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

TL;DR

We address implementing the non-interacting one-particle sector of Quantum Cellular Automata (QCA) to realize Quantum Walk (QW) and Quantum Walk search on NISQ hardware. The approach maps QCA time evolution to a circuit Quantum Electrodynamics (cQED) processor, using two-qubit XY-like gates (e.g., U^(2)(π/4)=√iSWAP) and tessellated update rules to build the global step, with resource scaling of $O(n)$ qubits and $O(n t)$ gates and depth $O(t)$. Simulations are conducted with Qiskit Aer (noiseless) and C12’s Callisto emulator (noisy), analyzing state-count distributions, the Hellinger fidelity, the ℓ¹ distance, hitting times, and success probabilities on $N$-cycles and $N imes N$ torus graphs, including a Dicke-state initialization for QW search. Results indicate that QCA on a cQED emulator can faithfully reproduce QW dynamics and provides a robust baseline for QW search under realistic noise, highlighting QCA as a promising framework for early NISQ demonstrations and guiding hardware development for scalable quantum walks.

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.

Figures (12)

• Figure 1: Left: One time-step of an 8-cycle QW (a) and QW search (b). The minimal tessellation cover of cycles is two: $\mathcal{T}_{0}$ (dashed line) and $\mathcal{T}_{1}$ (solid line), see Eq. (\ref{['eq:tess_1D']}). Each time-step is divided into two sub-steps, where two-qubit matrices $W_{j, j+1}(\theta_{j}=\frac{\pi}{4})$ (orange box) and $W_{j, j+1}(\theta_{j}=\frac{\pi}{2})$ (red box) are applied via the transition functions $W_{\mathcal{T}_{0}}$ and $W_{\mathcal{T}_{1}}$, associated to each tessellation, see Eq. (\ref{['eq:QCA_tr_func']}). Right: One time-step of a 4x4 torus QW (c) and QW search (d). The minimal tessellation cover of torus graphs is four: $\mathcal{T}_{0_{x}0_{y}}$ (black solid line), $\mathcal{T}_{1_{x}0_{y}}$ (black dashed line), $\mathcal{T}_{0_{x}1_{y}}$ (red solid line) and $\mathcal{T}_{1_{x}1_{y}}$ (red dashed line), see Eq. \ref{['eq:tess_2D']}. For clarity, only the indices of the first and last rows of the first torus in panel (c) are shown. A single time-step is divided into four sub-steps, where two-qubit matrices $W_{j, j+1}(\theta_{j}=\frac{\pi}{4})$ (orange box) and $W_{j, j+1}(\theta_{j}=\frac{\pi}{2})$ (red box) are applied via the transition functions $W_{\mathcal{T}_{0_{x}0_{y}}}$, $W_{\mathcal{T}_{1_{x}0_{y}}}$, $W_{\mathcal{T}_{0_{x}1_{y}}}$ and $W_{\mathcal{T}_{1_{x}1_{y}}}$, associated to each tessellation see Eq. (\ref{['eq:QCA_tr_func']}). The blue dot in b) and d) indicates the marked vertex in the search.
• Figure 2: Schematic of C12's spin qubit. An electron is hosted in a carbon nanotube (grey tube), suspended over five electrodes (blue squares) generating a double quantum dot (black squared well) trapping the particle (black arrowed dot), and immersed in an asymmetric magnetic field. One qubit gates are then realized by changing the dot's voltage (black wave) and two-qubit gates are realized by embedding two qubits within the same microwave cavity (red resonator plates).
• Figure 3: Hellinger Fidelity vs. time-step of the QW (left) and QW search (right) via QCA over 4 (black dots), 6 (orange triangles), 8 (blue diamonds), 10 (red pyramids), 12-cycle (black dashed dot line), 16-cycle (orange dashed squares line) and 4x4-torus (blue dashed cross line).
• Figure 4: Left: $\ell^{1}$ distance vs. time-step for QW (left) i) via QCA : 4 (black curve), 8 (orange curve), 16-cycles (blue curve) and 4x4 torus (red curve) on Callisto ; ii) via Shakel QFT: 4 (black: crosses on ibmqx2, triangles on ibmqx2, squares on ibmq_16_melbourne), 8-cycles (orange: crosses on ibmqx2 computer, diamonds on ibmqx2 processor, squares on ibmq_16_melbourne) via Shakeel QFT-based QW Shakel; iii) via Portugal Portugal: 8 (orange dots on ourense), 16-cycles (blue crosses on vigo processor, diamonds on ibmqx2 and 4x4 torus (red dots on vigo). Right: $\ell^{1}$ distance vs. time-steps for QW search (right) i) via QCA : 4 (black), 8 (orange), 16-cycles (blue) and 4x4 torus (red) on Callisto.
• Figure 5: Noiseless (Aer) and noisy (Callisto) simulations of QW search via QCA. First row: noiseless vertex probabilities vs. time-step (for 8 and 16-cycle some vertex curves are not shown for visual clarity). Second row: noiseless (orange) and noisy (blue) success probabilities vs. time-step. Third row: noiseless (orange) and noisy (blue) selectivity vs. time-step.