Lackadaisical quantum walks with multiple marked vertices
Raina bulv. 19, Riga, LV-1586, Latvia
The concept of lackadaisical quantum walk – quantum walk with self loops – was first introduced for discrete-time quantum walk on one-dimensional line . Later it was successfully applied to improve the running time of the spacial search on two-dimensional grid .
In this paper we study search by lackadaisical quantum walk on the two-dimensional grid with multiple marked vertices. First, we show that the lackadaisical quantum walk, similarly to the regular (non-lackadaisical) quantum walk, has exceptional configuration, i.e. placements of marked vertices for which the walk has no speed-up over the classical exhaustive search. Next, we demonstrate that the weight of the self-loop suggested in  is not optimal for multiple marked vertices. And, last, we show how to adjust the weight of the self-loop to overcome the aforementioned problem.
Quantum walks are quantum counterparts of classical random walks . Similarly to classical random walks, there are two types of quantum walks: discrete-time quantum walks (DTQW), introduced by Aharonov et al. , and continuous-time quantum walks (CTQW), introduced by Farhi et al. . For the discrete-time version, the step of the quantum walk is usually given by two operators – coin and shift – which are applied repeatedly. The coin operator acts on the internal state of the walker and rearranges the amplitudes of going to adjacent vertices. The shift operator moves the walker between the adjacent vertices.
Quantum walks have been useful for designing algorithms for a variety of search problems. To solve a search problem using quantum walks, we introduce the notion of marked elements (vertices), corresponding to elements of the search space that we want to find. We perform a quantum walk on the search space with one transition rule at the unmarked vertices, and another transition rule at the marked vertices. If this process is set up properly, it leads to a quantum state in which the marked vertices have higher probability than the unmarked ones. This method of search using quantum walks was first introduced in  and has been used many times since then.
Most of the papers studying quantum walks consider a search space containing a single marked element only. However, in contrary of classical random walks, the behavior of the quantum walk can drastically change if the search space contains more that one marked element. Ambainis and Rivosh  have studied DTQW on two-dimensional grid and showed that if the diagonal of the grid is fully marked then the probability of finding a marked element does not grow over time. Wong  analyzed the spatial search problem by CTQW on the simplex of complete graphs and showed that the placement of marked vertices can dramatically influence the required jumping rate of the quantum walk. Wong and Ambainis  analysed DTQW on the simplex of complete graphs and showed that if one of the complete graphs is fully marked then there is no speed-up over classical exhaustive search. Nahimovs and Rivosh [6, 5] studied DTQW on two-dimensional grid for various placements of multiple marked vertices and proved several gaps in the running time of the walk (depending on the placement of marked vertices). Additionally the authors have demonstrated placements of a constant number of marked vertices for which the walk have no speed-up over classical exhaustive search. They named such placements exceptional configurations. Nahimovs and Santos  have extended their work to general graphs.
The concept of lackadaisical quantum walk (quantum walk with self loops) was first studied for DTQW on one-dimensional line [8, 13]. Later on, Wong showed an example of how to apply the self-loops to improve the DTQW based search on the complete graph  and two-dimensional grid . The running time of the lackadaisical walk heavily depends on a weight of the self-loop. Saha et al. showed that the weight suggested by Wong for two-dimensional grid with a single marked vertex is not optimal for multiple marked vertices. They have demonstrated that for a block of marked vertices one should use the weight .
In this paper, we study search by discrete-time lackadaisical quantum walk on two-dimensional grid with multiple marked vertices. First, we show that the lackadaisical quantum walk, similarly to the regular (non-lackadaisical) quantum walk, has exceptional configuration, i.e. placements of marked vertices for which the walk have no speed-up over the classical exhaustive search. Next, we study an arbitrary placement of marked vertices and demonstrate that the weight suggested by Wong is not optimal for multiple marked vertices. The same holds for the weight suggested by Saha et al., which seems to work only for a block of marked vertices. Last, we analyze how to adjust the weight to overcome the aforementioned problem. We propose two better constructions – and – and discuss their boundaries of application.
2 Quantum walk on the two-dimensional grid
2.1 Regular (non-lackadaisical) quantum walk
Consider a two-dimensional grid of size with periodic (torus-like) boundary conditions. The locations of the grid are labeled by the coordinates for . The coordinates define a set of state vectors, , which span the Hilbert space associated with the position. Additionally, we define a 4-dimensional Hilbert space , spanned by the set of states , associated with the direction. We refer to it as the coin subspace. The Hilbert space of the quantum walk is .
The evolution of a state of the walk (without searching) is driven by the unitary operator , where is the flip-flop shift operator
and is the coin operator, given by the Grover’s diffusion transformation
The system starts in
which is uniform distribution over vertices and directions. Note, that this is a unique eigenvector of with eigenvalue .
To use quantum walk for search, we extend the step of the algorithm with a query to an oracle, making the step
Here is the query transformation which flips the sign at a marked vertex, irrespective of the coin state. Note that is a 1-eigenvector of but not of . If there are marked vertices, the state of the algorithm starts to deviate from . In case of a single marked vertex, after steps the inner product becomes close to . If one measures the state at this moment, he will find the marked vertex with probability . With amplitude amplification this gives the total running time of steps.
2.2 Lackadaisical quantum walk
In case of lackadaisical quantum walk the coin subspace of the walk is 5-dimensional Hilbert space spanned by the set of states . The Hilbert space of the quantum walk is .
The shift operator acts on a self loop as
The coin operator is
The system starts in
which is uniform distribution over vertices, but not directions. As before is a unique 1-eigenvector of .
The step of the search algorithm is . As it is shown in , in case of a single marked vertex, for the weight , after steps the inner product becomes close to . If one measures the state at this moment, he will find the marked vertex with probability, which gives improvement over the loopless algorithm.
3 Stationary states of the lackadaisical quantum walk
In this section we will show that the lackadaisical quantum walk, similarly to the regular (non-lackadaisical) quantum walk, has exceptional configurations, i.e. placements of marked vertices for which the walk have no speed-up over the classical exhaustive search.
Consider a state . Similarly one can define states , and . The defined states are orthogonal to . Consider an effect of the coin transformation on :
As one can see, the coin transformation inverts a sign of the state.
Now, consider a two-dimensional grid with two marked vertices and . Let be a state where the coin part of all unmarked vertices is , the coin part of is and the coin part of is (see Fig. 1), that is,
We claim that this state is not changed by a step of the algorithm.
Consider a grid of size with two adjacent marked vertices and . Then the state , given by Eq. (10), is not changed by the step of the algorithm, that is, .
Consider the effect of a step of the algorithm on . The query transformation flips the sign of marked vertices. The coin transformation has no effect on but flips the signs of and . Thus, does not change the amplitudes of unmarked vertices and twice flips the signs of amplitudes of marked vertices. Therefore, we have The shift transformation swaps the amplitudes of near-by vertices. For , it swaps with and with . Thus, we have . ∎
The initial state of the algorithm, given by Eq. (9), can be written as
for . The only part of the initial state which is changed by the step of the algorithm is
Let us establish an upper bound on the probability of finding a marked vertex.
Consider a grid of size with two adjacent marked vertices and . Then for any number of steps, the probability of finding a marked vertex is .
We have . The only part of the initial state changed by the step of the algorithm is . The basis states and have the biggest amplitudes of in the stationary state. Therefore, the maximum probability of finding a marked vertex is reached if the state becomes
for . Thus, is at most
Since the evolution is unitary, we have . Due to symmetry and should be equal, so the expression (14) reaches the maximum when .
We have and . Each of summands in the expression (14) is and, therefore, we have . ∎
That is the probability of finding a marked vertex is of the same order as for the classical exhaustive search.
Note that if we have a block of marked vertices we can construct a stationary state as long as we can tile the block by the sub-blocks of size and . For example, consider for . Then the stationary state is given by
For more details on constructions of stationary states for blocks of marked vertices on two-dimensional grid see . The paper focuses on the non-lackadaisical quantum walk, nevertheless, the results can be easily extended to the lackadaisical quantum walk.
4 Optimality of for multiple marked vertices
In  Wong showed that in case of a single marked vertex, for the weight , after steps the inner product becomes close to . If one measures the state at this moment, he will find the marked vertex with probability111The numerical results in  show that probability of finding a marked vertex is close to and approaches then goes to infinity.. The suggested value of , however, is optimal for a single marked vertex only. Saha et al. studied search for a block of marked vertices and showed that optimal weight in this setting is .
In this section we study search for an arbitrary placement of multiple marked vertices. The presented data is obtained from numerical simulations. The values listed in the tables are calculated in the following way. The number of steps of the algorithm is the smallest for which reaches 0 (becomes negative). By the probability we mean the probability of finding a marked vertex when is measured.
Tables 1 and 2 give the number of steps and the probability of finding a marked vertex for random placements of and marked vertices on grid for . As one can see the probability of finding a marked vertex is no more close to as it is for a single marked vertex.
|(0, 0), (23, 27)||153||0.586377681077719|
|(0, 0), (35, 68)||150||0.591030741055657|
|(0, 0), (30, 69)||151||0.588384716869901|
|(0, 0), (42, 4)||152||0.590451037529614|
|(0, 0), (84, 60)||151||0.584982804352049|
|(0, 0), (34, 52), (93, 53)||117||0.440756928151790|
|(0, 0), (26, 12), (22, 32)||126||0.434581723157292|
|(0, 0), (40, 94), (13, 62)||119||0.430688837061525|
|(0, 0), (7, 44), (7, 98)||131||0.430029225026132|
|(0, 0), (80, 78), (28, 31)||118||0.454915029501263|
Table 3 shows the number of steps and the probability for grid with the set of marked vertices
for weights of a self-loop suggested by Wong (the 2nd and the 3rd columns) and by Saha et al. (the last two columns). As one can see for both weights the probability goes down with the number of marked vertices.
We tried to adjust the value of to increase the probability of finding a marked vertex. We searched for a better value of in the form . The figures 2 and 3 show the probability of finding a marked vertex for grid with the sets of marked vertices and , respectively, for different values of .
As one can see the optimal value of for is close to and for is close to . The similar results were obtained for bigger grids with larger sets of marked vertices. Table 4 gives the optimal value of and the corresponding number of steps and the probability for grid with the set of marked vertices .
This raises a conjecture that the optimal weight of a self loop is . The table 5 shows the number of steps and the probability for grid with the set of marked vertices for the weight (the 2nd and the 3rd columns) and (the last two columns). As one can see results in a high probability of finding a marked vertices for a small number of marked vertices, however, the probability goes down with the number of marked vertices. On the other hand, gives a modest probability for a small number of marked vertices, but the probability grows with the number of marked vertices (and, moreover, seems to tend to a constant). Therefore, we would suggest to use the last of the proposed value of , especially for bigger grids and large number of marked vertices.
It worth noting, that the found values of result in high probability not only for sets of marked vertices, but work equivalently well for other placements of marked vertices, including a random placement.
In this paper, we have demonstrated the existence of exceptional configurations of marked vertices for search by lackadaisical quantum walk on two-dimensional grid. We also numerically showed that weights of the self-loop , suggested by previous papers [16, 11], are not optimal for multiple marked vertices (both weight seems to work in specific cases only). We proposed two values of resulting in a much higher probability of finding a marked vertex than previously suggested weights. Moreover, for the found values, the probability of finding a marked vertex does not decrease with number of marked vertices.
-  Y. Aharonov, L. Davidovich, and N. Zagury. Quantum random walks. Physical Review A, 48(2):1687–1690, 1993.
-  A. Ambainis, J. Kempe, and A. Rivosh. Coins make quantum walks faster. In Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms, pages 1099–1108, 2005.
-  A. Ambainis and A. Rivosh. Quantum walks with multiple or moving marked locations. In Proceedings of SOFSEM, pages 485–496, 2008.
-  E. Farhi and S. Gutmann. Quantum computation and decision trees. Physical Review A, 58:915–928, 1998.
-  N. Nahimovs and A. Rivosh. Exceptional configurations of quantum walks with Grover’s coin. In Proceedings of MEMICS, pages 79–92, 2015.
-  N. Nahimovs and A. Rivosh. Quantum walks on two-dimensional grids with multiple marked locations. In Proceedings of SOFSEM 2016, volume 9587, pages 381–391, 2016. arXiv:quant-ph/150703788.
-  N. Nahimovs and R.A.M. Santos. Adjacent vertices can be hard to find by quantum walks. In Proceedings of SOFSEM 2017, volume 10139, pages 256–267, 2017.
-  Inui Norio, Konno Norio, and Segawa Etsuo. One-dimensional three-state quantum walk. Physical Review E. Statistical Nonlinear and Soft Matter Physics, 72(5 Pt 2):168â–191, 2005.
-  R. Portugal. Quantum walks and search algorithms. Springer, New York, 2013.
-  D. Reitzner, D. Nagaj, and V. Buzek. Quantum walks. In Acta Physica Slovaca 61, volume 6, pages 603–725, 2011. arxiv.org/abs/1207.7283.
-  A. Saha, R. Majumdar, D. Saha, A. Chakrabarti, and S. Sur-Kolay. Search of clustered marked states with lackadaisical quantum walks. ArXiv e-prints, 2018. arXiv:1804.01446.
-  N. Shenvi, J. Kempe, and K. B. Whaley. A quantum random walk search algorithm. Physical Review A, 67(052307), 2003.
-  M. Stefanak, I. Bezdekova, and I Jex. Limit distributions of three-state quantum walks: the role of coin eigenstates. Physical Review A, 90(1):124–â129, 2014.
-  T. G. Wong. Grover search with lackadaisical quantum walks. Journal of Physics A Mathematical General, 48, 2015.
-  T. G. Wong. Spatial search by continuous-time quantum walk with multiple marked vertices. Quantum Information Processing, 15(4):1411–1443, 2016.
-  T. G. Wong. Faster search by lackadaisical quantum walk. Quantum Information Processing, 17, 2018.
-  T. G. Wong and A. Ambainis. Quantum search with multiple walk steps per oracle query. Phys. Rev. A, 92, Aug 2015.