and undirected edges. We consider the walks with coins that are (real) linear combinations of permutation matrices of order three. We show that the walks are periodic only for coins that are permutation or negative of a permutation matrix. Finally, we investigate the localization property of the walks through numerical simulations and observe that the walks localize for a wide range of coins for different sizes of the graphs.
... The Hadamard DTQW for this initial state has the following properties: (i) The dynamics is periodic of period T dyn = 8 (T dyn = 24) on the 4-cycle (8-cycle) [65]. (ii) Maximally entangled single-particle states-entanglement between position and coin-are generated after one step of the walk and then recurrently with period T MESPS = 4 (T MESPS = 12) on the 4-cycle (8-cycle) [59]. ...
... In this regard, we tested the proposed quantum circuit on actual quantum hardware, ibm_cairo, considering a Hadamard DTQW on the 4-and 8-cycles. Both are characterized by periodic dynamics [65] and by recurrent generation of maximally entangled singleparticle states [59]. We claim two main results. ...
Quantum walks have proven to be a universal model for quantum computation and to provide speed-up in certain quantum algorithms. The discrete-time quantum walk (DTQW) model, among others, is one of the most suitable candidates for circuit implementation due to its discrete nature. Current implementations, however, are usually characterized by quantum circuits of large size and depth, which leads to a higher computational cost and severely limits the number of time steps that can be reliably implemented on current quantum computers. In this work, we propose an efficient and scalable quantum circuit implementing the DTQW on the 2n-cycle based on the diagonalization of the conditional shift operator. For t time steps of the DTQW, the proposed circuit requires only O(n2+nt) two-qubit gates compared to the O(n2t) of the current most efficient implementation based on quantum Fourier transforms. We test the proposed circuit on an IBM quantum device for a Hadamard DTQW on the 4-cycle and 8-cycle characterized by periodic dynamics and by recurrent generation of maximally entangled single-particle states. Experimental results are meaningful well beyond the regime of few time steps, paving the way for reliable implementation and use on quantum computers.
... Equation (2) implies that the walker will move a step clockwise when the coin is in |0⟩c and a step counterclockwise when the coin is in |1⟩c, as shown in Fig. 1. Due to quantum superposition, the walker can move clockwise and counterclockwise simultaneously, resulting in some novel phenomena such as state revivals 57 in the QW on circles. The unitary operator for each step is defined as ...
Quantum walks are the quantum counterpart of classical random walks and have various applications in quantum information science. Polar molecules have rich internal energy structure and long coherence time and thus are considered as a promising candidate for quantum information processing. In this paper, we propose a theoretical scheme for implementing discrete-time quantum walks on a circle with dipole–dipole coupled SrO molecules. The states of the walker and the coin are encoded in the pendular states of polar molecules induced by an external electric field. We design the optimal microwave pulses for implementing quantum walks on a four-node circle and a three-node circle by multi-target optimal control theory. To reduce the accumulation of decoherence and improve the fidelity, we successfully realize a step of quantum walk with only one optimal pulse. Moreover, we also encode the walker into a three-level molecular qutrit and a four-level molecular ququart and design the corresponding optimal pulses for quantum walks, which can reduce the number of molecules used. It is found that all the quantum walks on a circle in our scheme can be achieved via optimal control fields with high fidelities. Our results could shed some light on the implementation of discrete-time quantum walks and high-dimensional quantum information processing with polar molecules.
... The replication phenomena in Dirac equation in fact may have applications in the physics of graphene [22]. There are theoretical works about revival phenomena in conformal field theories [23,24], in tight binding models [25], Loschmidt echoes [26][27][28][29][30] or in quantum walks on cycles [31]. In addition, this effect was studied for non linear generalizations of the Schrödinger equation which describes the dynamics of vortex filaments in [32,33]. ...
The Schrödinger equation on a circle with an initially localized profile of the wave function is known to give rise to revivals or replications, where the probability density of the particle is partially reproduced at rational times. As a consequence of the convolutional form of the general solution it is deduced that a piecewise constant initial wave function remains piecewise constant at rational times as well. For a sphere instead, it is known that this piecewise revival does not necessarily occur, indeed the wave function becomes singular at some specific locations at rational times. It may be desirable to study the same problem, but with an initial condition being a localized Dirac delta instead of a piecewise constant function, and this is the purpose of the present work. By use of certain summation formulas for the Legendre polynomials together with properties of Gaussian sums, it is found that revivals on the sphere occur at rational times for some specific locations, and the structure of singularities of the resulting wave function is characterized in detail. In addition, a partial study of the regions where the density vanishes, named before valley of shadows in the context of the circle, is initiated here. It is suggested that, differently from the circle case, these regions are not lines but instead some specific set of points along the sphere. A conjecture about the precise form of this set is stated and the intuition behind it is clarified.
... Under certain conditions, some coin operators may give rise to periodic quantum walks. Here, some necessary conditions for specific quantum walks with periodicity [26][27][28][29][30] have been discussed. In particular, for the lazy quantum walk, Konno and Kajiwara 29 studied the necessary condition that the coin operator to have a finite period. ...
Quantum hash function is an important area of interest in the field of quantum cryptography. Quantum hash function based on controlled alternate quantum walk is a mainstream branch of quantum hash functions by virtue of high efficiency and flexibility. In recent development of this kind of schemes, evolution operators determined by an input message depend on not only coin operators, but also direction-determine transforms, which usually are hard to extend. Moreover, the existing works ignore the fact that improper choice of initial parameters may cause some periodic quantum walks, and further collisions. In this paper, we propose a new quantum hash function scheme based on controlled alternate lively quantum walks with variable hash size and provide the selection criteria for coin operators. Specifically, each bit of an input message determines the magnitude of an additional long-range hop for the lively quantum walks. Statistical analysis results show excellent performance in the aspect of collision resistance, message sensitivity, diffusion and confusion property, and uniform distribution property. Our study demonstrates that a fixed coin operator, along with different shift operators, can effectively work on the design of a quantum hash function based on controlled alternate quantum walks, and shed new light on this field of quantum cryptography.
... The replication phenomena in Dirac equation in fact may have applications in the physics of graphene [20]. There are theoretical works about revival phenomena in conformal field theories [21], [22], in tight binding models [23], Loschmidt echoes [24] or in quantum walks on cycles [25]. In addition, this effect was studied for non linear generalizations of the Schrödinger equation which describes the dynamics of vortex filaments in [26], [27]. ...
... From here, the formula for R becomes obvious, recalling the definition of F . If 2π|n|/q < θ < π the latter big parenthesis is positive and then F (e(n/q), θ)/R = e(−n/2q) giving the first case in (25). On the other hand, if 0 ≤ θ < 2π|n|/q, it is negative and we recover the positivity introducing a factor e(±1/2). ...
... Then there is no singularity when G(a, a + k; q) = 0 and we get (23). If G(a, a + k; q) = 0, applying (11), (24) and (25) to (26), we see that (22) follows if we prove ...
The Schr\"odinger equation on a circle with an initially localized profile of the wave function is known to give rise to revivals or replications, where the probability density of the particle is partially reproduced at rational times. As a consequence of the convolutional form of the general solution it is deduced that a piecewise constant initial wave function remains piecewise constant at rational times as well. For a sphere instead, it is known that this piecewise revival does not necessarily occur, indeed the wave function becomes singular at some specific locations at rational times. It may be desirable to study the same problem, but with an initial condition being a localized Dirac delta instead of a piecewise constant function, and this is the purpose of the present work. By use of certain summation formulas for the Legendre polynomials together with properties of Gaussian sums, it is found that revivals on the sphere occur at rational times for some specific locations, and the structure of singularities of the resulting wave function is characterized in detail. In addition, a partial study of the regions where the density vanishes, named before valley of shadows in the context of the circle, is initiated here. It is suggested that, differently from the circle case, these regions are not lines but instead some specific set of points along the sphere. A conjecture about the precise form of this set is stated and the intuition behind it is clarified.
... While the position space in one-dimensional discretetime QW (DTQW) is infinite dimensional, we can as well define k-cycle DTQW with k-dimensional position space. For k-cycle DTQW, complete state revival -the walker returning to the initial position once after every t r stepswith particular choice of coin parameters has been shown for different k values such as 2, 3, 4, 5, 6, 8, and 10 [30][31][32][33]. This revival can be attributed to quantum recurrence theorem [34], which states that any closed quantum system with discrete energy eigenvalues, when it evolves in time, it will repeat itself as accurately as possible after a specific finite time. ...
... where 1 p is identity operator in the position space and 0 ≤ ≤ 1 [32]. The shift operator is defined aŝ ...
... For example, in a k-cycle DTQW with k assuming 4, 5 and 6, the position space probability distribution recurs completely after t r = 20, 60, and 28 steps, provided = (3 − √ 5)/8, (5 − √ 5)/10, and 2[1 − cos(π/7)]/3, respectively. [32] This is shown in Fig. 1, where the probability of finding the walker at the initial position (x = 1), denoted by P 1 (t), is plotted against the walk steps. Evidently, the walker returns to the initial position after every 60 steps. ...
The ability of quantum walks to evolve in a superposition of distinct quantum states has been used as a resource in quantum communication protocols. Under certain settings, the $k$-cycle discrete-time quantum walks\,(DTQW) are known to recur to its initial state after every $t_r$ steps. We first present a scheme to optically realize any $k$-cycle DTQW using $J$-plate, orbital angular momentum\,(OAM) sorters, optical switch, and optical delay line. This entangles the polarization and OAM degrees of freedom\,(DoF) of a single photon. Making use of this recurrence phenomena of $k$-cycle DTQW and the entanglement generated during the evolution, we present a new quantum direct communication protocol. The recurrence and entanglement in $k$-cycle walk are effectively used to retrieve and secure the information, respectively, in the proposed protocol. We investigate the security of the protocol against intercept and resend attack. We also quantify the effect of amplitude damping and depolarizing noises on recurrence and mutual information between polarization and OAM DoF of a single photon.
... Thus, one can consider full-state revivals of a QW, which means that both the coin system and the walker return to a given joint state. In fact, several studies have been conducted on state revivals in QWs [33][34][35][36][37]. In particular, the conditions for a quantum walker on a cyclic path to exhibit state revival are presented in Ref. [33]. ...
... In fact, several studies have been conducted on state revivals in QWs [33][34][35][36][37]. In particular, the conditions for a quantum walker on a cyclic path to exhibit state revival are presented in Ref. [33]. Also, two periodic state revivals in a single-photon one-dimensional QW governed by a time-dependent coin-flip operator were experimentally observed [34]. ...
The study of recurrences and revivals in quantum systems has attracted a great deal of interest because of its importance in the control of quantum systems and its potential use in developing new technologies. In this paper, we introduce a protocol to induce full-state revivals in a huge class of quantum walks on a d-dimensional lattice governed by a c-dimensional coin system. The protocol requires two repeated interventions in the coin degree of freedom. We also present a characterization of the walks that admit such a protocol. Moreover, we modify the quantity known as the Pólya number, typically used in the study of recurrences in classical random walks and quantum walks, to create a witness of the first revival of the walk.
... This is a concept of great interest [18,19]. Several studies have been conducted on state revivals in QWs [20][21][22][23][24]. In particular, the conditions for a quantum walker on a cyclic path to exhibit state revival are presented in Ref. [20]. ...
... Several studies have been conducted on state revivals in QWs [20][21][22][23][24]. In particular, the conditions for a quantum walker on a cyclic path to exhibit state revival are presented in Ref. [20]. Also, two periodic state revivals in a single-photon one-dimensional QW governed by a time-dependent coin-flip operator were experimentally observed [21]. ...
... Therefore, the only non-null elements of G are the ones in its secondary diagonal, i.e., g n,c−1−n . As a result, we can simplify the notation and write Eq. (20). ...
The study of recurrences and revivals in quantum systems has attracted a great deal of interest because of its importance in the control of quantum systems and its potential use in developing new technologies. In this work, we introduce a protocol to induce full-state revivals in a huge class of quantum walks on a $d$-dimensional lattice governed by a $c$-dimensional coin system. The protocol requires two repeated interventions in the coin degree of freedom. We also present a characterization of the walks that admits such a protocol. Moreover, we modify the quantity known as P\'olya number, typically used in the study of recurrences in classical random walks and quantum walks, to create a witness of the first revival of the walk.
