Staggered quantum walks with superconducting microwave resonators

Abstract

The staggered quantum walk model on a graph is defined by an evolution operator that is the product of local operators related to two or more independent graph tessellations. A graph tessellation is a partition of the set of nodes that respects the neighborhood relation. Flip-flop coined quantum walks with the Hadamard or Grover coins can be expressed as staggered quantum walks by converting the coin degree of freedom into extra nodes in the graph. We propose an implementation of the staggered model with superconducting microwave resonators, where the required local operations are provided by the nearest neighbor interaction of the resonators coupled through superconducting quantum interference devices. The tunability of the interactions makes this system an excellent toolbox for this class of quantum walks. We focus on the one-dimensional case and discuss its generalization to a more general class known as triangle-free graphs.Quantum walks are the quantum generalization of random walks, and form the building blocks in designing quantum search algorithms outperforming the similar classical ones [1]. The two main paradigms in this respect are the coined discrete-time quantum walk (DTQW) [2] and the continuoustime quantum walk (CTQW) [3]. In one-dimensional (1D) DTQWs, a two-level quantum system works as a coin, whose quantum property to exist in a superposition of states gives the distinct ballistic spreading of the walker encoded in a set of discrete states. In CTQWs, it is the excitation exchange between the neighboring sites, in a lattice, that directly works as a walker without the need of a coin. Typically a tight-biding Hamiltonian followed by a linear coupling between excitations in bosonic modes suffices to implement the CTQW model, making its implementation convenient (See Ref. [4] for an example with nanomechanical resonators). However, when the data structure is a lattice with dimension less than four, search algorithms based on the standard CTQW do not outperform the classical algorithms based on random walks [5].Recently, a general class of coinless discrete-time quantum walks was proposed-the staggered quantum walk (SQW) [6][7][8], which includes the quantum walks studied in Refs. [9,10] as particular cases. This model also includes as particular cases the flip-flop coined DTQWs with Hadamard and Grover coins and the entire Szegedy's quantum walk model [11]. In the language of graph theory, the required unitary operators (not Hamiltonians) can be obtained by a graphical method based on graph tessellations. A tessellation is a partition of the set of nodes into cliques; that is, each element of the partition is a clique. A clique is a subgraph that is complete, namely, all nodes of a clique are neighbors.We have proposed an extension of the SQW model, called SQW with Hamiltonians [12], which uses the graph tessellations to define local Hamiltonians instead of the local unitary evolution operators. The extended model includes the quantum walks analyzed in Ref.[13] as particular cases. The SQW with H...

Spatial Search via Memoryless Walk with Selfloop

“…They also have a simple structure, alternating two or more non-commuting reflections that can be derived from tessellations of the underlying graph. A proposal for implementing memoryless walks using superconducting microwave resonators is given by [MdOP17], and an implementation of memoryless walks on IBM quantum computers is in [AAMP20]. Staggered walks are a class of memoryless walks and were defined by Portugal et al in [PSFG16].…”

Discretization of continuous-time quantum walks via the staggered model with Hamiltonians

“…This model is endowed with many interesting features, remarkably, it provides a generalization of the Szegedy's quantum walk model [24] and exactly simulates the instances of flip-flop coined quantum walks that employ the Hadamard or Grover coins [19]. An implementation of this model in the class of triangle-free graphs has also been recently proposed [17].…”