One-dimensional discrete-time quantum walks with general coin
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One-dimensional discrete-time quantumwalks with general coin
Mahesh N. Jayakody, Chandrakala Meena and Priodyuti Pradhan
Abstract —Quantum walk (QW) is the quantum analog of the random walk. QW is an integral part of the development ofnumerous quantum algorithms. Hence, an in-depth understanding of QW helps us to grasp the quantum algorithms. Werevisit the one-dimensional discrete-time QW and discuss basic steps in detail by incorporating the most general coinoperator. We investigate the impact of each parameter of the general coin operator on the probability distribution of thequantum walker. We show that by tuning the parameters of the general coin, one can regulate the probability distribution ofthe walker. We provide an algorithm for the one-dimensional quantum walk driven by the general coin operator. The studyconducted on general coin operator also includes the popular coins – Hadamard, Grover, and Fourier coins.
Index Terms —Quantum walk, Hadamard coin, Grover coin, Fourier coin, Quantum entanglement. ✦ NTRODUCTION W E have entered the quantum technology erathat puts quantum mechanical principles intopractice [1]. Quantum mechanics is a fundamentaltheory in physics that describes the physical propertiesof nature at the scale of atoms and subatomic particles.Making use of the properties in the realm of atoms andsubatomic particles, quantum technology provides apromising enhancement in our day-to-day utilities [2].For instance, the atomic clock used in the satellitesis a direct application of quantum mechanics, whichsynchronizes the time with high precision over theglobe to provide a reliable GPS navigation systemto smartphone users [3]. In recent times, quantumtechnologies have started empowering fast computing,secure communication, accurate imaging in healthcare,and advancing radar technology in military services[4]. In this article, we explore a frequently used toolin quantum technologies, quantum walk (QW) [5], [6],which has diverse applications in the development ofalgorithms [7].QWs are quantum analogs of classical randomwalks [8], [9]. The very first conceptual proposal,which could be recognized as a QW, was published ina seminal work by R. P. Feynman related to quantummechanical computers [10]. Nonetheless, it is generally • Mahesh N. Jayakody is associated with Faculty of En-gineering, Bar-Ilan University, Ramat Gan, Israel. E-mail:[email protected] • Chandrakala Meena is associated with the Council of Scientific andIndustrial Research – National Chemical Laboratory (CSIR-NCL)Pune, Dr Homi Bhabha Rd, Ward No. 8, NCL Colony, Pashan,Pune, Maharashtra 411008. E-mail: [email protected] • Priodyuti Pradhan is associated with the Complex Network Dy-namics Lab, Mathematics Department, Bar-Ilan University, RamatGan, Israel. E-mail: [email protected] accepted that the first paper which explicitly definesthe QWs was published by Aharonov et al. in 1993[11]. Principally, QWs contribute to theoretical andpractical improvements in quantum algorithms [12],and in quantum computing [13]. For instance, QWimproves cryptoanalysis [14], complex network anal-ysis [15], secure data transfer in 5G internet of things(IoT) communications and in wireless networking withedge computing platforms [16], [17]. Algorithms basedon QWs provide exponential speedup for some or-acle problems compared to any classical algorithms[18]. Moreover, QW based algorithms give polynomialspeedups over classical algorithms for problems suchas element distinctness [19], and in triangle finding[20]. QWs are used to model transport in biologicalsystems [21], physical phenomena such as Andersonlocalization [22] and also in understanding the topo-logical phases [23]. QWs have also been utilized to de-velop a single particle graph isomorphism algorithmthat can successfully distinguish all pairs of graphs,including all strongly regular graphs with up to vertices [24].There are two broad classes of QWs known asdiscrete-time QWs (DTQW) and continuous-time QWs(CTQW), each of which has significant distinctions intheir mathematical formalism [7]. One of their signifi-cant differences is that DTQW is defined on a discrete-time domain, and in contrast, CTQW is defined on acontinuous-time domain. However, it has been shownthat DTQWs can be transformed into a CTQWs undercertain conditions [25]. Here, we focus on the DTQWson the integer line. Such QWs are referred to as one-dimensional discrete-time QWs or simply as QWs ona line. It has the simplest structure of a QW and helps OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, FEBRUARY 16, 2021 2 us to understand the higher dimensional walks ongraphs [26]. One-dimensional QWs have three majorcomponents – a) coin operator, b) shift operator, andc) initial coin state. Starting from an initial state andby successively applying the coin and the shift opera-tors, we get the probability distribution of a quantumwalker. It is a well-known fact that the final behaviorof the walker for the Hadamard coin depends on theinitial coin state [27]. In the Hadamard walk, we geta symmetric probability distribution for an unbiasedinitial coin state, and for a biased initial coin state,the final probability distribution becomes asymmetric.Even though studies of QWs on a line have beendone extensively, the coin operators used in thosestudies are mostly limited to the Hadamard, Grover,and Fourier coins, and an exploration of the generalcoin operator is still lacking from the engineeringperspective [27]. Our contribution can be summarizedas follows • We explore the probability distribution of theone-dimensional discrete-time QWs with thegeneral coin. • We show that by tuning the general coin’sparameters, we can make the final probabilitydistribution asymmetric even for an unbiasedinitial coin state. • Our analysis reveals that the rotation and asingle phase parameter of the general coin playan essential role in controlling the probabilitydistribution of the walker. • We show that our numerical results are in goodagreement with that of the analytical deriva-tions. • We provide an algorithm for the one-dimensional QWs on line. • We give some intuitions of the quantum entan-glement in the context of QWs. • We present steps of the QW as simple as possi-ble and illustrate the matrix notations.We fabricate the article as follows, in section 2, webegin our discussion with the classical random walkand subsequently discuss the QW from the quantummechanical point of view in order to understand howthe concepts of the classical random walk are imple-mented in the quantum regime. In section 3, we trans-form the abstract quantum mechanical notations intoconcrete matrix form for the implementation. Further,in section 4, we discuss the results & discussion andprovide an algorithm for the one-dimensional QWdriven by the most general coin. Finally, in section5, we conclude our results and discuss some openproblems for further research.
ATHEMATICAL F RAMEWORK OF QW S Let us consider the one-dimensional discrete-time clas-sical random walk (RW) in which a particle on theinteger line jumps either to the right or left dependingupon the outcome of a coin toss [9]. At a given time,the motion of the particle on the integer line consistsof two sequential processes, the first being the tossingof the coin and the second being the shifting of theparticle that depends upon the outcome of the coin.In other words, RW is a random process that capturesthe dynamical behavior of the particle. For the sakeof simplicity, we assume the particle is placed at theorigin. We can then move the particle on the integerline for a desired number of steps by using the cointoss and the shifting rule. Suppose we record theposition of the particle at each step. Then by usingthe recorded data, we can generate the probabilitydistribution of the walker and which converges to thefamous Bell curve [28].
It is possible to define an analog of RW in the quantumregime as well. For that, we need to define the quan-tum version of the RW by incorporating the rules ofquantum mechanics. We can identify two componentsof an RW – (a) the integer line on which the particleis moving and (b) the outcome of the coin toss. Ourtask is to define the quantum version of the RWby transforming these components into the quantumregime.According to the postulates of quantum mechanics,we can analyze the dynamical behavior of a physicalsystem by associating a complex Hilbert space to thesystem [29]. And we represent any state of the physicalsystem by an element of the complex Hilbert space. Forinstance, let us consider an isolated atom comprised ofa single proton and an electron having four distinctenergy states. This can be considered as a physicalsystem. Now, we attach a complex Hilbert space to oursystem. We represent each energy state by using thebasis elements of the Hilbert space. Since we have fourdistinct energy states, we need a basis set with fourelements. Thus, we associate a Hilbert space spannedby this basis set to our system to study the dynamicsin terms of quantum mechanics. Note that, for four-level energy states, we need four-dimensional Hilbertspace. However, an atom can have an infinite numberof energy states, and hence to represent such a system,we need infinite-dimensional Hilbert space. Further,consider another isolated physical system that consistsof a single particle having two possible spin states.For instance, an isolated electron can be considereda physical system with two possible spin states called’spin up’ and ’spin down’. Like in the previous case,
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Symbols Description Z integer numbers C complex numbers ⊗ tensor product |·i column vector or ket h·|·i inner product H c Hilbert space associated to coin space H x Hilbert space associated to position space H = H x ⊗ H c Hilbert space associated to walker state space C , C coin operator and associated matrix S , S shift operator and associated matrix | x i ∞ x = −∞ Position states α x ( t ) | H i + β x ( t ) | T i Coin states as superposition of two states | H i and | T i| ψ ( x, t ) i state of the particle at position x at time t . | ψ t i = P ∞ x = −∞ | ψ ( x , t ) i state of the particle at time t . P ( x , t ) probability of finding the particle at time t with position x . P ( t ) = P ∞ x = −∞ P ( x , t ) total probability | α | , | β | probability at t = 0 TABLE 1List of symbols and notations. to study the dynamics of this single particle system,we need to attach a Hilbert space that spans from abasis set with two elements. Next, consider a scenarioin which one needs to study the behavior of boththe systems, the isolated atom, and the isolated singleparticle, together as a whole system. Then one needs tocombine the Hilbert spaces attached to each system ina reasonable manner. The operation of tensor products( ⊗ ) of Hilbert spaces comes into the picture to resolvethis problem.In connection to the example of the isolated atomand the isolated single particle, let us discuss how todevelop the mathematical model for QWs in detail. Asmentioned earlier, at a given time, a particle moving inan RW has a specific position on the integer line and aspecific coin outcome. To define the quantum versionof it, we attach the position and the coin outcome ofthe quantum particle with two suitable Hilbert spaces.Like in the example of an atom and an electron, we canthink positions of the particle as the energy states ofthe atom and spin states of the electron as the outcomeof the coin toss. Since the set of integers is infinite,the quantum walker can occupy an infinite number ofposition states. Therefore, the Hilbert space attachedto the position must have an infinite dimension. Thatis, the basis set of the Hilbert space attached to theposition must be infinite. Let H x be the Hilbert spacespanned by the basis set {| x i} x ∈ Z . Further, the coinstate has only two possible outcomes (head and tail).Hence, we can attach the coin state to a Hilbert spacethat spans from a basis set containing two elements.Let H c be the Hilbert space spanned by the basis set {| H i , | T i} . Note that the coin has two states. A quan-tum system that has two states is termed as a two-levelsystem [30]. In the context of quantum information,any two-level system is considered as a qubit [30]. Hence, one can view the coin of the quantum walkeras a qubit.By attaching the position state and the coin state ofthe quantum particle to H x and H c , one can representthe wave function or state ( | ψ ( x , t ) i ∈ H = H x ⊗ H c )of the quantum particle at any position x and time t as | ψ ( x , t ) i = ( α x ( t ) | H i + β x ( t ) | T i ) ⊗ | x i (1)where the positions state is represented by | x i and thecoin state is represented by the superposition of twobasis states of | H i and | T i as [7] α x ( t ) | H i + β x ( t ) | T i (2)where α x ( t ) and β x ( t ) are some complex numbers.Then the probability of finding the quantum walker atposition x at time t can be calculated as P ( x , t ) = h ψ ( x , t ) | ψ ( x , t ) i = | α x ( t ) | + | β x ( t ) | (3)under the condition P ( x , t ) ≤ . Further, we can findthe total wave function or state of the QW at time t from Eq. (1) as | ψ t i = ∞ X x = −∞ | ψ ( x , t ) i = ∞ X x = −∞ ( α x ( t ) | H i + β x ( t ) | T i ) ⊗ | x i (4)Now, with the help of the total wave function, wecan calculate the total probability distribution of thequantum walker as follows P t = h ψ t | ψ t i = ∞ X x = −∞ | α x ( t ) | + | β x ( t ) | = 1 (5)Our interest lies in understanding the total probabilitydistribution in detail, which captures the dynamics ofthe quantum walker. From Eq. (4), we can see that OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, FEBRUARY 16, 2021 4 particle can occupy infinite number of places. How-ever, for real-world implementation, we transform theinfinite number of position states into a finite number.
In quantum mechanics, the time evolution of anyphysical process in a closed system is defined in termsof a unitary transformation that alters a given initialstate ( | ψ i i ) of the system to a final state ( | ψ f i ) ina reversible manner [31]. Thus, one needs to utilizeunitary operators in describing how the state of aquantum mechanical system changes with time. There-fore, when implementing a QW it is essential to definea unitary operators that mimic the sequential processesof coin toss and shifting. The unitary operator thatperform the coin toss operation is referred as thegeneral coin operator for an one-dimensional QW andit is defined as C = cos θ | H ih H | + e iφ sin θ | H ih T | + e iφ sin θ | T ih H | − e i ( φ + φ ) cos θ | T ih T | (6)where θ ∈ [0 , π ) and φ , φ ∈ [0 , π ) [32]. We refer θ as rotation and φ & φ as phase parameters of general coinoperator. For instance, when θ = 45 ◦ , φ = φ = 0 ◦ ,the coin becomes the famous Hadamard coin operator.Further, for θ = 90 ◦ , φ = φ = 0 ◦ , we obtain theGrover coin and when θ = 45 ◦ , φ = φ = 90 ◦ , we getthe Fourier coin [33]. Most of the previous researchhave focused only on these three coins and for theother values of θ , φ , and φ the walker probabilitydistribution remains elusive. In this article, we explic-itly explore the general coin operator for all possiblecombinations of rotation and phase parameters andinvestigate the total probability distribution. Now, letus define the unitary operator that perform the shiftoperation as S = | H ih H | ⊗ (cid:18) ∞ X x = −∞ | x + 1 ih x | (cid:19) + | T ih T | ⊗ (cid:18) ∞ X x = −∞ | x − ih x | (cid:19) (7)The mathematical operation ⊗ is termed as tensorproduct. In plain language, the meaning of tensorproducts is that the operations on each space areexecuted separately. Note that the operator S treats thecoin states and the position state separately. Therefore,in comparison with RW, the unitary operator C is usedfor the quantum coin toss, and the unitary operator S is used to shift the particle on the integer line.Hence, a single-step progression of the QW on a linecan be written as a sequential process in which thequantum coin is tossed at first, and then the walkeris moved conditionally upon the outcome of the coin.The unitary operator that corresponds to a single-step evolution of the QW on a line is given by U = S ( C ⊗ I ) (8)where I is identity operator. Let us see an exampleof how the evolution of the QW occurs under theoperation of U . Suppose at time t = 0 the positionof the particle is x , and the initial state of the coin is α x (0) | H i + β x (0) | T i (Eq. (2)). Then the state of theparticle is represented as a tensor product of coin andposition states and we denote it as | ψ i = | ψ ( x , i = (cid:18) α x (0) | H i + β x (0) | T i (cid:19) ⊗| x i (9)where | α x (0) | + | β x (0) | = 1 . If we consider | α x (0) | = | β x (0) | in the above expression, weget an unbiased initial coin state. In contrast, when | α x (0) | = | β x (0) | , we get a biased initial coin state.Our main results consider only the unbiased initialcoin state. However, for the sake of simplicity, inthis section we use the biased initial coin state where | β x (0) | = 0 . Hence, the initial state of the particle inEq. (9) can be rewritten as | ψ i = | H i ⊗ | x i (10)which says that the walker is residing at position x with the coin state of Head . Now, we apply U on theinitial state in Eq. (10) and we get (detail in SI Eq. S1) | ψ i = U| ψ i = cos θ | H i ⊗ | x + 1 i + e iφ sin θ | T i ⊗ | x − i (11)Initially the quantum walker was at | x i position. How-ever, after the first step quantum walker occupies twopositions | x − i and | x + 1 i where | cos θ | , and | e iφ sin θ | are the probabilities of finding the walkerat those positions respectively. Further, we apply U on | ψ i and get | ψ i as (detail in SI Eq. S2) | ψ i = U| ψ i = cos θ | H i ⊗ | x + 2 i + (cid:18) e i ( φ + φ ) sin θ | H i + e iφ sin θ cos θ | T i (cid:19) ⊗ | x i− e i ( φ +2 φ ) sin θ cos θ | T i ⊗ | x − i (12)One can observe that after the second step quan-tum walker occupies three position states | x − i , | x i , and | x + 2 i where | cos θ | , | e i ( φ + φ ) sin θ | + | e iφ sin θ cos θ | and | − e i ( φ +2 φ ) sin θ cos θ | are theprobabilities of finding the walker at those positionsrespectively. Fig. 1 illustrates the progression of QWfor the first two steps. In general, we represent thestate of the walker at time t as | ψ t i = U t | ψ i (13) OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, FEBRUARY 16, 2021 5
Fig. 1. Progression of QW for the first and second steps initiating from the state | H i ⊗ | x i . Here, | x − i , . . . | x + 2 i are the positionstates of the walker on the integer line and | H i is the initial coin state. We observe that by applying the coin and shift operator on theinitial coin state | H i at position state | x i , walker can reside two places simultaneously ( | x − i and | x +1 i ) with probabilities sin θ and cos θ respectively. After the second step, walker resides on | x − i , | x i and | x +2 i with probabilities sin θ cos θ , sin θ +sin θ cos θ and cos θ respectively. Note that in QWs, we always need to consider thestate of the coin before it is tossed. However, in theclassical random walks, we never pay attention to thismatter. In classical random walks, we never worryabout checking whether we keep
Head or T ail onthe top side before we toss the coin. What we do iswe keep any side of the coin (
Head or T ail ) on thetop side and toss the coin. Then, depending on theoutcome, we shift the particle. However, in QWs, acoin toss is done by applying the coin operator onthe current coin state of the particle. Hence, differentinitial coin states give different outcomes . This is one ofthe subtle differences between QWs and RWs.
ATRIX REPRESENTATIONS
Any two Hilbert spaces whose orthonormal bases havethe same cardinality are isomorphic [34]. This allowsus to map the position Hilbert space ( H x ) and thecoin Hilbert space ( H c ) into the corresponding matrixHilbert space. As discussed earlier, the position Hilbert space at-tached to the QW has an infinite dimension. Hence,when mapping the H x into the matrix Hilbert space,we need infinite-dimensional Hilbert space. However,employing an infinite-dimensional Hilbert space giverise to practical issues in the implementations. Weresolve this problem by fixing the number of positionstates to a finite value. Let N be an arbitrary positive integer. Hence, we represent the set of position states {| − N i , . . . , | i , . . . , | N i} on the integer line wherethe walker can reside. This can be done by mappinga set of position states to the standard basis set of n = (2 N + 1) dimensional Hilbert space. When as-signing the basis elements to position states, we followa certain order. As a rule of thumb, we set the state | − N i to the basis element of (1 0 · · · T andthen map rest of the position states accordingly. Forexample, if N = 2 , we have position states. Werepresent these position states using the standard basisof -dimensional Hilbert space (Fig. 2). Similarly, wemap the coin Hilbert space ( H c ) into its correspond-ing matrix Hilbert space. Since there are only twocoin states, we need a 2-dimensional Hilbert space torepresent H c . We map coin states | H i and | T i to thestandard bases (1 0) T and (0 1) T respectively. Now,we express the state of the walker in Eqs. (10-12) incolumn representation (SI section 2) as follows where Fig. 2. One-to-one mapping of the position states ( {| − i , | − i , | i , | i , | i} ) to the standard basis elements of five dimen-sional matrix Hilbert space. The first basis vector is attached tothe left most corner position and accordingly other basis vectorsare attached to the position states. OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, FEBRUARY 16, 2021 6 | x i = | i and we get | ψ i = , | ψ i = θ e iφ sin θ (14) | ψ i = e i ( φ + φ ) sin θ θ − e i ( φ +2 φ ) sin θ cos θ e iφ sin θ cos θ (15)By looking at the element of column vector represen-tation of the state vector in Eqs. (14-15), we can findthe position of the quantum particle and also the prob-ability of finding the particle at a certain position. Wecan identify that the first five elements of each columnvectors (first block) hold the coefficients correspondto | H i . The next five elements of each column vector(second block) correspond to | T i . Furthermore, byrepresenting the linear combination form of the basisfor the two blocks of each column vector separately,we can get the information of particle’s position andprobability of being at that position (Fig. 3). Now wecan also easily compare the elements of the columnvectors and the coefficients of the graphical represen-tation given in (Fig. 1). Fig. 3. Decomposing a state ( | ψ t i ) of the QW. First five elementsof | ψ t i contains the coefficients corresponding to the coin state( | H i ) and the next five elements contain the coefficients cor-responding to the coin state ( | T i ). By rewriting first block andsecond block separately as a linear combination of basis set givenin Fig. (2) one can identify the location where the coin coefficientsreside. Moreover, from the QW state vector, we can alsoobserve an interesting phenomenon, i.e., quantumentanglement, which is a unique type of correlationshared between coin and position states [35]. It isimportant to note that the shift operator (Eq. 7) cre-ates the correlation between the coin and positionstates. Consider the three QW states given in Eqs.(14-15). Although the walk is initialized in such away that the coin and position states are separable( | ψ i = | H i ⊗ | x i ), after the first and second steps,the coin and position states of the walk are no longerseparable but entangled (details in SI). Hence, QWsbecome a potential testing platform to understandquantum entanglement. In the above, we discuss about the column represen-tation of the state vector in terms of basis elementsof Hilbert spaces. Here, we use the chosen bases toexplicitly construct the matrix corresponding to coinand shift operators and denote as C and S . We consider | H i = (1 0) T and | T i = (0 1) T in Eq. (6), and get C = (cid:18) cos θ e iφ sin θe iφ sin θ − e i ( φ + φ ) cos θ (cid:19) (16)where θ ∈ [0 , π ) and φ , φ ∈ [0 , π ) . Hence, fordifferent values of θ , φ and φ , we get different coins.Next, we consider the shift operator S given in (7).It contains two components that belong to coin spaceand position space. The terms | H ih H | and | T ih T | areapplied on the coin state. The terms P x | x + 1 ih x | and P x | x − ih x | are applied on the position states.Let us write the component of S that applied on theposition states as M = P x | x + 1 ih x | . Now, takethe conjugate transpose of the operator M and weget M † = P x | x ih x + 1 | . By changing the variable x → x − we can write M † = P x | x − ih x | .Since, the position states are infinite, practically it isinconvenient to represent the operators M and M † asblock matrices. Hence, we fix the number of positionstates to {| − N i , . . . , | i , . . . , | N i} where N be a finitepositive integer. Then the standard block matrix of theoperator M can be defined as M = . . . . . . . . . ... ... ... . . . ... ... . . . (2 N +1) × (2 N +1) As all the elements of M are real numbers, M † = M T .Hence, using the matrix form of M and M † , one can OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, FEBRUARY 16, 2021 7 express shift operator in (7) as block matrix form S = (cid:18) (cid:19) ⊗ . . . . . . . . . ... ... ... . . . ... ... . . . + (cid:18) (cid:19) ⊗ . . . . . . . . . ... ... ... . . . ... ... . . . having (4 N + 2) × (4 N + 2) dimensions. From theabove explicit matrix formulation, we can also expressthe evolution of the quantum walk for a given initialstate as follows, | ψ t i = U t | ψ i (17)where U = S ( C ⊗ I ) is the transition matrix (detailsin SI section 4). Note that position states are denotedby {| − N i , . . . , | i , . . . , | N i} , coin state are denotedby {| H i , | T i} and the state of the system in differenttime is demoted by {| ψ i , | ψ i , . . . , | ψ t i} and H x ∈ C N +1 , H c ∈ C and H ∈ C N +2 , respectively. ESULTS AND D ISCUSSION
In this section, we explore the probability distributionof the QW by employing a systematic investigation.The motion of the quantum walker on a line dependsupon five parameters – two parameters ( α and β ) fromthe initial coin state (Eq. (2)) and three parametersfrom the general coin operator ( θ , φ , and φ ) (Eq. (6)).By changing these five parameters, we can regulatethe shape of the total probability distribution of thewalker (Eq. (5)). It is a known fact that by changingthe initial coin state parameters of QW driven by theHadamard coin, one can change the probability distri-bution of the walker (SI Fig. S1). However, we havevery little knowledge about how the total probabilitydistribution changes when we vary the parameters ofthe general coin operator while keeping a fixed initialcoin state. In this study, we fix the initial coin stateto an unbiased state and analyze the impact of thegeneral coin parameters on the probability distributionof the QW for a finite number of steps. Our aim is toidentify the impact of these parameters ( θ , φ , and φ )on the total probability distribution of the quantumwalker, which can help to regulate the dynamics. Here, we consider the general form of the initial stateof the quantum walker in Eq. (9) as | ψ i = ( α | H i + β | T i ) ⊗ | i (18) where α ∈ C , β ∈ C are the initial coefficient value atposition | i at time t = 0 . The meaning of this initialstate is that, in the beginning, we place the quantumwalker at | i position state with the probabilities | α | and | β | of having | H i and | T i states respectively. Bychoosing different values for α and β , we can define aninfinite number of initial states. However, in this studywe focus on the initial states of the form | ψ i = (cid:18) √ | H i − i √ | T i (cid:19) ⊗ | i (19)Here, the probability of having | H i and | T i in initialcoin state is equal ( | α | = 1 / and | β | = 1 / ) i.e., anunbiased initial state. Now, we explore the total prob-ability distribution for the general coin (Eq. (6)) withall possible combinations of θ , φ and φ . However, forthe simplicity first we vary θ and φ = φ = 0 ◦ remainfixed to zero. Starting from the initial state in Eq. (19) and we applycoin and shift operators (Eqs. (6) and (7)). As θ variesin general coin operator from ◦ to ◦ , we obtaindifferent probability distributions of the walker (Fig.4). For instance, when θ = 90 ◦ , ◦ the particle getslocalized to the initial position. Further, for the valuesof θ = 0 ◦ or ◦ , the distribution has two peakswith the same height, that is, the quantum walkergets localized to two extreme positions with an equalprobability. On the other hand, when θ = 45 ◦ and θ = 225 ◦ , the walker can reside with non-zero proba-bilities in between the two extreme points. Further, forthe pairs of θ and θ + π , we get the mirror image ofthe same distribution. In other words, for different θ values, we get different total probability distributionsof the walker.To understand the observations described above,we check the coin and the first few states of thewalker. Recall that all the graphs (Fig. 4) are plotted fordifferent θ values while keeping φ = 0 ◦ and φ = 0 ◦ .Hence, for θ = 90 ◦ , ◦ and φ = φ = 0 ◦ the coins(Eq. (16)) take the form C = (cid:18) (cid:19) , C = − (cid:18) (cid:19) (20)One can observe that the coins in (20) differs onlyby a factor of minus sign. This minus sign act as anoverall phase factor. In probability calculation suchoverall phase factor has no contribution. Thus, we get OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, FEBRUARY 16, 2021 8 -200 -100 0 100 200 x i P i - - x i . . P i -200 -100 0 100 200 x i P i -200 -100 0 100 200 x i P i - - x i .
04 0 . P i -200 -100 0 100 200 x i P i - - x i .
04 0 . P i - -
100 0 100 200 x i . . P i Fig. 4. Probability distributions of QW for different θ ∈ [0 , π ) values and under the conditions of φ = 0 ◦ , φ = 0 ◦ and t = 100 . the same effect from both the coins. Now, writing firstfew states of the walker using the coin in (20), we have | ψ i = (cid:18) α | H i + β | T i (cid:19) | i| ψ i = U | ψ i = α | T i| − i + β | H i| i| ψ i = U | ψ i = (cid:18) α | H i + β | T i (cid:19) | i = | ψ i| ψ i = U | ψ i = α | T i| − i + β | H i| i = | ψ i ... | ψ t i = U | ψ i = (cid:18) α | H i + β | T i (cid:19) = | ψ i (21)From (21) we can observe that when θ = 90 ◦ or ◦ the coefficients of the initial states α and β togglesbetween the coin states | H i and | T i and as a resultthe walker is confined to the position states | i when t is even with the probability distribution, P t = P ( , t ) = | α | + | β | Thus, for an unbiased initial coin state (Eq. (19)) withan even time step, we can observe localization of thewalker at | i position when θ = 90 ◦ or ◦ .Another observation we made in Fig. 4 is thatwhen θ = 0 ◦ , ◦ we get two sharp peaks with equal heights at left and right most corners of the plot.Following the previous approach, for θ = 0 ◦ , ◦ and φ = φ = 0 ◦ the coins takes the following form C = (cid:18) − (cid:19) , C = − (cid:18) − (cid:19) (22)Like in the previous case, coins in (22) differs only bya factor of minus sign and we get the same effect fromboth the coins in probability calculation. By writing thestates using the coins given in (22) we get | ψ i = (cid:18) α | H i + β | T i (cid:19) | i| ψ i = α | H i| i + ( − β | T i|− i| ψ i = α | H i| i + ( − β | T i|− i| ψ i = α | H i| i + ( − β | T i|− i ... | ψ t i = α | H i| N i + ( − t β | T i|− N i (23)where N = t . Thus, the total probability of finding thewalker at the left and right most corner positions, P t = P ( N , t ) + P ( − N , t ) = | α | + | β | Thus, when θ = 0 ◦ , ◦ we get two sharp peakswith equal heights at left and right most corners. OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, FEBRUARY 16, 2021 9
Furthermore, for coins ( C i and C j ) corresponding to θ and θ + π differ only by a factor of minus sign. Thatis, both coins differ only by an overall phase factor( C i = − C j ). We know from Eqs. (5) and (17) P t = h ψ t | ψ t i = h ψ | ( U t ) † U t | ψ i = h ψ | (cid:18) ( S ( − C ⊗ I )) t (cid:19) † ( S ( − C ⊗ I )) t | ψ i = ( − t h ψ | (cid:18) ( S ( C ⊗ I )) t (cid:19) † ( S ( C ⊗ I )) t | ψ i = h ψ | (cid:18) ( S ( C ⊗ I )) t (cid:19) † ( S ( C ⊗ I )) t | ψ i As mentioned earlier, in probability calculation, suchan overall phase factor has no contribution. Thus, weget the same effect from both coins. Hence, we get thesame distribution for the pairs of θ and θ + π (Fig. 4). -200 -100 0 100 20000.10.2-200 -100 0 100 20000.30.6 P i -200 -100 0 100 200 x i x i P i -200 -100 0 100 20000.040.08-200 -100 0 100 200 x i θ = 0 θ = 15 θ = 45θ = 60 θ = 75 θ = 90 Fig. 5. Probability distributions of QW when θ varies form ◦ to ◦ where φ = φ = 0 ◦ and t = 100 . We observe that as θ value increases the walker will be localized to the initial position( | i ). Finally, we magnify the θ value in between ◦ and ◦ and observe the total probability distribution ofthe quantum walker (Fig. 5). One can observe thatwhen θ = 0 ◦ the quantum walker tends to stay atthe opposite end positions with equal probabilities.However, when the value of θ increases, the proba-bility of finding the walker in the left and rightmostposition decreases. At the same time, as the θ increases,the distribution tends to shrink towards the initialposition, and when θ = 90 ◦ walker gets completelylocalized to the initial position. Therefore, from theabove investigation, one can learn that for a fixednumber of time steps, the maximum spreading of thewalker over the line can be obtained for θ values close to ◦ while keeping φ = φ = 0 ◦ . In contrast, when θ isclose to ◦ and φ = φ = 0 ◦ , one can confine the walkercloser to the initial position . This section is devoted to study the impact of thephase parameters ( φ and φ ) on the total probabilitydistribution for each θ value. Using Eq. (19), we canfind the total probability distribution corresponding tothe initial state as follows P = | α | + | β | (24)where | α | = 1 / and | β | = 1 / . By applying thegeneral coin operator and shift operator given in (6)and (7) on the initial state in (19), we can write thestate at t = 1 as follows | ψ i = α (1) | H i| i + β − (1) | T i|− i (25)where α (1) = α cos θ + βe iφ sin θβ − (1) = αe iφ sin θ − βe i ( φ + φ ) cos θ Hence, the total probability distribution at time t = 1 is given by P = ( | α | + | β | ) cos θ + ( | α | + | β | ) sin θ + αβ ∗ e − iφ sin θ cos θ + α ∗ βe iφ sin θ cos θ − αβ ∗ e − iφ sin θ cos θ − α ∗ βe iφ sin θ cos θ where α ∗ and β ∗ are complex conjugates. As α and β are constant, probability distribution at time t = 1 does not have any impact from the φ . Now, the stateof the quantum walker at an arbitrary time k can bewritten from Eq. (4) as | ψ k i = X x ∈ Z ( α x ( k ) | H i + β x ( k ) | T i ) ⊗ | x i Then, the corresponding probability distribution attime k takes the form of (Eq. (5)) P k = X x ∈ Z | α x ( k ) | + | β x ( k ) | From the inductive hypothesis P k does not contain φ .Now, by applying the coin ( C ) and shift ( S ) operatorsgiven in (6) and (7) on | ψ k i , the state at time k + 1 canbe written as | ψ k +1 i = X x ∈ Z ( α x ( k +1) | H i + β x ( k +1) | T i ) ⊗| x i (26)where we get the following relationship (proof in SIsection 5) α x ( k + 1) = α x − ( k ) cos θ + β x − ( k ) e iφ sin θβ x ( k + 1) = α x +1 ( k ) e iφ sin θ − β x +1 ( k ) e iφ + φ cos θ (27) OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, FEBRUARY 16, 2021 10 -200 -100 0 100 200 x i P i - -
100 0 100 200 x i . . P i -200 -100 0 100 200 x i P i -200 -100 0 100 200 x i P i -200 -100 0 100 200 x i P i Fig. 6. Impact of φ on the probability distribution of the QW when θ = 45 ◦ . As φ increases from ◦ to ◦ the difference between thetwo peaks increases and the maximum difference appears when φ = 90 ◦ (distribution becomes asymmetric). The initial distributionis restored as φ increases from ◦ to ◦ . From Eq. (26) the probability distribution at k + 1 canbe calculated as P k +1 = X x | α x ( k + 1) | + | β x ( k + 1) | = X x (cid:18) | α x − ( k ) | + | β x +1 ( k ) | (cid:19) cos θ + X x (cid:18) | α x +1 ( k ) | + | β x − ( k ) | (cid:19) sin θ + X x (cid:18) α x − ( k ) β ∗ x − ( k ) e − iφ + α ∗ x − ( k ) β x − ( k ) e iφ − α x +1 ( k ) β ∗ x +1 ( k ) e − iφ − α ∗ x +1 ( k ) β x +1 ( k ) e iφ (cid:19) sin θ cos θ According to the expression given in (24), the proba-bility distribution at t = 0 does not contain any termsof φ . Hence, probability distribution at t = 1 doesnot contain any terms of φ . Thus, we can say thatif the probability distribution P k at time k does notcontain the parameter of φ the distribution P k +1 attime k + 1 does not contain φ which we can observefrom the induction step. Finally, from the principleof mathematical induction, we conclude that the totalprobability distribution ( P t ) at any arbitrary time t isfree from φ . The numerical verification for the inertimpact of φ is illustrated in SI. However, it is obviousthat θ and φ have an impact on the total probabilitydistribution. Hence, we continue our study by setting φ = 0 ◦ .Now, we examine the impact of φ on the distributionin detail. Let us choose θ = 45 ◦ and vary φ from ◦ to ◦ (Fig. 6). We wish to emphasize that thecombination of θ = 45 ◦ , φ = 0 ◦ and φ = 0 ◦ givesfamous Hadmard coin. In Hadamard’s walk, we cansee two sharp peaks of equal heights (distributionis symmetric). In between the two sharp peaks, wehave comparatively smaller probabilities of finding thewalker. As φ increases from ◦ to ◦ the differencebetween the two sharp peaks tends to increase andreaches its maximum when φ = 90 ◦ (distributionbecomes asymmetric). Further, as we increase φ , thedifference between the two sharp peaks tends to de-crease, and when φ = 180 ◦ , the initial distributionis restored. We observe that for each θ value, byvarying φ we get similar behavior in the probabilitydistribution as in Fig. 6. We wish to emphasize that ouranalysis unveils – for each φ ∈ (0 , π ) , an asymmetryin the probability distribution even when the initialcoin state is unbiased. From the above investigations,we learn that rotation and one of the phase parametersplay a crucial role in the final probability distribution of thewalker, and by tuning them, we can regulate the dynamicalbehavior of the walker. Now, we present an algorithm for the QW on a line(Algorithm 1). We use the recurrence relation (Eq.
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Algorithm 1
1D Quantum Walk( N , θ , φ , φ , α , β ) n ← N + 3 mid ← N + 2 [Coin operator] C , ← cos θC , ← sin θe iφ C , ← sin θe iφ C , ← − cos θe i ( φ + φ ) [Amplitude Matrix] A ← ← [Initial Conditions of Amplitude Matrix] B ,mid ← αB ,mid ← β [Amplitude Calculation] for k ← to N dofor m ← to N + 2 do A ,m ← C , × B ,m − + C , × B ,m − A ,m ← C , × B ,m +1 + C , × B ,m +1 end for B ← A end for [Probability Density Function] P ← for m ← to n do P m ← B ,m × B ∗ ,m + B ,m × B ∗ ,m x m ← − n +12 + m end forreturn P (27)) of the coefficient for the probability calculationto develop the algorithm. The variables N , α , β , θ , φ , and φ are the input parameters of the algorithm.First, a positive integer value is assigned to the variable N = t . The arrays’ size is defined by the variable N ,and the position corresponding to the origin ( | i ) ismarked by the variable mid . With the help of N , thenumber of position states ( n ) and the origin is calcu-lated. For unbiased initial coin state, we assign α = √ and β = − i √ . Next, we initialize the general coin withthe rotation and shift parameters, i.e., with specificvalues of θ , φ , and φ . To measure the probabilitydistribution, we store the coefficient value of both thecoin states | H i and | T i at each position in the higherdimensional array B ∈ C × n which is initialized tozero. Then the updating of coefficient value (Eq. (27))in B is written in the form of a nested loop where weapply the coin to each element of the array. Finally,the probability distribution of the QW is calculatedby employing the Eq. (5) where B ∗ ij is the complexconjugate of B ij . The data and codes used in this paperare available at GitHub repository [36]. ONCLUSION
This article revisits the framework of discrete-timeQW on a line and presents it comprehensively froman engineering perspective. We keep all the steps assimple as possible. We make the abstract notationsinto more concrete matrix notations. We explore theprobability distribution of the discrete-time quantumwalks with the general coin. We show that by fixingthe initial coin state to an unbiased state and by regu-lating the parameters of the general coin operator, onecan tune the probability distribution of the quantumwalker. The general coin operator of the QWs on a lineconsists of three parameters (one rotation and two-phase parameters). By changing the parameters me-thodically, we get different coin operators.
Our analysisreveals that the rotation and a single phase parameter ofthe general coin play an essential role in controlling thewalker’s probability distribution. Our investigation uncov-ers that, for an unbiased initial coin state, we can make theprobability distribution asymmetric by tuning a single phaseparameter of the general coin operator.
We show that ournumerical results are in good agreement with that ofthe analytical derivations. Among the coin operatorsthat we have studied, Hadamard, Grover, and Fouriercoins are special cases of the general coin operator.Additionally, we sketch the basic intuition of the quan-tum entanglement in the context of Qws. Finally, weprovide an algorithm for the one-dimensional QWs.As future extensions of this study, one can focuson (1) the dynamics of the QWs driven by the generalcoin operator on the graph, (2) the spectral propertiesof the unitary operator that comprise the general coinoperator, (3) how the entanglement between coin andposition states varies when the coin parameters arechanged and (4) how the parameters of general coinoperator can be utilized to develop tools to cater thefuture Engineering developments. A CKNOWLEDGMENT
Mahesh N. Jayakody acknowledges the presidentialscholarship of Bar-Ilan university for PhD scholarsand the research funding received from Dr. EliahuCohen (Faculty of Engineering, Bar-Ilan University)Moreover, he is thankful to Dr. Eliahu Cohen for hav-ing useful discussions and to Prof. Asiri Nanayakkara(National Institute of Fundamental Studies, Sri Lanka)for sharing QW code. Priodyuti Pradhan is indebtedto Prof. Baruch Barzel for providing the postdoctoralresearch grant and acknowledges Bar-Ilan Universityfor the Kolman-Soref postdoctoral fellowship. R EFERENCES [1] Mohseni, M., Read, P., Neven, H., Boixo, S., Denchev, V.,Babbush, R., ... & Martinis, J. (2017). Commercialize quantumtechnologies in five years. Nature News, 543(7644), 171.
OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, FEBRUARY 16, 2021 12 [2] Dowling, J. P., & Milburn, G. J. (2003). Quantum technology:the second quantum revolution. Philosophical Transactionsof the Royal Society of London. Series A: Mathematical,Physical and Engineering Sciences, 361(1809), 1655-1674.[3] Bresson, A., Bidel, Y., Bouyer, P., Leone, B., Murphy, E., & Sil-vestrin, P. (2006). Quantum mechanics for space applications.Applied Physics B, 84(4), 545-550.[4] Ac´ın, Ac´ın, A., Bloch, I., Buhrman, H., Calarco, T., Eichler, C.,Eisert, J., ... & Wilhelm, F. K. (2018). The quantum technolo-gies roadmap: a European community view. New Journal ofPhysics, 20(8), 080201.[5] Jayakody, M. N., & Nanayakkara, A. (2019). Full staterevivals in higher dimensional quantum walks. PhysicaScripta, 94(4), 045101.[6] Alderete, C. H., Singh, S., Nguyen, N. H., Zhu, D., Balu,R., Monroe, C., ... & Linke, N. M. (2020). Quantum walksand Dirac cellular automata on a programmable trapped-ionquantum computer. Nature communications, 11(1), 1-7.[7] Wang, J., and Manouchehri, K. (2013). Physical implementa-tion of quantum walks. Springer Berlin.[8] Montanaro, A. (2016). Quantum algorithms: an overview. npjQuantum Information, 2(1), 1-8.[9] Xia, F., Liu, J., Nie, H., Fu, Y., Wan, L., & Kong, X. (2019).Random walks: A review of algorithms and applications.IEEE Transactions on Emerging Topics in ComputationalIntelligence, 4(2), 95-107.[10] Feynman, R. P. (1986). Quantum mechanical computers.Found. Phys., 16(6), 507-532.[11] Aharonov, Y., Davidovich, L., and Zagury, N. (1993). Quan-tum random walks. Physical Review A, 48(2), 1687.[12] Kempe, J. (2003). Quantum random walks: an introductoryoverview. Contemporary Physics, 44(4), 307-327.[13] Childs, A. M., Gosset, D., and Webb, Z. (2013). Univer-sal computation by multiparticle quantum walk. Science,339(6121), 791-794.[14] Abd el-Latif, A. A., Abd-el-Atty, B., Amin, M., & Iliyasu, A.M. (2020). Quantum-inspired cascaded discrete-time quan-tum walks with induced chaotic dynamics and crypto-graphic applications. Scientific reports, 10(1), 1-16.[15] Biamonte, J., Faccin, M., & De Domenico, M. (2019). Com-plex networks from classical to quantum. CommunicationsPhysics, 2(1), 1-10.[16] Abd El-Latif, A. A., Abd-El-Atty, B., Venegas-Andraca, S. E.,Elwahsh, H., Piran, M. J., Bashir, A. K., ... & Mazurczyk, W.(2020). Providing end-to-end security using quantum walksin IoT networks. IEEE Access, 8, 92687-92696.[17] Abd El-Latif, A. A., Abd-El-Atty, B., Mazurczyk, W., Fung,C., & Venegas-Andraca, S. E. (2020). Secure data encryptionbased on quantum walks for 5G Internet of Things scenario.IEEE Transactions on Network and Service Management,17(1), 118-131.[18] Childs, A. M., Schulman, L. J., and Vazirani, U. V. (2007).Quantum algorithms for hidden nonlinear structures. In48th Annual IEEE Symposium on Foundations of ComputerScience (FOCS’07) (pp. 395-404). IEEE.[19] Ambainis, A. (2007). Quantum walk algorithm for elementdistinctness. SIAM Journal on Computing, 37(1), 210-239.[20] Magniez, F., Santha, M., and Szegedy, M. (2007). Quantumalgorithms for the triangle problem. SIAM Journal on Com-puting, 37(2), 413-424.[21] Hoyer, S., Sarovar, M., and Whaley, K. B. (2010). Limits ofquantum speedup in photosynthetic light harvesting. NewJournal of Physics, 12(6), 065041.[22] Xue, P., Qin, H., and Tang, B. (2014). Trapping photons onthe line: controllable dynamics of a quantum walk. Scientificreports, 4, 4825.[23] Kitagawa, T., Broome, M. A., Fedrizzi, A., Rudner, M. S.,Berg, E., Kassal, I., ... and White, A. G. (2012). Observationof topologically protected bound states in photonic quantumwalks. Nature communications, 3, 882. [24] Douglas, B. L., and Wang, J. B. (2008). A classical approachto the graph isomorphism problem using quantum walks.Journal of Physics A: Mathematical and Theoretical, 41(7),075303.[25] Strauch, F.W, (2006) Connecting the discrete andcontinuous-time quantum walks. Physical Review A, 74,030301.[26] Aharonov, D., Ambainis, A., Kempe, J., & Vazirani, U. (2001,July). Quantum walks on graphs. In Proceedings of thethirty-third annual ACM symposium on Theory of comput-ing (pp. 50-59).[27] Venegas-Andraca, S. E. (2012). Quantum walks: a com-prehensive review. Quantum Information Processing, 11(5),1015-1106.[28] Weiss, G. H. (1983). Random walks and their applications:Widely used as mathematical models, random walks playan important role in several areas of physics, chemistry, andbiology. American Scientist, 71(1), 65-71.[29] Harris, E. G. (1995). Introduction to Quantum Mechanics byDavid J. Griffiths. AMERICAN JOURNAL OF PHYSICS, 63,767-767.[30] Thaller, B., 2005.
Advanced visual quantum mechanics .Springer Science & Business Media.[31] Bowers, P.L., 2020.
Lectures on Quantum Mechanics: A Primerfor Mathematicians . Cambridge University Press.[32] BTregenna, B., Flanagan, W., Maile, R., & Kendon, V. (2003).Controlling discrete quantum walks: coins and initial states.New Journal of Physics, 5(1), 83.[33] Gratsea, A., Metz, F., & Busch, T. (2020). Universal andoptimal coin sequences for high entanglement generationin 1D discrete time quantum walks. Journal of Physics A:Mathematical and Theoretical, 53(44), 445306.[34] Hunter, J.K. and Nachtergaele, B., (2001).
Applied analysis .World Scientific Publishing Company.[35] Audretsch, J. (2008).
Entangled systems: new directions inquantum physics . John Wiley & Sons.[36] Our codes and data are available at the following linkhttps://github.com/priodyuti/qw codes data. r X i v : . [ qu a n t - ph ] F e b Supplementary Information: One-dimensional discrete-timequantum walks with general coin
Mahesh N. Jayakody, Chandrakala Meena, and Priodyuti Pradhan
Here, we provide detail steps of the walker state calculation in Eqs. (11) and (12), where we consider ψ = | H i ⊗ | x i and use Eqs. (6) and (7) and get | ψ i = U| ψ i = S ( C ⊗ I )( | H i ⊗ | x i )= S ( C| H i ⊗ | x i )= S (cid:18) cos θ | H ih H | H i + e iφ sin θ | H ih T | H i + e iφ sin θ | T ih H | H i − e i ( φ + φ ) cos θ | T ih T | H i ⊗ | x i (cid:19) = S (cid:18) cos θ | H i + e iφ sin θ | T i ) ⊗ | x i (cid:19) = S (cid:18) cos θ | H i ⊗ | x i + e iφ sin θ | T i ⊗ | x i (cid:19) = (cid:18)X x | H ih H | ⊗ | x + 1 ih x | + | T ih T | ⊗ | x − ih x | (cid:19)(cid:18) cos θ | H i ⊗ | x i + e iφ sin θ | T i ⊗ | x i (cid:19) = (cid:18)X x (cos θ | H ih H | H i ⊗ | x + 1 ih x | x i + (cos θ | T ih T | H i ⊗ | x − ih x | x i )+ e iφ sin θ | H ih H | T i ⊗ | x + 1 ih x | x i + e iφ sin θ | T ih T | T i ⊗ | x − ih x | x i (cid:19) = cos θ | H i ⊗ | x + 1 i + e iφ sin θ | T i ⊗ | x − i (S1) -200 -100 0 100 200 x i P i -200 -100 0 100 200 x i x i (a) (b) (c) Fig. S1.
Compare the probability distribution of the quantum walker where we use unbiased vs. biased initial coinstate ( α | H i + β | T i ) in Eq. (18) with Hadamard coin ( θ = 45, φ = 0, φ = 0). For t = 100 (a) α = √ and β = i √ (b) α = 1 and β = 0 and (c) α = 0 and β = 1. ψ i = U| ψ i = S ( C ⊗ I ) (cid:26) cos ( θ ) | H i ⊗ | x + 1 i + e iφ sin ( θ ) | T i ⊗ | x − i (cid:27) = S (cid:26) cos ( θ ) C| H i ⊗ | x + 1 i + e iφ sin ( θ ) C| T i ⊗ | x − i (cid:27) = S (cid:26) cos θ H ih H | H i + e iφ cos θ sin θ | H ih T | H i + e iφ cos θ sin θ | T ih H | H i − e i ( φ + φ ) cos θ | T ih T | H i ⊗ | x + 1 i (cid:27) + S (cid:26) e iφ sin θ cos θ H ih H | T i + e i ( φ + φ ) sin θ sin θ | H ih T | T i + e iφ sin θ | T ih H | T i − e i ( φ +2 φ ) cos θ sin θ | T ih T | T i ⊗ | x − i (cid:27) = S (cid:26) (cos θ H i + e iφ cos θ sin θ | T ) ⊗ | x + 1 i} + S{ ( e i ( φ + φ ) sin θ | H i − e i ( φ +2 φ ) cos θ sin θ | T i ) ⊗ | x − i (cid:27) = (cid:18)X x | H ih H | ⊗ | x + 1 ih x | + | T ih T | ⊗ | x − ih x | (cid:19)(cid:18) cos θ | H i + e iφ cos θ sin θ | T i (cid:19) ⊗ | x + 1 i + (cid:18)X x | H ih H | ⊗ | x + 1 ih x | + | T ih T | ⊗ | x − ih x | (cid:19)(cid:18) e i ( φ + φ ) sin θ | H i − e i ( φ +2 φ ) cos θ sin θ | T i (cid:19) ⊗ | x − i = cos θ | H i ⊗ | x + 2 i + (cid:18) e i ( φ + φ ) sin θ | H i + e iφ sin θ cos θ | T i (cid:19) ⊗ | x i − e i ( φ +2 φ ) sin θ cos θ | T i ⊗ | x − i (S2) Here, we give column vector representation of the abstract state notations of the quantum walk. | ψ i = | H i ⊗ | i = (cid:18) (cid:19) ⊗ = | ψ i = cos θ | H i ⊗ | i | {z } | ψ ( , i + e iφ sin θ | T i ⊗ | − i | {z } | ψ ( − , i = cos θ (cid:18) (cid:19) ⊗ + e iφ sin θ (cid:18) (cid:19) ⊗ = θ e iφ sin θ ψ i = cos θ | H i ⊗ | i + (cid:18) e i ( φ + φ ) sin θ | H i + e iφ sin θ cos θ | T i (cid:19) ⊗ | i − e i ( φ +2 φ ) sin θ cos θ | T i ⊗ | − i = cos θ (cid:18) (cid:19) ⊗ + (cid:18) e i ( φ + φ ) sin θ (cid:18) (cid:19) + e iφ sin θ cos θ (cid:18) (cid:19)(cid:19) ⊗ − e i ( φ +2 φ ) sin θ cos θ (cid:18) (cid:19) ⊗ = e i ( φ + φ ) sin θ θ − e i ( φ +2 φ ) sin θ cos θ e iφ sin θ cos θ Quantum entanglement is a unique type of correlation shared between components of a quantum system[1]. The concept of entanglement comes into the picture when we consider a system that comprises severalsubsystems. In other words, entanglement can be viewed as a special type of correlation between eachsubsystems. The mathematical framework of entanglement depends upon the fact that whether the systemof interest is closed or open. That is, whether the system of interest is isolated from its environment or not.Here, we define the quantum entanglement with respect to a closed system that includes two subsystems [2].Consider two arbitrary quantum systems A and B , with respective Hilbert spaces H A and H B of d A and d B dimensions. The composite Hilbert space that comprise H A and H B is given by H = H A ⊗ H B with thedimension of d = d A d B . Let {| a i} d A − a =0 and {| b i} d B − b =0 be bases of H A and H B respectively. Then any stateof the composite system | φ i AB ∈ H can be written as | φ i AB = d A − X a =0 d B − X b =0 λ a,b ( | a i ⊗ | b i ) (S3)where λ a,b ∈ C and P d A − a =0 P d B − b =0 | λ a,b | = 1. Suppose there exists a set of complex numbers { µ a } d A − a =0 and { γ b } d B − b =0 such that for each a and b , λ a,b = µ a γ b and | φ i A = P d A − a =0 µ a | a i and | φ i B = P d B − b =0 γ b | b i arerespective states of H A and H B with P d A − a =0 | µ a | = 1 and P d B − b =0 | γ b | = 1. Then | φ i AB is a separablestate. Otherwise | φ i AB is an entangled state. Further, if the separable condition is satisfied then | φ i AB canbe decomposed as | φ i AB = | φ i A ⊗ | φ i B . For instance, consider the Hilbert spaces of H A and H B with twodimensions. Let {| i A , | i B } and {| i B , | i B } be the bases of H A and H B respectively. Consider a state | φ i AB ∈ H of the form | φ i AB = 12 ( | i A ⊗ | i B ) + 12 ( | i A ⊗ | i B ) + 12 ( | i A ⊗ | i B ) + 12 ( | i A ⊗ | i B ) (S4)Then, we have λ , = λ , = λ , = λ , = . Choose µ = µ = γ = γ = √ so that λ , = µ γ , λ , = µ γ , λ , = µ γ , λ , = µ γ . Define | φ i A = 1 √ | i A + | i A ) | φ i B = 1 √ | i B + | i B ) (S5)3ote that | φ i A ∈ H A and | φ i B ∈ H B . Then we have | φ i A ⊗ | φ i B = (cid:18) √ | i A + | i A ) (cid:19) ⊗ (cid:18) √ | i B + | i B ) (cid:19) = 12 ( | i A ⊗ | i B ) + 12 ( | i A ⊗ | i B ) + 12 ( | i A ⊗ | i B ) + 12 ( | i A ⊗ | i B )= | φ i AB (S6)Thus | φ i AB is not an entangled state but a separable state. Now consider a state of the following form | ψ i AB = 1 √ | i A ⊗ | i B ) + 1 √ | i A ⊗ | i B ) (S7)It is not possible to find a set of complex numbers { µ a } and { γ b } such that the state | ψ i AB can be decomposedinto two separate states from H A and H B . Thus, | ψ i AB is called an entangled states.Now we use the above definition of the entanglement to check whether we can observe entanglement in | ψ i (Eq. 13). | ψ i = cos θ | H i ⊗ | i + e iφ sin θ | T i ⊗ | − i = cos θ (cid:18) (cid:19) ⊗ + e iφ sin θ (cid:18) (cid:19) ⊗ = θ e iφ sin θ (S8)Suppose | ψ i is separable and there exist | φ c i ∈ H c and | φ x i ∈ H x such that | ψ i = | φ c i ⊗ | φ x i (S9)Now we express | φ c i and | φ x i in terms of the bases of their respective Hilbert spaces. Then we have | φ c i = µ (cid:18) (cid:19) + µ (cid:18) (cid:19) | φ x i = γ + γ + γ + γ + γ (S10)where µ , µ ∈ C such that | µ | + | µ | = 1 and γ , γ , γ , γ , γ ∈ C such that | γ | + | γ | + | γ | + | γ | + | γ | =1. Since we have assumed that | ψ i is separable we can write | ψ i = µ γ (cid:18) (cid:19) ⊗ + µ γ (cid:18) (cid:19) ⊗ + µ γ (cid:18) (cid:19) ⊗ + µ γ (cid:18) (cid:19) ⊗ + . . . (S11)4y comparing terms in (S8) and (S11) we get µ γ = cos θµ γ = e iφ sin θγ = γ = γ = 0 (S12)In addition, we have another two equations as follows | µ | + | µ | = 1 | γ | + | γ | = 1 (S13)From (S12) one can claim that any of the µ , µ , γ , γ are non-zero. Thus we have | ψ i = µ γ (cid:18) (cid:19) ⊗ + µ γ (cid:18) (cid:19) ⊗ + µ γ (cid:18) (cid:19) ⊗ + µ γ (cid:18) (cid:19) ⊗ = µ γ µ γ µ γ µ γ (S14)But from (S8) we have | ψ i = θ e iφ sin θ (S15)Comparing elements in (S14) and (S15) we get the following relations µ γ = 0 µ γ = 0 (S16)This will lead to a contradiction. Thus, | ψ i is entangled. Similarly, we can also find that | ψ i is also entangledwith coin and position state. Although we start with a separable state at time t = 0, as time progress coinand position states become more and more entangled. Several studies have been conducted to investigatequantum entanglement in the context of QWs and more details can be found in [3, 4, 5]. For N = 2, we have five position states | − i , | − i , | i , | i and | i . The coin states are given by | H i and | T i . By using the column matrix representation of position and coin states, we can write the matricescorresponding to coin operator C and shift operator ¯S as follows C = (cid:18) a bc d (cid:19) (S17)5here a = cos θ , b = e iφ sin θ , c = e iφ sin θ and d = − e i ( φ + φ ) cos θ such that θ ∈ [0 , π ) and φ , φ ∈ [0 , π ). ¯S = (cid:18) (cid:19) ⊗ + (cid:18) (cid:19) ⊗ = (S18)By combining the matrices C and ¯S , one can derive the unitary operator ¯U for the QW as follows ¯U = ¯S ( C ⊗ I ) = (cid:18) a bc d (cid:19) ⊗ = a b a b a b a b
00 0 0 0 a bc d c d c d c d
00 0 0 0 c d = a ba b a b a b a b c d c d c d
00 0 0 0 c dc d (S19)6ow, we apply the unitary operator ¯U on the initial state | ψ i given in Eq. (13) and get | ψ i = ¯U | ψ i = a ba b a b a b a b c d c d c d
00 0 0 0 c dc d = θ e iφ sin θ (S20)Further we apply ¯U one more time on | ψ i , and get | ψ i = ¯U | ψ i = bc a bd ab bc a bd aba bc ab bd a bc ab bd
00 0 a bc ab bdac cd bc d ac cd bc d
00 0 ac cd bc d cd ac d bc cd ac d bc = e i ( φ + φ ) sin θ θ − e i ( φ +2 φ ) sin θ cos θ e iφ sin θ cos θ (S21)Hence, applying ¯U on | ψ i we can obtain | ψ t i for any t . However, the block matrix of the unitary operatorincreases in size when the position space increases. Hence, we use the recursive formula in Eq. (27) to developan algorithm (Algorithm 1) to simulate the quantum walks in practical scenarios. The state of the quantum walker at an arbitrary time k can be written as | ψ k i = ∞ X x = −∞ ( α x ( k ) | H i + β x ( k ) | T i ) ⊗ | x i (S22)By applying the coin ( C ) and shift ( S ) operators on | ψ k i in Eqs. (6) and (7), we can write the state of thequantum walker at time k + 1 as follows | ψ k +1 i = S ( C ⊗ I ) | ψ k i = S ( C ⊗ I ) ∞ X x = −∞ ( α x ( k ) | H i + β x ( k ) | T i ) ⊗ | x i ! = ∞ X x = −∞ ( α x ( k ) cos θ + β x ( t ) e iφ sin θ | H i ⊗ | x + 1 i + ∞ X x = −∞ (cid:0) α x ( k ) e iφ sin θ − β x ( k ) e iφ + φ cos θ (cid:1) | T i ⊗ | x − i (S23)Note that the summation over x goes from −∞ to ∞ . Thus, by changing x + 1 → x and x − → x we canwrite 7 ψ k +1 i = ∞ X x = −∞ (cid:0) α x − ( k ) cos θ + β x − ( k ) e iφ sin θ (cid:1) | H i ⊗ | x i + ∞ X x = −∞ (cid:0) α x +1 ( k ) e iφ sin θ − β x +1 ( k ) e iφ + φ cos θ (cid:1) | T i ⊗ | x i (S24)Alternatively, the state of the quantum walker at time k + 1 can be written in the following form as well | ψ k +1 i = ∞ X x = −∞ ( α x ( k + 1) | H i + β x ( k + 1) | T i ) ⊗ | x i (S25)By comparing the corresponding coefficients in (S24) and (S25) we can write the following recurrence formula α x ( k + 1) = α x − ( k ) cos θ + β x − ( k ) e iφ sin θβ x ( k + 1) = α x +1 ( k ) e iφ sin θ − β x +1 ( k ) e iφ + φ cos θ (S26)This complete the proof. If we set φ and φ to non-zero values, we can observe that the height of one peak tend to reduce andthe other peak tend to increase proportionally in the probability distribution (Fig. 6). It indicates that theparticle is biased towards one side of the line. Hence, we use the difference between the heights of these twopeaks as a measure to quantify the impact of φ and φ on the total probability distribution of the QW. Wemeasure the impact of φ and φ on the probability distribution as follows ∆ = max x P ( x , t ) − max y = x P ( y , t ) (S27)where P ( x , t ) is the probability at position x at time t (Eq. (3)). Fig. S2 shows a set of phase diagrams fromwhich we can extract the information about the impact of φ and φ on the distribution. From Fig. S2, onecan observe that the parameter φ has no impact on the probability distribution. That is, when we vary φ while keeping φ for a fixed value, there are no difference in the shape of the total probability distribution.However, the coin parameter φ has a significant impact on the shape of the probability distribution. Thephase diagram for other θ values can be shown in Fig. S3. References [1] Xi, Z., Li, Y., & Fan, H. (2015). Quantum coherence and correlations in quantum system. Scientific reports, 5(1),1-9.[2] Audretsch, J. (2008).
Entangled systems: new directions in quantum physics . John Wiley & Sons.[3] Carneiro, I., Loo, M., Xu, X., Girerd, M., Kendon, V., & Knight, P. L. (2005). Entanglement in coined quantumwalks on regular graphs. New Journal of Physics, 7(1), 156.[4] Abal, G., Donangelo, R., & Fort, H. (2007). Asymptotic entanglement in the discrete-time quantum walk. arXivpreprint arXiv:0709.3279.[5] Singh, S., Balu, R., Laflamme, R., & Chandrashekar, C. M. (2019). Accelerated quantum walk, two-particleentanglement generation and localization. Journal of Physics Communications, 3(5), 055008. ig. S2. Phase space diagram for different θ ∈ [0 , π ) and φ , φ ∈ [0 , π ) values when t = 100. Shades of coloursrepresent the magnitude of the difference between the two peaks at left and right most corners of the distribution.Colours of purple and yellow denote the minimum and maximum difference respectively. It is clearly visible thatchange in φ phase parameter has no effect on the difference but change in φ phase parameter has a significantimpact. ig. S3. Phase space for different θ values.values.