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arXiv:2302.07581 · Physica Scripta

Quantum walk state transfer on a hypercube

M. Štefaňák, S. Skoupý

Published Aug 24, 2023• 5 citations
arXivHTMLPDF

Abstract

Abstract We investigate state transfer on a hypercube by means of a quantum walk where the sender and the receiver vertices are marked by a weighted loops. First, we analyze search for a single marked vertex, which can be used for state transfer between arbitrary vertices by switching the weighted loop from the sender to the receiver after one run-time. Next, state transfer between antipodal vertices is considered. We show that one can tune the weight of the loop to achieve state transfer with high fidelity in shorter run-time in comparison to the state transfer with a switch. Finally, we investigate state transfer between vertices of arbitrary distance. It is shown that when the distance between the sender and the receiver is at least 2, the results derived for the antipodes are well applicable. If the sender and the receiver are direct neighbours the evolution follows a slightly different course. Nevertheless, state transfer with high fidelity is achieved in the same run-time.

DOI
10.1088/1402-4896/acf3a2

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Abstract We investigate state transfer on a hypercube by means of a quantum walk where the sender and the receiver vertices are marked by a weighted loops. First, we analyze search for a single marked vertex, which can be used for state transfer between arbitrary vertices by switching the weighted loop from the sender to the receiver after one run-time. Next, state transfer between antipodal vertices is considered. We show that one can tune the weight of the loop to achieve state transfer with high fidelity in shorter run-time in comparison to the state transfer with a switch. Finally, we investigate state transfer between vertices of arbitrary distance. It is shown that when the distance between the sender and the receiver is at least 2, the results derived for the antipodes are well applicable. If the sender and the receiver are direct neighbours the evolution follows a slightly different course. Nevertheless, state transfer with high fidelity is achieved in the same run-time.
Search and state transfer between hubs, i.e., fully connected vertices, on otherwise arbitrary connected graph is investigated. Motivated by a recent result of Razzoli et al. [J. Phys. A: Math. Theor. 55, 265303 (2022)] on the universality of hubs in continuous-time quantum walks and spatial search, we extend the investigation to state transfer and also to the discrete-time case. We show that the continuous-time quantum walk allows for perfect state transfer between multiple hubs if the numbers of senders and receivers are close. Turning to the discrete-time case, we show that the search for hubs is successful provided that the initial state is locally modified to account for a degree of each individual vertex. Concerning state transfer using discrete-time quantum walk, it is shown that between a single sender and a single receiver one can transfer two orthogonal states in the same run time. Hence, it is possible to transfer an arbitrary quantum state of a qubit between two hubs. In addition, if the sender and the receiver know each other location, another linearly independent state can be transferred, allowing for exchange of a qutrit state. Finally, we consider the case of transfer between multiple senders and receivers. In this case we cannot transfer specific quantum states. Nevertheless, quantum walker can be transferred with high probability in two regimes---either when there is a similar number of senders and receivers, which is the same as for the continuous-time quantum walk, or when the number of receivers is considerably larger than the number of senders. Our investigation is based on dimensional reduction utilizing the invariant subspaces of the respective evolutions and the fact that for the appropriate choice of the loop weights the problem can be reduced to the complete graph with loops.
Perfect state transfer between two marked vertices of a graph by means of a discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that, for the coined quantum walks on a star graph and a complete graph with self-loops, perfect state transfer between the sender and receiver vertex is achieved for an arbitrary number of vertices NNN in O(N)O(\sqrt{N})O(N​) steps of the walk. Finally, we show that Szegedy's walk with queries on a complete graph allows for state transfer with unit fidelity in the limit of large NNN.
We investigate coined quantum-walk search and state-transfer algorithms, focusing on the complete MMM-partite graph with NNN vertices in each partition. First, it is shown that by adding a loop to each vertex, the search algorithm finds the marked vertex with unit probability in the limit of a large graph. Next, we employ the evolution operator of the search with two marked vertices to perform a state transfer between the sender and the receiver. We show that when the sender and the receiver are in different partitions, the algorithm succeeds with fidelity approaching unity for a large graph. However, when the sender and the receiver are in the same partition, the fidelity does not reach exactly 1. To solve this problem, we propose a state-transfer algorithm with an active switch, whose fidelity can be estimated based on the single vertex search alone.
Abstract Previously it was shown that (almost) perfect state transfer can be achieved on the complete bipartite graph by a discrete-time coined quantum walk based algorithm when both the sender and receiver vertices are in the same partition of the graph and when the sender and receiver are in opposite partitions of the same size. By changing the coin operator, we analyze the state transfer problem and we show that it is still possible to achieve state transfer with high fidelity even when the sender and receiver are in different partitions with different sizes. Moreover, it is also possible to use an active switch approach using lackadaisical quantum walks where the marked vertex is switched between the sender and receiver during the algorithm.
We consider pretty good state transfer in coined quantum walks between antipodal vertices on the hypercube [Formula: see text]. When [Formula: see text] is a prime, this was proven to occur in the arc-reversal walk with Grover coins. We extend this result by constructing weighted Grover coins that enable pretty good state transfer on every [Formula: see text]. Our coins are real and require modification of the weight on only one arc per vertex. We also generalize our approach and establish a sufficient condition for pretty good state transfer to occur on other graphs.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.
We propose a protocol for perfect state transfer between any pair of vertices in a hypercube. Given a pair of distinct vertices in the hypercube, we determine a subhypercube that contains the pair of vertices as antipodal vertices. Then a switching process is introduced for determining the subhypercube of a memory-enhanced hypercube that facilitates perfect state transfer between the desired pair of vertices. Furthermore, we propose a physical architecture for the pretty good state transfer implementation of our switching protocol with fidelity arbitrary close to unity, using superconducting transmon qubits with tunable couplings. The switching is realized by the control over the effective coupling between the qubits resulting from the effect of ancilla qubit couplers for the graph edges. We also report an error bound on the fidelity of state transfer due to faulty implementation of our protocol.