Dissipative ground-state preparation of a quantum spin chain on a trapped-ion quantum computer

TL;DR

This work presents a dissipative framework for ground-state preparation on a quantum computer by employing a CPTP map $\Gamma_K$ whose unique fixed point is the ground state $|E_0\rangle\langle E_0|$. The authors derive a Kraus representation valid for arbitrary time step $\tau$ and implement the protocol on a trapped-ion device to prepare a 1D transverse-field Ising model with up to $N=19$ spins using a single ancilla, demonstrating robust convergence despite substantial circuit depth. Zero-noise extrapolation further mitigates hardware noise, enabling energy estimates that closely match noiseless simulations within statistical uncertainties. The results highlight the practical viability and resilience of dissipative ground-state preparation on NISQ devices and lay groundwork for extending this approach to broader quantum many-body Hamiltonians.

Abstract

We demonstrate a dissipative protocol for ground-state preparation of a quantum spin chain on a trapped-ion quantum computer. As a first step, we derive a Kraus representation of a dissipation channel for the protocol recently proposed by Ding et al. [Phys. Rev. Res. 6, 033147 (2024)] that still holds for arbitrary temporal discretization steps, extending the analysis beyond the Lindblad dynamics regime. The protocol guarantees that the fidelity with the ground state monotonically increases (or remains unchanged) under repeated applications of the channel to an arbitrary initial state, provided that the ground state is the unique steady state of the dissipation channel. Using this framework, we implement dissipative ground-state preparation of a transverse-field Ising chain for up to 19 spins on the trapped-ion quantum computer Reimei provided by Quantinuum. Despite the presence of hardware noise, the dynamics consistently converges to a low-energy state far away from the maximally mixed state even when the corresponding quantum circuits contain as many as 4110 entangling gates, demonstrating the intrinsic robustness of the protocol. By applying zero-noise extrapolation, the resulting energy expectation values are systematically improved to agree with noiseless simulations within statistical uncertainties.

Figures (9)

• Figure 1: (a) Example of the filter function $\tilde{f}(\omega)$ in the frequency domain. The dashed vertical lines indicate $\omega=a$ and $\omega=b$ with $a <b<0$. (b) Same as (a), but shown over a wider frequency range. (c) Fourier transform $f(s)$ of the filter function $\tilde{f}(\omega)$. The quantities $|f(s)|$, ${\rm Re}f(s)$, and ${\rm Im}f(s)$ are shown as solid, dashed, and dash-dotted curves, respectively. The solid vertical lines mark $s=2\pi n/(b-a)$ for integer $n$, where $f(s)=0$ except at $s=0$. (d-f) Same as (a-c), but with time discretization. In (d), the red arrows at $\omega=a$ and $\omega=b$ indicate the edge broadening of width $\pi/S_s$, arising from truncation of the time-integration range. In (e), the red arrow indicates the aliasing period $2\pi/\Delta_s$, corresponding to the discrete sampling points of $f(s_l)$ shown in (f). In (f), the large and small arrows indicate the time-discretization parameters $S_s$ and $\Delta_s$, respectively. The discretized values $f(s_l)$ shown in (f) are used in the quantum experiments for the $N=6$ system.
• Figure 2: Quantum circuit used to estimate the expectation value ${\rm Tr}[\hat{\rho}(m)\hat{O}]$. The topmost qubit, initialized in the state $|0\rangle$, serves as the ancilla, while the remaining qubits, initialized in the state $\hat{\rho}(0)$, serves as the system qubits.
• Figure 3: Energy $E(m)$ as a function of the time steps $m$ for (a) $N=4$ and (b) $N=6$. The parameters $J=-1$ and $B_X=-1.2$ are used for the transverse-field Ising model in both cases. Blue circles, red filled triangles, and orange inverted triangles show results from noiseless simulations (Noiseless), the noisy emulator (Reimei-E), and the quantum hardware ( Reimei), respectively. For each data point, Noiseless results are obtained using 1000 measurement shots, whereas Reimei and Reimei-E results are obtained using 100 measurement shots. Error bars indicate the standard deviations. The exact ground-state energy $E_0$ is shown as a magenta horizontal line.
• Figure 4: Energy $E(m)$ as a function of the time step $m$ for the $N=6$ transverse-field Ising model with parameters $J=-1$ and $B_X=-1.2$. Results for noise-scaling factors $G=1$, $3$, and $5$ are obtained on Reimei. For each data point, Noiseless results use $1000$ measurement shots, whereas the Reimei results for $G=1$, $3$, and $5$ use $100$ measurement shots. "ZNE exp" and "ZNE lin" denote the ZNE estimates obtained via exponential and linear extrapolations, respectively. Error bars indicate the standard deviations and, for the ZNE exp and ZNE lin data, also include the uncertainty associated with the extrapolation. The exact ground-state energy $E_0$ is indicated by the magenta horizontal line.
• Figure 5: Energy $E(m)$ as a function of the time step $m$ for the $N=19$ transverse-field Ising model with parameters $J=-1$ and $B_X=-1.2$. Results for noise-scaling factors $G=1$, $3$, and $5$ for $m\leqslant 30$ are obtained on Reimei. For comparison, results obtained with Reimei-E for $G=1$ at $m \geqslant 30$ are also shown as orange open triangles. For each data point, Noiseless results use $1000$ measurement shots, whereas the Reimei results for $G=1$, $3$, and $5$ and the Reimei-E results for $G=1$ at $m\geqslant 30$ use $100$ measurement shots. "ZNE exp" and "ZNE lin" denote the ZNE estimates obtained via exponential and linear extrapolations, respectively. Error bars indicate the standard deviations and, for the ZNE exp and ZNE lin data, also include the uncertainty associated with the extrapolation. The exact ground-state energy $E_0$ is indicated by the magenta horizontal line.