We demonstrate an implementation of the hitting time of a discrete time quantum random walk on cubelike graphs using IBM's Qiskit platform. Our implementation is based on efficient circuits for the Grover and Shift operators. We verify the results about the one-shot hitting time of quantum walks on a hypercube as proved in [https://link.springer.com/article/10.1007/s00440-004-0423-2]. We extend the study to another family of cubelike graphs called the augmented cubes [https://onlinelibrary.wiley.com/doi/abs/10.1002/net.10033]. Based on our numerical study, we conjecture that for all families of cubelike graphs there is a linear relationship between the degree of a cubelike graph and its hitting time which holds asymptotically. That is, for any cubelike graph of degree
Δ, the probability of finding the quantum random walk at the target node at time
2πΔ approaches 1 as the degree
Δ of the cubelike graph approaches infinity.
We study the one-shot and concurrent hitting for the discrete-time Grover-coined quantum walk on cubelike graphs
G=Cay(Z2d,Ω) of degree
Δ=∣Ω∣. Starting from the vertex labeled
0, we identify
σ=⨁ω∈Ωω as a natural target vertex; for the hypercube,
σ is precisely the antipodal vertex.
For families with
Δ→∞, let
T be an integer having the same parity as
Δ and satisfying $ \left|T-\frac{πΔ}{2}\right|\leq 1. $ We show that the probability
pT(σ) of finding the walker at
σ when it is measured at time
T satisfies
pT(σ)=1−O(Δ−1/5). Thus the target is found with probability tending to one after
Θ(Δ) steps.
For the concurrently measured walk, let
HTConc(σ) denote the probability that the target is detected at or before time
T when it is tested after every step. We prove
pT(σ)≤THTConc(σ), which implies
HTConc(σ)=Ω(Δ−1) over the same time scale.
The proof uses the Walsh-Fourier decomposition, an exact two-dimensional reduction of each Fourier mode, and a universal second-moment identity for the associated character sums. Our results extend Kempe's hypercube hitting phenomenon (J. Kempe, Probab. Theory Relat. Fields 133, 215-235, 2005) to arbitrary cubelike generating sets and establish the conjectured asymptotic hitting behavior for cublelike and augmented cubes in Mulherkar, Rajdeepak and Sunitha (Int. J. Quantum Inf. 20,2250020, 2022)
This paper presents an implementation of quantum random walk using the Qiskit library, given the quantum circuits extracted from the literature. We started by implementing an instance of Coined Discrete Quantum Walk on an 8-node cyclic graph. Observing the limitations of this implementation, we experimented a different approach by proposing an implementation for Szegedy Quantum Walk in the IBM Qiskit library [1]. The proposed solution could be used as a building block to construct quantum implementation of other classic algorithms.
It has been proved by Kempe that discrete quantum walks on the hypercube (HC) hit exponentially faster than the classical analog. The same was also observed numerically by Krovi and Brun for a slightly different property, namely, the expected hitting time. Yet, to what extent this striking result survives in more general graphs is to date an open question. Here, we tackle this question by studying the expected hitting time for quantum walks on HCs that are embedded into larger symmetric structures. By performing numerical simulations of the discrete quantum walk and deriving a general expression for the classical hitting time, we observe an exponentially increasing gap between the expected classical and quantum hitting times, not only for walks on the bare HC, but also for a large family of embedded HCs. This suggests that the quantum speedup is stable with respect to such embeddings.
In recent years, quantum walks have been widely researched and have shown exciting properties. One such is a quadratic speed-up in hitting time compared to its classical counterpart. In this paper, we design a quantum circuit for the MNRS algorithm, which finds a marked node in a graph with a quantum walk, and use it to find a hitting time for the marked nodes in the walk. We do this by implementing the circuit on IBM quantum simulators and show that the execution on a noise-free simulator results in hitting times that agree with the theoretical expectations. We also run the algorithm on a mock backend that simulates the noise of the IBM Melbourne computer. As expected, the noise has an extensive impact on the output, resulting in outcomes far from the noise-free simulation.
Cubelike graphs are the Cayley graphs of the elementary Abelian group
Z2n (e.g., the hypercube is a cubelike graph). We study perfect state transfer between two particles in quantum networks modeled by a large class of cubelike graphs. This generalizes the results of Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)] and Facer et al. [Phys. Rev. A 92, 187902 (2008)].