Decoherence on Szegedy s Quantum Walk

Transcription

1 Decoherence on Szegedy s Quantum Walk Raqueline A. M. Santos, Renato Portugal. Laboratório Nacional de Computação Científica (LNCC) Av. Getúlio Vargas 333, , Petrópolis, RJ, Brazil raqueline@lncc.br, portugal@lncc.br. Abstract: Quantum walks have been used for developing quantum algorithms that outperform their classical analogues. The quantum hitting time plays an important role as the stopping time for quantum walk based algorithms that search marked elements. It is known that experimental implementations of quantum systems face decoherence problems. The interactions with the environment will possibly destroy or reduce the quantum coherence. Thus, it is important to analyze the effects of decoherence on quantum walks. In this work we use a decoherence model that is inspired on percolation. We analyze this decoherence model on Szegedy s quantum walk. By performing averages over all possible evolution operators affected by the decoherence we show that it is possible to define a decoherent quantum hitting time. Keywords: quantum walks, decoherence, quantum hitting time 1 Introduction In Computer Science, random walks or Markov chains are used in randomized algorithms, specially in search algorithms that search a marked vertex in a graph. The expected time to reach a vertex for the first time, known as hitting time, plays an important role in those algorithms as the running time to find a solution. For instance, we can see applications for the k-sat and the graph connectivity problem [15]. Quantum walks or quantum Markov chains are the quantum analogue of classical random walks. They are obtained through a process of quantization: the state of the quantum system is described by a vector on a Hilbert space and the system s evolution is governed by a unitary operator if the system is totally isolated from interactions with the world around it. There are discrete and continuous time quantum walks. Both have been used for developing quantum algorithms that outperform their classical versions [3, 20, 4, 7]. We can see two different formalisms for the discrete time quantum walks. The first one, introduced by Aharonov et al. [1], adds an additional space that is related to the coin. The second formalism was developed by Szegedy [21] and it is described by reflection operators in an associated bipartite graph obtained from the original one by a process of duplication. Szegedy [21] showed that the quantum hitting time has a quadratic improvement over the classical one to detect a set of marked vertices for ergodic and symmetric Markov chains. Santos and Portugal [18] showed that this quadratic improvement remains valid to finding a vertex, in a set of marked vertices, on the complete graph. Based on Szegedy s work, Magniez et al. [14] developed a quantum algorithm to finding a marked vertex on reversible, state-transitive Markov chains restricted to the case of only one marked vertex. Recently, Krovi et al. [11] showed that the quadratic speed-up to finding a marked vertex also holds for any reversible Markov chain using a new interpolating algorithm. When implementing quantum systems, there is no doubt we must face decoherence problems. As the system may not be completely isolated, interactions with the environment are possible and can destroy or reduce the quantum coherence. This generally undesired effect might occurs to quantum walks as well. In this way, it is crucial to understand how the decoherence affects them. 618

2 The decoherence is generally modeled as a non-unitary evolution of the quantum walk. This can be achieved by adding an extra non-unitary operation (a measure operator, for example), or we can change the coin or shift operators by non-unitary operators. In the literature, various works study the influence of decoherence over the different models of quantum walks. Brun et al. [5] showed how the coined quantum walk behaves as a classical random walk by the decoherence on the coin operator. Kendon e Tregenna [9] studied the computational consequences of the decoherence on the coin. Romanelli et al. [17] worked on the one-dimensional quantum walk and they considered the possibility of having broken links between the vertices. This technique was later generalized for the bidimensional case by Oliveira et al. [16]. Alagic e Russel [2] analyzed the effect of making independent measures on the continuos time quantum walk on the hypercube. A review on decoherence on quantum walks can be seen in [8]. The decoherence on Szegedy s formalism was studied by Chiang and Gomez [6], who analyzed the sensibility to perturbation due to system s precision limitations by adding a symmetric matrix E, representing the noise, to the transition probability matrix of the graph. The quadratic speed-up vanishes when the magnitude of the noise E Ω(δ(1 δǫ)), where δ is the spectral gap of the transition probability matrix of the graph and ǫ is the ratio between the number of marked vertices and the number of vertices of the graph. In this context, we propose a new model of decoherence on Szegedy s quantum walk. Our decoherence model is inspired on percolattion graphs. The Refs. [12, 13] analyzed the behavior of the coined quantum walk on percolation lattices. In this case, we have edges or vertices randomly missing on the graph. In our case, the dynamics acts different from Refs. [12, 13, 17, 16] because, at each time step, we can introduce defects in the graph whether by the introduction of new edges or by breaking the links between two vertices. By performing averages over all possible evolution operators affected by the decoherence we are able to define a decoherent quantum hitting time that comes naturally from the definition without decoherence. The paper is organized as follows. In Sec. 2 we review Szegedy s quantum walk and the definition for the quantum hitting time. In Sec. 3 we analyze our decoherence model on Szegedy s quantum walk. In Sec. 4 we draw the conclusions. 2 Szegedy s Quantum Walk Szegedy [21] has proposed a quantum walk driven by reflection operators in an associated bipartite graph obtained from the original one by a process of duplication. Let Γ(X,E) be a connected, undirected and non-bipartite graph, where X is the set of vertices and E is the set of edges. The stochastic matrix P associated with this graph is defined such that p xy is the inverse of the outdegree of the vertex x. Define a bipartite graph associated with Γ(X,E) through a process of duplication. X and Y are the sets of vertices of same cardinality of the bipartite graph. Each edge {x i,x j } in E of the original graph Γ(X,E) is converted into two edges in the bipartite graph {x i,y j } and {y i,x j }. To define a quantum walk in the bipartite graph, we associate with the graph a Hilbert space H n2 = H n H n, where n = X = Y. The computational basis of the first component is { x : x X } and of the second { y : y Y }. The computational basis of H n2 is { x,y : x X,y Y }. The quantum walk on the bipartite graph is defined by the evolution operator U P given by where U P := R B R A, (1) R A = 2AA T I n 2, (2) R B = 2BB T I n 2. (3) 619

3 The operators A : H n H n2 and B : H n H n2 are defined as follows A = x X B = y Y Φx x, (4) Ψy y, (5) where Φx = x y Y pxy y, (6) Ψy = ( x X pyx x ) y. (7) In the bipartite graph, an application of U P corresponds to two quantum steps of the walk, from X to Y and from Y to X. We have to take the partial trace over the space associated with Y to get the state on the set X. 2.1 Quantum Hitting Time Instead of using the stochastic matrix P, Szegedy defined the quantum hitting time by using a modified evolution operator U P associated with a modified stochastic matrix P, that is given by { p xy = pxy, x M; (8) δ xy, x M. M is the set of marked vertices. The initial condition of the quantum walk is ψ(0) = 1 n x X y Y pxy x,y. (9) Note that ψ(0) is an eigenvector of UP with eigenvalue 1. However, ψ(0) is not an eigenvector of U P in general. Definition 2.1 [21] The quantum hitting time H P,M of a quantum walk with evolution operator U P given by Eq. (1) and initial condition ψ(0) is defined as the least number of steps T such that F(T) 1 m n, (10) where m is the number of marked vertices, n is the number of vertices of the original graph and F(T) is F(T) = 1 U t ψ(0) P ψ(0) 2, (11) where U t P is the evolution operator after t steps using the modified stochastic matrix. The quantum hitting time was calculated analytically for the complete graph and the cycle [18, 19]. The graphs in Fig. 1 show the behavior of the function F(T) for the complete graph and the cycle when n = 100 and m = 15. F(T) grows rapidly through the dashed line 1 m n, then oscillates around its limiting value. The quantum hitting time can be seen in the graph at time T such that F(T) = 1 m n. 620

4 F(T) 1-m/n F(T) 1-m/n 2,0 2,0 1,5 1,5 1,0 1,0 0,5 0, T (a) Complete graph H P,M T (b) Cycle H P,M Figure 1: Graphs of the function F(T) (solid line) and 1 m n graph and the cycle when n = 100 and m = 15. (dashed line) for the complete 3 Decoherence on the Quantum Hitting Time According to Kesten [10], percolation is a simple probabilistic model which exhibits a phase transition. Consider a 2D lattice, for example, which we view as a graph with edges between neighboring vertices. All edges are, independently of each other, chosen to be open (the edge exists) with probability p and closed (the edge is missing) with probability 1 p. A basic question in this model is What is the probability that there exists an open path, i.e., a path all of whose edges are open, from the origin to a destination vertex in the graph? Percolation can be generalized to percolation on any graph and we can consider site percolation, when we have vertices randomly missing, or bond percolation, for the case of edges randomly missing. Our decoherence model is inspired on percolation graphs. The dynamics of the proposed model acts different because we consider that changes on the graph can occur at each time step, due to insertion or removal of edges between two vertices. The link between two vertices of the graph has a fixed probability, p, of being removed or created. These modifications on the topology of the graph leads to changes on the transition probability matrix associated to the graph, which eventually modifies the evolution operator. Therefore, instead of having an usual walk evolving as ψ(t) = UP t ψ(0), now we have, ψ(t) = UPt U Pt 1 U P1 ψ(0) =: U Pt ψ(0). (12) where P t = {P 1,...,P t 1,P t } and U Pt = U Pt U Pt 1 U P1. P i s are not necessarily equal. In this context, it is useful to define an operator that will gather the behavior of the operators affected by the decoherence. Then, let Ū dec := Pr(P)U P, (13) P be the operator obtained by doing an average over all possible evolution operators affected or not by the decoherence. The following result show that the average over all possible sequences P, with size T, according to its probability distribution, is equal to ŪT dec. Lemma 3.1 Consider t T, then P T Pr( P T )U Pt = Ūt dec. (14) 621

5 Proof Since Pr( P T ) = T i=1 Pr(P i), we have, Pr( T P T )U Pt U Pt 1 U P1 = Pr(P i )U Pt U Pt 1 U P1 P T P T i=1 = ( T ) Pr(P i ) U Pt U Pt 1 U P2 Pr(P 1 )U P1 P T P T 1 P 2 i=2 P1 = Pr(P T )Pr(P T 1 ) Pr(P t+1 )Ūt dec P T P T 1 P t+1 = Ūt dec In order to define the quantum hitting time for the evolution with decoherence we should make an average over all possible sequences P. Define, F dec (T) := ( Pr( P 1 ) T ) U Pt ψ(0) ψ(0) 2. (15) PT Now we prove that F dec (T) is equivalent to F(T) of Eq. (11) when the evolution operator is Ū dec. Theorem 3.2 Proof F dec (T) = 1 Ūt dec ψ(0) ψ(0) 2. (16) F dec (T) = ( Pr( 1 ) P T ) U Pt ψ(0) ψ(0) 2 PT = ( Pr( 1 ( P T ) 2 2 ψ(0) )) U Pt ψ(0) PT = ψ(0) Pr( P T )U Pt ψ(0) PT (17) By Lemma 3.1, we have F dec (T) = 1 = 1 ( ) 2 2 ψ(0) Ū t ψ(0) dec Ūt dec (18) ψ(0) ψ(0) 2. The occurrence probability of a given P i is determined as follows. If 0 < p < 1, then Pr(P i ) = (1 p) ac a dp a d, where a c = n(n 1) 2 is the number of edges of the complete graph with n vertices and a d is the number of edges removed plus the number of edges included to obtain P i from P. If p = 0, Pr(P i = P ) = 1, and Pr(P i P ) = 0. And, if p = 1, we have Pr(P i = P ) = 1, and Pr(P i P ) = 0, where P is the complement of P. Now, we can naturally define the quantum hitting time with decoherence, using the expression of F dec obtained in Theorem

6 Definition 3.3 The quantum hitting time H dec P,M of a quantum walk with evolution operator U P given by Eq. (1) and initial condition ψ(0) is defined as the least number of steps T such that F dec (T) 1 m n. (19) We notice that when p = 0, we have the original definition, since Ūdec = U P. 4 Conclusions We have proposed a new model of decoherence on quantum walks inspired on percolation graphs. This model is characterized by the possibility of insertion or removal of edges at each time step. By applying this model on Szegedy s quantum walk we can notice that the probability matrix of the graph can be changed at each time step and, consequently, the evolution operator. Thus, the state of the walker on a given instant of time is obtained by the application of possible different evolution operators to the initial state. We were able to define a decoherent hitting time by using a new operator that is obtained by performing an average over all possible evolution operators affected or not by the decoherence. Future works might analyze the behavior of the decoherent quantum hitting time, whether numerically or algebraically, verifying the consequences on the speed-up obtained by the version without decoherence. Acknowledgments We thank F. Marquezino for fruitful discussions. R.A.M. Santos acknowledges a CAPES fellowship and R. Portugal acknowledges CNPq. References [1] Y. Aharonov, L. Davidovich, and N. Zagury. Quantum random walks. Physical Review A, 48(2): , [2] G. Alagic and A. Russell. Decoherence in quantum walks on the hypercube. Physical Review A, [3] A. Ambainis. Quantum walk algorithm for element distinctness. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, [4] A. Ambainis, J. Kempe, and A. Rivosh. Coins make quantum walks faster. In Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms, pages , [5] T. A. Brun, H. A. Carteret, and A. Ambainis. Quantum to classical transition for random walks. Physical Review Letters, 91(130602), [6] C.-F. Chiang and G. Gomez. Hitting time of quantum walks with perturbation. Quantum Information Processing, 2012, [7] A. Childs and J. Goldstone. Spatial search by quantum walk. Physical Review A, 70(022314), [8] V. Kendon. Decoherence in quantum walks - a review. Mathematical Structures in Computer Science, 17(6): , [9] V. Kendon and B. Tregenna. Decoherence can be useful in quantum walks. Physical Review A, 67(042315),

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