Continuous-time quantum walk-based ansätze on neutral atom hardware

TL;DR

The paper addresses the gap between abstract CTQW-based quantum algorithms and near-term hardware by implementing CTQW-based variational ansätze on a neutral-atom platform, Aquila, using constrained independent-set subspaces enforced by Rydberg blockade.It introduces phase-walk ansätze that interleave CTQW-based mixing with phase encoding, analyzes both product and bracelet target states, and develops analytic and spectral-gap-based optimization strategies to achieve high-fidelity state preparation.Experimentally, it demonstrates substantial amplification and near-optimal preparation of product states and entangled bracelet states, with scaling trends consistent with non-adiabatic CTQW predictions, while revealing hardware-imposed limits such as finite coherence time and blockade-induced phase errors.The results establish a practical pathway for translating CTQW advantages into current analog quantum devices, highlight the role of spectral gaps in enabling non-adiabatic shortcuts, and point to future improvements in hardware and graph-engineered CTQWs.Overall, this work provides a concrete demonstration of CTQW dynamics as a resource for state preparation on neutral-atom hardware and offers a framework for leveraging constrained subspaces to realize super-quadratic convergence within NISQ-era devices.

Abstract

Continuous-time quantum walks offer provable speedups for certain computational problems, yet translating these advantages to near-term hardware remains challenging. We present the first experimental demonstration of variational ansätze based on continuous-time quantum walks on an analog neutral-atom processor. For unentangled targets, we derive closed-form expressions for near-optimal control parameters that transfer directly to hardware with minimal calibration. Experiments on QuEra's Aquila processor provide the first observation of the super-quadratic convergence characteristic of efficient quantum walk algorithms, visible at low circuit depth, with theory predicting stronger speedups as hardware improves. For entangled targets, specifically symmetric superpositions in the Rydberg-blockaded subspace, we introduce an optimization protocol exploiting spectral properties of the walk dynamics. The required evolution time scales inversely with the spectral gap, offering an advantage over adiabatic protocols that scale to the square of the spectral gap. We demonstrate this scaling behavior on Aquila and verify that the prepared states are coherent superpositions via quench dynamics. This constitutes the first preparation of such symmetric entangled states on neutral-atom hardware. Our results establish a practical pathway from abstract quantum walk algorithms to analog quantum processors, demonstrating that the dynamics underlying their potential for super-quadratic quantum speedup are accessible on current devices.

Figures (12)

• Figure 1: An example walk graph for independent set constrained subspace walks. Each vertex is an independent set in a ring of 7 vertices labeled by bitstring $z$ and representing basis vector $|z\rangle$, with red dots indicating inclusion of that physical vertex in the set. There is an edge between vertices iff the Hamming distance of the bitstring label is 1.
• Figure 2: Success probability of preparing product states $\vert z^* \rangle = \vert 0^{N - 2h}(01)^h \rangle$, where $h = 1$ to $\lfloor N/2 \rfloor$, using optimized walk times $\tau_0$ and $\tau_1$, for systems sizes $N = 5$ to $23$. Top-left shows success probability across ansatz depths $p = 1$ to $5$ colored by Hamming weight ($h$). Bottom-left highlights the reference states $z^{\rm half}$ and $z^{\rm MIS}$ (see \ref{['sec:target_states']}). Right shows success probability as a function of $\tau_0$ and $\tau_1$ for the $N=9$ instance of $z^{\rm half}$ at depths $p = 1$, $2$, and $3$. The optimal points (given in Table \ref{['tab:product']}) are marked with a red 'X', and theoretical predictions from \ref{['eq:tau_0_star', 'eq:tau_1_star']} is indicated by a yellow '+'.
• Figure 3: Bracelet state preparation. Top: Population of the $N=9$ target bracelet state $\vert [z^{\rm half}] \rangle$ during a continuous-time quantum walk from $\vert 0 \rangle$, sampled at $\tau_j=j\,\Delta\tau$ with $\Delta\tau=0.02$, over $\tau\in[0,20]$. Local maxima $\tau^{(m)}_{\rm{peak}}$ above the threshold $1/|D_N|$ are marked by a pink 'X', which serve as fixed parameters in variational optimization of phases $\gamma$. Bottom: Accumulated walk time $\tau_{\mathrm{eff}}$ required to reach success probability $\ge 0.98$ against the inverse resolvable spectral minimum $1/\Delta_{\rm{min}}(\kappa^*)$ (see \ref{['eq:resolvable_spectral_minimum', 'eq:bracelet_tau_eff']}) for all dihedrally distinct targets $\vert [z^*] \rangle$ with $N=5,\dots,12$. Here $\Delta_{\rm{min}}(\kappa)$. We fix $\kappa^*\approx 7.2\pm 0.4$ from a stability plateau by maximizing $\mathrm{mean}(r)-2\,\mathrm{std}(r)$ over sliding $\kappa$ windows and taking the window center. Maximum-independent-set (MIS) targets are circled in black.
• Figure 4: Representative Analog programs to implement quantum walk state preparation. Top plots the analog waveforms for product state preparation. Purple is the waveform for the Rabi frequency, which implements the quantum walk Hamiltonian where the total area of each trapezoid implements a walk of unitless time $\tau_i$. Red is the waveform for the local detuning, which implements the phase-separator $Z$ term on a subset of sites; each triangle has an integrated area of $\pi/2$. The Rabi term is turned off during the phase accumulation to avoid any non-commutative terms. Middle plots the analog waveforms for bracelet state preparation. The red dashed line is the Rabi phase, which is U(1) gauge equivalent to the phase separator via phase jumps. Bottom plots the atom positions for 7, 13, and 17 atoms, with a background grid spacing of $2\mu$m. Observe that the top of each circle and bottom of $N=13$ and $17$ are flattened to conform to the row constraints of Aquila wurtz2023aquila.
• Figure 5: Probability distributions for preparation of the $z^{\rm half}$ product state and the corresponding bracelet state at $N=7$, obtained from numerical simulation of the ideal CTQW dynamics ("Perfect"), noiseless emulation of the Rydberg Hamiltonian ("Emulation"), the raw probability distribution from 1000 shots on Aquila, and the reconstructed distribution ("EM") that accounts for measurement error. Top shows the product state distribution. Middle shows the bracelet state distribution. The table below summarizes the corresponding success probabilities: the State column labels the target state; Ansatz Depth is the number of walk layers; the Naïve count column gives the fraction of bitstrings matching the target directly from raw counts; the Perfect column reports the preparation probability under ideal CTQW dynamics; the Emulation column shows the same quantity using noiseless Rydberg-atom emulation. The EM (H) column gives the success probability from Aquila hardware data after mitigation of measurement errors, while EM (E) reports the corresponding value from 1000 emulated shots convolved with the noisy measurement channel. Agreement between EM (E) and the Emulation column validates that the EM procedure correctly recovers the underlying distribution. Data for all target states are shown in \ref{['tab:product', 'tab:bracelet']} of \ref{['app:table_details']}.