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.
Keywords
- quantum walks
- invariance
- symmetry
- Dirac equation
- gauge transform
1. Introduction
In their origins [1–5], quantum walks (QWs) were thought as the quantum-mechanical generalization of the standard random walk in one dimension: the mathematical model describing the motion of a particle which follows a path that consists of a succession of jumps with fixed length whose direction depends on the random outcome of flipping a coin. In the quantum version, the coin toss is replaced by the action of a unitary operator upon some intrinsic degree of freedom of the system, a quantum observable with only two possible eigenvalues: for example, the spin of an electron, the polarization of a photon, or the chirality of a molecule.
After this preliminary analysis, it became clear that the similitude between these two processes was mainly formal and that random and QWs displayed divergent properties [6]. The most remarkable of these discrepancies is perhaps the ability of unbiased QWs to spread over the line, not as the square root of the elapsed time, the fingerprint of any diffusion process, but with constant speed [7]. This higher rate of percolation enables the formulation of quantum algorithms [8, 9] that can tackle some problems in a more efficient way than their classical analogs: For instance, QWs are very promising resources for optimal searching [10–12]. Today, QWs have exceeded the boundaries of quantum computation and attracted the attention of researchers from other fields as, for example, information theory or game theory [13–16].
As a consequence of this wide interest, diverse extensions of the discrete-time QW on the line have been considered in the past. Most of these variations are related with the properties of the unitary coin operator [17], backbone of the novel features of the process. Thus, one can find in the literature QWs whose evolution depends on more than one coin [18–20], QWs that suffer from decoherence [21, 22], or QWs driven by inhomogeneous, site-dependent coins [23–28]. There are also precedents where the temporal variability of the QW is explicit: in the form of a recursive rule for the coin selection, as in the so-called Fibonacci QWs [29, 30], through a given function that determines the value of the coin parameters [31–33], or by means of an auxiliary random process that modifies properties of the coin [34].
The main goal in most of these seminal papers is to find out new and exciting features that the considered modifications introduce in the behavior of the system, like the emergence of quasiperiodic patterns or the induction of dynamic localization. Recent works [35–37], however, have also regarded the issue from the opposite point of view, by exploring the conditions under which the evolution of the system results unchanged. In particular, Montero [37] considers the case of a discrete-time QW on the line with a time-dependent coin, a unitary operator with changing phase factors.
These phase factors are three parameters that appear in the definition of the coin operator whose relevance has been sometimes ignored in the past: When these phases are static magnitudes, they are superfluous [38], but if they are dynamic quantities, they can substantially modify the evolution of the system. This fact does not close the door to the possibility that a set of well-tuned variable phase factors can keep the process unchanged from a probabilistic perspective. This defines a control mechanism that can compensate externally induced decoherence and introduces a nontrivial invariance to be added to other well-known symmetries of QWs [39–41].
In this Chapter, we will review the approach taken in [37] and consider a generalization of it. Now, the evolution of the discrete-time quantum walker on the line will be subjected to the introduction of a fully inhomogeneous coin operator: The properties of the unitary operator will depend both on the location and on the present time through the action of the aforementioned phase factors. This extra variability leads to additional constraints to be satisfied by these magnitudes if one wants to guarantee that the properties of the motion of the walker remain unaltered. Finally, we will connect our results with those appearing in the study of Di Molfetta et al. [36], where the authors considered how the inclusion of time- and site-dependent phase factors in the coin operator of a quantum walk on the line may induce some dynamics which, in the continuous limit, can be linked with the propagation of a Dirac spinor coupled to some external electromagnetic field. We will also explore the implications of this mapping here.
2. Fundamentals of QWs
We begin this Chapter with a survey of the fundamental concepts required in the designing of discrete QWs on the line. In its simplest version, the particle represented by the walker can occupy detached and numerable locations on a one-dimensional space. This space of positions may be just a topological space (a graph or a chain, for instance) or can be endowed with a metric. In such a case, it is usual to consider that the sites are separated by a fixed distance
There is another kind of QW, called continuous quantum walk, in which the walker can modify its position at any time: this is the quantum counterpart of continuous-time random walk. The evolution of processes belonging to this category is ruled by a Hamiltonian and the corresponding Schrödinger equation. In spite they are different, discrete, and continuous QWs share common traits [42].
Up to this point, there is no significant difference between random and quantum walks. The major distinction is found in the nature of the random event that determines the progress of the particle. While in a world governed by the laws of classical mechanics, randomness is the way in which we describe the uncertain effect of multiple (and usually uncontrollable) external agents acting upon a system, in the realms of quantum mechanics randomness is not an exogenous ingredient. This means that we can use some internal degree of freedom in the quantum system with two possible eigenvalues (the spin, the polarization, or the chirality) as a proxy for the coin and understand that any change in this inner property is the result of the act of tossing. Therefore, to represent the state of the walker, we need two different Hilbert spaces:
where we have introduced the wave-function components
Now, we have to consider the mechanism that connects these two properties, position and quirality, which eventually leads to a model for the dynamics of
In a second step, the shift operator With the present definition, the problem is spatially homogeneous and the system displays translational invariance. Therefore, alternative shift rules may be considered with equivalent results, as in the case of directed quantum walks [43, 44], where the particle can either remain still in the place or proceed in a fixed direction but never move backward.
Therefore, the state of the system at a later time
and the complete evolution of the system is determined once
Needless to say that the linearity and the translational invariance of the problem ensure that the solution for a general initial state can be recovered by direct superposition of the evolution of Eq. (7), Eqs. (14) to (17) later.
The similarities and dissimilarities between classical and QWs must be grounded on the analysis of the probability mass function (PMF) of the process,
where
On the basis of the values of the moduli of
or the value of
another interesting magnitude that can be connected with the local magnetization of the system if the internal degree of freedom has its origin in the spin of the particle [45].
2.1. General solution
The evolution operator
and
whose general solution [38] can be written in a compact way by using
and the nonzero components of the wave function at time
since
and
where
and
It is noted that in this picture the evolution of each component depends only on their own initial values. In fact, it can be shown [38] that
Even though the expression for
Equation (16) is recovered from the above relationship once one considers the initial condition
Observe how
This conclusion is illusory, however. It is well known [19] that
2.2. Stationary PMF
Figure 1 also shows us that the disparity in the bias is not the most striking aspect that distinguishes QWs from their classical analogues. These differences can be appreciated more easily when one considers the stationary limit [34]. It can be shown [38] that for
in the range
Regarding the expectation value of the position of the quantum walker,
its magnitude does not stem from the location of the largest maximum of
the expectation value of the position of the walker will increase linearly with time:
as it can be checked in Figure 2. The converse is not true [40, 41]: in order to get quantum walkers that show an exact symmetry in the parity one has to demand that
but also that Eq. (23) implies
equations that have only three main families of solutions [38], being the most relevant of them the one corresponding to
3. Inhomogeneous QWs
The fact that not only
Consider a general inhomogeneous, time-dependent unitary operator
where
In this case, the information supplied by the initial state of the system is not so important: Assume that
with
In practice, this means that we can modify
The recursive equations of the wave-function components under the present dynamics induced by
and
Since we have a specific interest in revealing a new kind of invariance, we will introduce
and
Therefore, our task is to find out nontrivial relationships connecting both set of parameters. Regarding this, note that
4. Invariance
The properties of the system enumerated up to this point are based on the moduli of the components of the wave function. This means, in particular, that if one has that
and
If we assume the validity of Eqs. (32) and (33) and replace these expressions in Eqs. (28) and (29), the conditions to recover Eqs. (30) and (31) are
These equations lead to the following prescription to modify the phases leaving invariant the moduli of the components of the wave function:
4.1. Invariance of global observables
The first conclusion that can be drawn from Eqs. (34)–(36) is that there is an infinite variety of choices for
This assumption simplifies enormously Eqs. (34)–(36):
One particular choice that satisfies the above requirements is
a possible solution of Eqs. (37) and (38). The above expressions lead to the following homogeneous update rule for
where
We illustrate in Figure 3 the invariance of
We can sketch a complementary picture that may help in the understanding the behavior of
onto the
where
4.2. Exact invariance
Obviously, we can go further and demand exact invariance in the problem. This can be achieved by setting
As is shown below, these equations can be expressed in terms of finite differences which in turn lead to partial derivatives. In fact, in the expression of
Equations (46) to (48) read, as we have anticipated,
This means, in particular, that we can transform an inhomogeneous coin into a time-dependent one
4.3. Continuous limit
Let us express Eqs. (34) to (36) in a slightly different way. Consider the discrete difference operators
and similarly for
Observe how the expression connecting
At this point, it is appropriate to note that we are not taking into account the issue of the parity of indexes
However, our interest in this Section is to analyze the continuous limit,
We need to relate
where
and
The exact invariance,
whose respective space-time components must change according to the formulas
Note that
and, when one introduces these relationships into Eqs. (67) and (68), one obtains the standard gauge transformations for the components of the potential
a transform that keeps invariant the electric field
If we reconsider the example introduced at the end of Section 4.2,
we can conclude that it corresponds to a case in which the electric field
In the most general case, when
which departs from the gauge invariance of potential
Clearly,
that results in the invariance of the electric field. A possible choice is to demand that both
Another alternative solution to Eq. (78) has appeared above, in Section 4.1. The equivalent expressions for Eqs. (37) and (38) in the continuous limit read:
what provides another solution to Eq. (78). Note that in this case Eqs. (65) and (66) show not merely covariance but perfect invariance in the mass-less case,
5. Conclusion
Along this Chapter, we have analyzed some interesting aspects of discrete-time QWs on the line, specifically those related with the emergence of invariance. In the first part, we have elaborated a succinct but comprehensive review covering the main features of the most elementary version of this process, when the unitary operator which assumes the function of the coin in the classical analog is kept fixed. We have described the dynamics that determines the evolution of the walker, supplied explicit formulas for assessing the precise state of the system at any time and approximate expressions that capture the main traits of the process in the stationary limit. These equations have been very useful to pinpoint the role played by the different parameters on the solution to the problem and put into context the generalization considered afterward.
The second part of the Chapter contemplates the situation in which the coin is time and site dependent. In particular, we have focused our interest on the phase parameters that define the unitary operator and determined the constraints that must be imposed in these changing phases if one wants to obtain invariance. This invariance can be demanded at two different levels: one can require that the invariance connects states belonging to the same ray of the Hilbert space or a milder condition, that the transformation modifies unevenly the two wave-function components. In this latter case, global properties (e.g., the probability that the particle is in a particular place or in a given spin state) remain unaltered but some other local quantum properties depending on the relative phase of these components can become modified.
The Chapter ends by analyzing the introduced invariance in the continuous limit. This approach unveils that the evolution of a time- and site-inhomogeneous quantum walk can be understood in terms of the dynamics of a particle coupled to an electromagnetic field and that the new symmetry shown by the walker can be interpreted as a manifestation of the well-known gauge invariance of electromagnetism.
Acknowledgments
The author acknowledges partial support from the Spanish Ministerio de Economa y Competitividad (MINECO) under Contract No. FIS2013-47532-C3-2-P and from Generalitat de Catalunya under Contract No. 2014SGR608.
References
- 1.
Aharonov Y, Davidovich L, Zagury N: Quantum random walks, Physical Review A. 1993; 48 :1687–1690. DOI: 10.1103/PhysRevA.48.1690 - 2.
Travaglione B C, Milburn G J: Implementing the quantum random walk, Physical Review A. 2002; 65 :032310. DOI: 10.1103/PhysRevA.65.032310 - 3.
Konno N: Quantum random walks in one dimension, Quantum Information Processing. 2003; 1 :345–354. DOI: 10.1023/A:1023413713008 - 4.
Kempe J: Quantum random walks: an introductory overview, Contemporary Physics. 2003; 44 :307–327. DOI: 10.1080/00107151031000110776 - 5.
Venegas-Andraca S E: Quantum walks: a comprehensive review, Quantum Information Processing. 2012; 11 :1015–1106. DOI: 10.1007/s11128-012-0432-5 - 6.
Childs A, Farhi E, Gutmann S: An example of the difference between quantum and classical random walks, Quantum Information Processing. 2003; 1 :35–43. DOI: 10.1023/A:1019609420309 - 7.
Ambainis A, Bach E, Nayak A, Vishwanath A, Watrous J. One-dimensional quantum walks. In: Proceedings of the thirty-third annual ACM symposium on Theory of computing (STOC ’01); 06–08 July 2001; Heraklion. New York: ACM; 2001. p. 37–49. DOI: 10.1145/380752.380757 - 8.
Shor P W: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing. 1997; 26 :1484–1509. DOI: 10.1137/S0097539795293172 - 9.
Farhi E, Gutmann S: Quantum computation and decision trees, Physical Review A. 1998; 58 :915–928. DOI: 10.1103/PhysRevA.58.915 - 10.
Shenvi N, Kempe J, Whaley K B: Quantum random-walk search algorithm, Physical Review A. 2003; 67 :052307. DOI: 10.1103/PhysRevA.67.052307 - 11.
Agliari E, Blumen A, Nülken O: Quantum-walk approach to searching on fractal structures, Physical Review A. 2010; 82 :012305. DOI: 10.1103/PhysRevA.82.012305 - 12.
Magniez F, Nayak A, Roland J, Santha M: Search via quantum walk, SIAM Journal on Computing. 2011; 40 :142–164. DOI: 10.1137/090745854 - 13.
Flitney A P, Abbott D, Johnson N F: Quantum walks with history dependence, Journal of Physics A. 2004; 37 :7581–7591. DOI: 10.1088/0305-4470/37/30/013 - 14.
Bulger D, Freckleton J, Twamley J: Position-dependent and cooperative quantum Parrondo walks, New Journal of Physics. 2008; 10 :093014. DOI: 10.1088/1367-2630/10/9/093014 - 15.
Chandrashekar C M, Banerjee S: Parrondo’s game using a discrete-time quantum walk, Physics Letters A. 2011; 375 :1553–1558. DOI: 10.1016/j.physleta.2011.02.071 - 16.
Romanelli A, Hernández G: Quantum walks: Decoherence and coin-flipping games, Physica A. 2011; 390 :1209–1220. DOI: 10.1016/j.physa.2010.12.006 - 17.
Chandrashekar C M, Srikanth R, Laflamme R: Optimizing the discrete time quantum walk using a SU(2) coin, Physical Review A. 2008; 77 :032326. DOI: 10.1103/PhysRevA.77.032326 - 18.
Brun T A, Carteret H A, Ambianis A: Quantum walks driven by many coins, Physical Review A. 2003; 67 :052317. DOI: 10.1103/PhysRevA.67.052317 - 19.
Tregenna B, Flanagan W, Maile R, Kendon V: Controlling discrete quantum walks: coins and initial states, New Journal of Physics. 2003; 5 :83. DOI: 10.1088/1367-2630/5/1/383 - 20.
Venegas-Andraca S E, Ball J L, Burnett K, Bose S: Quantum walks with entangled coins, New Journal of Physics. 2005; 7 :221. DOI: 10.1088/1367-2630/7/1/221 - 21.
Brun T A, Carteret H A, Ambianis A: Quantum random walks with decoherent coins, Phyical Review A. 2003; 67 :032304. DOI: 10.1103/PhysRevA.67.032304 - 22.
Kendon V, Tregenna B: Decoherence can be useful in quantum walks, Physical Review A. 2003; 67 :042315. DOI: 10.1103/PhysRevA.67.042315 - 23.
Wójcik A, Łuczak T, Kurzyn’ski P, Grudka A, Bednarska M: Quasiperiodic dynamics of a quantum walk on the line, Physical Review Letters. 2004; 93 :180601. DOI: 10.1103/PhysRevLett.93.180601 - 24.
Romanelli A, Auyuanet A, Siri R, Abal G, Donangelo R: Generalized quantum walk in momentum space, Physica A. 2005; 352 :409–418. DOI: 10.1016/j.physa.2005.01.026 - 25.
Shikano Y, Katsura H: Localization and fractality in inhomogeneous quantum walks with self-duality, Physical Review E. 2010; 82 :031122. DOI: 10.1103/PhysRevE.82.031122 - 26.
Konno N, Łuczak T, Segawa E: Limit measures of inhomogeneous discrete-time quantum walks in one dimension, Quantum Information Processing. 2013; 12 :33–53. DOI: 10.1007/s11128-011-0353-8 - 27.
Zhang R, Xue P, Twamley J: One-dimensional quantum walks with single-point phase defects, Physical Review A. 2014; 89 :042317. DOI: 10.1103/PhysRevA.89.042317 - 28.
Xue P, Qin H, Tang B, Sanders C: Observation of quasiperiodic dynamics in a one-dimensional quantum walk of single photons in space, New Journal of Physics. 2014; 16 :053009. DOI: 10.1088/1367-2630/16/5/053009 - 29.
Ribeiro P, Milman P, Mosseri R: Aperiodic quantum random walks, Physical Review Letters. 2004; 93 :190503. DOI: 10.1103/PhysRevLett.93.190503 - 30.
Romanelli A: The Fibonacci quantum walk and its classical trace map, Physica A. 2009; 388 :3985–3990. DOI: 10.1016/j.physa.2009.06.022 - 31.
Romanelli A: Driving quantum-walk spreading with the coin operator, Physical Review A. 2009; 80 :042332. DOI: 10.1103/PhysRevA.80.042332 - 32.
Romanelli A, Segundo G: The entanglement temperature of the generalized quantum walk, Physica A. 2014; 393 :646–654. DOI: 10.1016/j.physa.2013.08.050 - 33.
Bañuls M C, Navarrete C, Pérez A, Roldán E, Soriano J C: Quantum walk with a time-dependent coin, Physical Review A. 2006; 73 :062304. DOI: 10.1103/PhysRevA.73.062304 - 34.
Ahlbrecht A, Vogts H, Werner A H, Werner R F: Asymptotic evolution of quantum walks with random coin, Journal of Mathematical Physics. 2011; 52 :042201. DOI: 10.1063/1.3575568 - 35.
Di Molfetta G, Brachet M, Debbasch F: Quantum walks as massless Dirac fermions in curved space-time, Physical Review A. 2013; 88 :042301. DOI: 10.1103/PhysRevA.88.042301 - 36.
Di Molfetta G, Brachet M, Debbasch F: Quantum walks in artificial electric and gravitational fields, Physica A. 2014; 397 :157–168. DOI: 10.1016/j.physa.2013.11.036 - 37.
Montero M: Invariance in quantum walks with time-dependent coin operators, Physical Review A. 2014; 90 :062312. DOI: 10.1103/PhysRevA.90.062312 - 38.
Montero M: Quantum walk with a general coin: exact solution and asymptotic properties, Quantum Information Processing. 2015; 14 :839–866. DOI: 10.1007/s11128-014-0908-6 - 39.
Chandrashekar C M, Srikanth R, Banerjee S: Symmetries and noise in quantum walk, Physical Review A. 2007; 76 :022316. DOI: 10.1103/PhysRevA.76.022316 - 40.
Asbóth J K: Symmetries, topological phases, and bound states in the one-dimensional quantum walk, Physical Review B. 2012; 86 :195414. DOI: 10.1103/PhysRevB.86.195414 - 41.
Kitagawa T: Topological phenomena in quantum walks: Elementary introduction to the physics of topological phases, Quantum Information Processing. 2012; 11 :1107–1148. DOI: 10.1007/s11128-012-0425-4 - 42.
Konno N: Limit theorem for continuous-time quantum walk on the line, Physical Review E. 2005; 72 :026113. DOI: 10.1103/PhysRevE.72.026113 - 43.
Hoyer S, Meyer D A: Faster transport with a directed quantum walk, Physical Review A. 2009; 79 :024307. DOI: 10.1103/PhysRevA.79.024307 - 44.
Montero M: Unidirectional quantum walks: evolution and exit times, Physical Review A. 2014; 88 :012333. DOI: 10.1103/PhysRevA.88.012333 - 45.
Souza A M C, Andrade R F S: Coin state properties in quantum walks, Scientific Reports. 2013; 3 :1976. DOI: 10.1038/srep01976
Notes
- There is another kind of QW, called continuous quantum walk, in which the walker can modify its position at any time: this is the quantum counterpart of continuous-time random walk. The evolution of processes belonging to this category is ruled by a Hamiltonian and the corresponding Schrödinger equation. In spite they are different, discrete, and continuous QWs share common traits [42].
- With the present definition, the problem is spatially homogeneous and the system displays translational invariance. Therefore, alternative shift rules may be considered with equivalent results, as in the case of directed quantum walks [43, 44], where the particle can either remain still in the place or proceed in a fixed direction but never move backward.
- Eq. (23) implies | ψ + ( + 1 , 1 ) | 2 = | ψ − ( − 1 , 1 ) | 2 = 1 / 2 , see Eqs. (26) and (27) below. In other words, this is the condition that ensures the absence of bias in the “initial velocities.”