Noisy simulations of Quantum Walk and Quantum Walk search via Quantum Cellular Automata on a semiconducting spin processor emulator

Abstract

In this work we map NISQ-friendly implementations of the non-interacting QCA to a circuit Quantum Electrodynamics (cQED) hardware. We perform both noiseless and noisy simulations of the QCA one particle sector, namely the Quantum Walk, on $N$-cycles and $N \times N$ torus graphs. Moreover, within this framework, we also investigate the search problem and present a circuit for preparing the W state (i.e., the Dicke state with hamming weight one) using only N-1 $\sqrt{\text{iSWAP}}$ gates and no ancilla qubits. The noiseless simulations are conducted with the Qiskit Aer simulator, while the noisy simulations with C12 Quantum Electronics' in-house noisy emulator, \textit{Callisto}. We benchmark the performance of our implementations by analyzing the simulations via relevant metrics and quantities such as the state count distributions, the Hellinger Fidelity, the $\ell^{1}$ distance, the hitting time, and success probability. Our results demonstrate that the QCA framework, in combination with cQED processors, holds promise as an effective platform for early NISQ implementations of Quantum Walk and Quantum Walk Search algorithms.