Optimizing the discrete time quantum walk using a SU(2) coin
aa r X i v : . [ qu a n t - ph ] M a r Optimizing the discrete time quantum walk using a SU (2) coin C. M. Chandrashekar, R. Srikanth,
2, 3 and Raymond Laflamme
1, 4 Institute for Quantum Computing, University of Waterloo, ON, N2L 3G1, Canada Poornaprajna Institute of Scientific Research, Devanahalli, Bangalore 562 110, India Raman Research Institute, Sadashiva Nagar, Bangalore, India Perimeter Institute for Theoretical Physics, Waterloo, ON, N2J 2W9, Canada
We present a generalized version of the discrete time quantum walk, using the SU (2) operationas the quantum coin. By varying the coin parameters, the quantum walk can be optimized formaximum variance subject to the functional form σ ≈ N and the probability distribution inthe position space can be biased. We also discuss the variation in measurement entropy with thevariation of the parameters in the SU (2) coin. Exploiting this we show how quantum walk can beoptimized for improving mixing time in an n -cycle and for quantum walk search. I. INTRODUCTION
The discrete time quantum walk has a very similarstructure to that of the classical random walk - a coin flipand a subsequent shift - but the behaviour is strikinglydifferent because of quantum interference. The variance σ of the quantum walk is known to grow quadraticallywith the number of steps N , σ ∝ N , compared to thelinear growth, σ ∝ N , for the classical random walk[1, 2, 3, 4]. This has motivated the exploration for a newand improved quantum search algorithms, which undercertain conditions are exponentially fast compared to theclassical analog [5]. Environmental effects on the quan-tum walk [6] and the role of the quantum walk to speedup the physical process, such as the quantum phase tran-sition have been explored [7]. Experimental implementa-tion of the quantum walk has been reported [8] and var-ious other schemes for a physical realization have beenproposed [9].The quantum walk of a particle initially in asymmetric superposition state | Ψ in i using a single-variable parameter θ in the unitary operator, U θ ≡ (cid:18) cos( θ ) sin( θ )sin( θ ) − cos( θ ) (cid:19) , as quantum coin returns the sym-metric probability distribution in the position space. Thechange in the parameter θ is known to affect the varia-tion in the variance, σ [3]. It has been reported thatobtaining a symmetric distribution depends largely onthe initial state of the particle [3, 4, 10].In this paper, the discrete time quantum walk has beengeneralized using the SU (2) operator with three Caley-Klein parameters ξ , θ and ζ as the quantum coin. Weshow that the variance can be varied by changing the pa-rameter θ , σ ≈ (1 − sin( θ )) N and the parameters ξ and ζ introduce asymmetry in the position-space probabilitydistribution even if the initial state of the particle is insymmetric superposition. This asymmetry in the proba-bility distribution is similar to the distribution obtainedfor a walk on a particle initially in a non-symmetric su-perposition state. We discuss the variation of measure-ment entropy in position space with the three parame-ters. Thus, we also show that the quantum walk can beoptimized for the maximum variance, for applications in search algorithm, improving mixing time in an n -cycleor general graph and other processes using a generalized SU (2) quantum coin. The combination of the measure-ment entropy and three parameters in the SU (2) coin canbe optimized to fit the physical system and for the rele-vant applications of the quantum walk on general graphs.This paper discuss the effect of SU (2) coin on quantumwalk with particle initially in symmetric superpositionstate. The SU (2) coin will have a similar influence ona particle starting with other initial states but with anadditional decrease in the variance by a small amount.The paper is organized as follows. Section II introducesto the discrete time quantum (Hadamard) walk. Sec-tion III discusses the generalized version of the quantumwalk using the arbitrary three-parameter SU (2) quan-tum coin. The effect of three parameters on the varianceof the quantum walker is discussed, and the functionaldependence of the variance due to parameter θ is shown.The variation of the entropy of the measurement in po-sition space after implementing the quantum walk usingdifferent values of θ is discussed in Sec. IV. Section Vand VI discuss optimization of the mixing time of thequantum walker on the n − cycle and the search using aquantum walk. Section VII concludes with a summary. II. HADAMARD WALK
To define the one-dimensional discrete time quantum(Hadamard) walk we require the coin
Hilbert space H c and the position Hilbert space H p . The H c is spannedby the internal (basis) state of the particle, | i and | i ,and the H p is spanned by the basis state | ψ i i , i ∈ Z .The total system is then in the space H = H c ⊗ H p .To implement the simplest version of the quantum walk,known as the Hadamard walk, the particle at origin inone of the basis state is evolved into the superposition ofthe H c with equal probability, by applying the Hadamardoperation, H = √ (cid:18) − (cid:19) , such that,( H ⊗ )( | i ⊗ | ψ i ) = 1 √ | i + | i ] ⊗ | ψ i ( H ⊗ )( | i ⊗ | ψ i ) = 1 √ | i − | i ] ⊗ | ψ i . (1)The H is then followed by the conditional shift operation S : conditioned on the internal state being | i ( | i ) theparticle moves to the left (right), S = | ih | ⊗ X i ∈ Z | ψ i − ih ψ i | + | ih | ⊗ X i ∈ Z | ψ i +1 ih ψ i | (2)The operation S evolves the particle into the superposi-tion in position space. Therefore, each step of quantum(Hadamard) walk is composed of an application of H anda subsequent S operator to spatially entangle H c and H p .The process of W = S ( H ⊗ ) is iterated without resort-ing to the intermediate measurements to realize a largenumber of steps of the quantum walk. After the firsttwo steps of implementation of W , the probability dis-tribution starts to differ from the classical distribution.The probability amplitude distribution arising from theiterated application of W is significantly different fromthe distribution of the classical walk. The particle withinitial coin state | i ( | i ) drifts to the right (left). Thisasymmetry arises from the fact that the Hadamard oper-ation treats the two states | i and | i differently, multi-plies the phase by − | i . To obtainleft-right symmetry in the probability distribution, (b)in Fig. (1 b), one needs to start the walk with the par-tilce in the symmetric superposition state of the coin, | Ψ in i ≡ √ [ | i + i | i ] ⊗ | ψ i . III. GENERALIZED DISCRETE TIMEQUANTUM WALK
The coin toss operation in general can be written as anarbitrary three parameter SU (2) operator of the form, U ξ,θ,ζ ≡ (cid:18) e iξ cos( θ ) e iζ sin( θ ) e − iζ sin( θ ) − e − iξ cos( θ ) (cid:19) , (3)the Hadamard operator, H = U , π , . By replacing theHadamard coin with an operator U ξ,θ,ζ , we obtain thegeneralized quantum walk. For the analysis of the gener-alized quantum walk we consider the symmetric superpo-sition state of the particle at the origin. By varying theparameter ξ and ζ the results obtained for walker startingwith one of the basis (or other nonsymmetric superposi-tion) state can be reproduced. A particle at origin in asymmetric superposition state | Ψ in i , when subjected toa subsequent iteration of W ξ,θ,ζ = S ( U ξ,θ,ζ ⊗ ) imple-ments a generalized discrete time quantum walk on a line.Consider an implementation of W ξ,θ,ζ , which evolves the −100 −50 0 50 10000.020.040.060.080.10.120.140.16 P r obab ili t y Particle position(a) = (0 ° , 15 ° , 0 ° )(b) = (0 ° , 45 ° , 0 ° )(c) = (0 ° , 60 ° , 75 ° )(d) = (75 ° , 60 ° , 0 ° ) FIG. 1: The spread of probability distribution for differentvalue of θ using operator U ,θ, , is wider for (a) = (0 , π , , π , ζ shifts the distribution to right, (c) = (0 , π , π ) and ξ shifts tothe left, (d) = ( π , π , walker to, W ξ,θ,ζ | Ψ in i = 1 √ (cid:0) e iξ cos( θ ) + ie iζ sin( θ ) (cid:1) | i| ψ − i + (cid:0) e − iζ sin( θ ) − ie − iξ cos( θ ) (cid:1) | i| ψ +1 i ] . (4)If ξ = ζ , Eq. (4) has left-right symmetry in the positionprobability distribution, but not otherwise. We thus findthat the generalized SU (2) operator as a quantum coincan bias a quantum walker in spite of the symmetry ofinitial state of the particle. We return to this point below.It is instructive to consider the extreme values of theparameters in the U ξ,θ,ζ . If ξ = θ = ζ = 0, U , , = Z ,the Pauli Z operation, then W ξ,θ,ζ ≡ S and the twosuperposition states, | i and | i , move away from eachother without any diffusion and interference having high σ = N . On the other hand, if θ = π , then U , π , = X ,the Pauli X operation, then the two states cross eachother going back and forth, thereby remaining close toposition i = 0 and hence giving very low σ ≈
0. Thesetwo extreme case are not of much importance, but theydefine the limits of the behavior. Intermediate values ofthe θ between these extremes show intermediate driftsand quantum interference. In Fig. (1) we show the sym-metric distribution of quantum walk at different valuesof θ by numerically evolving the density matrix. Fig. (2)shows the variation of σ with increase in θ for quantumwalk of different number steps with the operator, U ,θ, .The change in the variance for different value of the θ isattributed to the change in the value of C θ , a constant fora given θ , σ = C θ N , Fig. (4). Therefore, starting fromthe Hadamard walk ( θ = π ; ξ = ζ = 0), the variance canbe increased ( θ < π ) or decreased ( θ > π ) respectively.In the analysis of Hadamard walk on the line in [4], it θ σ
50 steps100 steps250 steps500 steps
FIG. 2: A comparison of variation of σ with θ for differentnumber of steps of walk using operator U ,θ, using numericalintegration. is shown that after N steps, the probability distributed isspread over the interval [ − N √ , N √ ] and shrink quickly out-side this region. The moments have been calculated forasymptotically large number of steps N and the varianceis shown to vary as σ ( N ) = (cid:16) − √ (cid:17) N [4].The expression for the variance of the quantum walkusing U ,θ, as a quantum coin can be derived by us-ing the approximate analytical function for the proba-bility distribution P ( i ) that fit the envelop of the quan-tum walk distribution obtained from the numerical inte-gration technique for different values of θ . For a quan-tum walk using U ,θ, as quantum coin, after N stepsthe probability distribution is spread over the interval( − N cos( θ ) , N cos( θ )) [3]. This is also verified by the an-alyzing the distribution obtained using the numerical in-tegration technique. By assuming the value of the prob-ability to be 0 beyond | N cos( θ ) | , the function that fitsthe probability distribution envelop is, Z P ( i ) di ≈ Z N cos( θ ) − N cos( θ ) [1 + cos (2 θ )] e K ( θ ) “ i N θ ) − ” √ N di ≈ , (5)where, K ( θ ) = √ N cos( θ )[1 + cos (2 θ )][1 + sin( θ )] [12].Fig. (3) shows the probability distribution obtained byusing the Eq. (5). The interval ( − N cos( θ ) , N cos( θ ))can be parametrised as a function of φ , i = f ( φ ) = N cos( θ ) sin( φ ) where φ range from − π to π . For a walkwith coin U ,θ, , the mean of the distribution is zero andhence the variance can be analytically obtained by eval-uating, σ ≈ Z N cos( θ ) − N cos( θ ) P ( i ) i di = Z π − π P ( f ( φ ))( f ( φ )) f ′ ( φ ) dφ. (6) −100 −50 0 50 10000.020.040.060.080.10.120.140.160.18 Particle position P r obab ili t y θ = 15 ° θ = 45 ° θ = 60 ° FIG. 3: The probability distribution obtained using using Eq.(5) for different value of θ . The distribution is for 100 steps. θ C θ = σ / N σ /N when η = 0 ° θ ) σ /N when η = 90 ° FIG. 4: Variation of C θ when η = | ξ − ζ | = 0 ◦ from numericalintegration to the function (1 − sin( θ )) to which it fits. Theeffect of maximum biasing, η = 90 ◦ on C θ is also shown andits effect is very small. σ ≈ Z π − π (1 + cos (2 θ )) √ N e K ( θ ) ( sin ( φ ) − ) ( N cos( θ ) sin( φ )) × ( N cos( θ ) cos( φ )) dφ = N (1 − sin( θ )) . (7) σ = C θ N ≈ (1 − sin( θ )) N . (8)We also verify from the results obtained through numer-ical integration that C θ = (1 − sin( θ )), Fig. (4).Setting ξ = ζ in U ξ,θ,ζ introduces asymmetry, biasingthe walker. Positive ζ contributes for constructive inter-ference towards right and destructive interference to theleft, whereas vice versa for ξ . The inverse effect can benoticed when the ξ and ζ are negative. As noted above,for ξ = ζ , the evolution will again lead to the symmetricprobability distribution. Apart from a global phase, onecan show that the coin operator U ξ,θ,ζ ≡ U ξ − ζ,θ, ≡ U ,θ,ζ − ξ . (9)In Fig. (1) we show the biasing effect for ( ξ, θ, ζ ) =(0 ◦ , ◦ , ◦ ) and for (75 ◦ , ◦ , ◦ ). The biasing does notalter the width of the distribution in the position spacebut probability goes down as a function of cos( η ) on oneside and up as a function of sin( η ) on the other side.Where η = | ξ − ζ | . The mean value ¯ i of the distribution,which is zero for U ,θ, , attains some finite value withnon-vanishing η , this contributes for an additional termin Eq. (6), σ ≈ Z N cos( θ ) − N cos( θ ) P ( i )( i − ¯ i ) di. (10)this contributes to a small decrease in the variance of thebiased quantum walker, Fig. (4).It is understood that, obtaining symmetric distribu-tion depends largely on the initial state of the particleand this has also been discussed in [3, 4, 10, 11]. Butusing U ξ,θ,ζ as coin operator, and examining the walkevolution shows how non-vanishing ξ and ζ introducebias. For example, the position probability distributionin Eq. (4) corresponding to the left and right positionsare [1 ± sin(2 θ ) sin( ξ − ζ )], which would be equal, andlead to a symmetric distribution, if and only if ξ = ζ .The evolution of the state after n steps, [ W ξ,θ,ζ ] n | Ψ in i is | Ψ( n ) i = n X m = − n ( A m,n | i| ψ m i + B m,n | i| ψ m i ) (11)and proceeds according to the iterative relations, A m,n = e iξ cos θA m − ,n − + e iζ sin θB m − ,n − (12a) B m,n = e − iζ cos θA m − ,n − − e − iξ sin θB m − ,n − . (12b)A little algebra reveals that the solutions A m,n and B m,n to Eqs. (12) can be decoupled (after the initialstep) and shown to satisfy A m,n +1 − A m,n − = cos θ ( e iζ A m − ,n − e iξ A m +1 ,n )(13a) B m,n +1 − B m,n − = cos θ ( e iξ B m − ,n − e iζ B m +1 ,n ) . (13b)For spatial symmetry from an initially symmetric super-position, the walk should be invariant under an exchangeof labels 0 ↔
1, and hence should evolve A m,n and B m,n alike (as in the Hadamard walk [14]). From Eq. (13), wesee that this happens if and only if ξ = ζ . IV. ENTROPY OF MEASUREMENT
As an alternative measure of position fluctuation tovariance, we consider the Shannon entropy of the walker m ea s u r e m en t en t r op y , H ( i ) θ
50 steps100 steps250 steps500 steps
FIG. 5: Variation of entropy of measurement H ( i ) with θ fordifferent number of steps. The decrease in H ( i ) is not drastictill θ is close to 0 or π . position probability distribution p i obtained by tracingover the coin basis: H ( i ) = − X i p i log p i . (14)The quantum walk with a Hadamard coin toss, U , π , ,has the maximum uncertainty associated with the prob-ability distribution and hence the measurement entropyis maximum. For ξ = ζ = 0 and low θ , operator U ,θ, is almost a Pauli Z operation, leading to localization ofwalker at ± N . At θ close to π , with ξ = ζ = 0, U ap-proaches Pauli X operation, leading to localization closeto the origin, and again, low entropy. However, as θ ap-proaches π , the splitting of amplitude in position spaceincreases towards the maximum. The resulting enhanceddiffusion is reflected in the relatively large entropy at π ,as seen in Fig. (5). Fig. (5) is the measurement entropywith variation of θ in the coin U ,θ, for different num-ber of steps of quantum walk. The decrease in entropyfrom the maximum by changing θ on either side of π isnot drastic untill the θ is close to 0 or π . Therefore formany practical purposes, the small entropy can be com-pensated for by the relatively large C θ , and hence σ . Formany other purposes, such as mixing of quantum walkon an n -cycle Cayley graph, it is ideal to adopt a lowervalue of θ . The effect of ξ and ζ on the measuremententropy is of very small magnitude. These parametersdo not affect the spread of the distribution and the vari-ation in the height reduces the entropy by a very smallfraction. −50 0 509.89.859.99.951010.05 x 10 −3 P r obab ili t y Particle position θ = 15 ° θ = 45 ° θ = 75 ° FIG. 6: A comparison of mixing time M of the probabilitydistribution of a quantum walker on a n -cycle for differentvalue of θ using coin operation U ,θ, , where n , the numberof position, is 101. Mixing is faster for lower value of θ . Thedistribution is for 200 cycles. V. QUANTUM WALK ON THE n -CYCLE ANDMIXING TIME The n -cycle is the simplest finite Cayley graph with n vertices. This example has most of the features of thewalks on the general graphs. The classical random walkapproaches a stationary distribution independent of itsinitial state on a finite graph. Unitary (i.e., non-noisy)quantum walk, does not converge to any stationary dis-tribution. But by defining a time-averaged distribution, P ( i, T ) = 1 T T − X t =0 P ( i, t ) , (15)obtained by uniformly picking a random time t between 0and ( T − t time steps and measuringto see which vertex it is at, a convergence in the prob-ability distribution can be seen even in quantum case.It has been shown that the quantum walk on an n -cyclemixes in time M = O ( n log n ), quadratically faster thanthe classical case which is O ( n ) [13]. From Eq. (6) weknow that the quantum walk can be optimized for max-imum variance and wide spread in position space, be-tween ( − N cos( θ ) , N cos( θ )) after N steps. For a walkon an n -cycle, choosing θ slightly above 0 would give the maximum spread in the cycle during each cycle. Maxi-mum spread during each cycle distributes the probabilityover the cycle faster and this would optimize the mixingtime. Thus optimizing mixing time with lower value of θ can in general be applied to most of the finite graphs.For optimal mixing time, it turns out to be ideal to fix ξ = ζ in U ξ,θ,ζ , since biasing impairs a proper mixing.Fig. (6) is the time averaged probability distribution ofa quantum walk on an n -cycle graph after n log n timewhere n is 101. It can be seen that the variation of theprobability distribution over the position space is leastfor θ = 15 ◦ compared to θ = 45 ◦ and θ = 75 ◦ . VI. QUANTUM WALK SEARCH
A fast and wide spread defines the effect of the searchalgorithm. For the basic algorithm using discrete timequantum walk, two quantum coins are defined, one fora marked vertex and the other for an unmarked vertex.The three parameter of the SU (2) quantum coin can beexploited for an optimal search. VII. CONCLUSION
In this paper we have generalized the Hadamard walkto a general discrete time quantum walk with a SU (2)coin. We conclude that the variance of quantum walk canbe optimized by choosing low θ without loosing much onmeasurement entropy. The parameters ξ and ζ intro-duce asymmetry in the position space probability distri-bution starting even from an initial symmetric superpo-sition state. This asymmetry in the probability distri-bution is similar to the distribution obtained for a walkon a particle initially in a non-symmetric superpositionstate. Optimization of quantum search and mixing timeon an n -cycle using low θ is possible. The combination ofthe parameters of the SU (2) coin and the measuremententropy can be optimized to fit the physical system andfor the relevant applications of the quantum walk on ageneral graph. Acknowledgement
CMC would like to thank the Mike and Ophelia Lazaridisfellowship for support. CMC and RL also acknowl-edge the support from CIFAR, NSERC, ARO/LPS grantW911NF-05-1-0469, and ARO/MITACS grant W911NF-05-1-0298. [1] Y. Aharonov, L. Davidovich and N. Zagury, Phys. Rev.A , 1687, (1993).[2] J. Kempe, Contemp. Phys. , 307 (2003).[3] Ashwin Nayak and Ashvin Vishwanath, Technical Reportquant-ph/0010117, Oct. 2000. [4] Andris Ambainis et al. , Proc. 33rd STOC , pages 60-69,New York, NY, 2001. ACM.[5] A. M. Childs et al. , in
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