Digital dissipative state preparation for frustration-free gapless quantum systems

TL;DR

The theory predicts that a transient cooling dynamics directly reveals the system's universal critical properties and the state preparation time is linear in the inverse of the finite-size gap when the system's dynamical critical exponent is larger or equal the effective spatial dimension explored by the quasiparticles.

Abstract

Preparing algebraically correlated ground states of quantum many-body systems is an important, yet challenging task for quantum simulation. We introduce a protocol that employs local projective measurements and unitary feedback for frustration-free gapless systems. Our approach prepares a priori unknown ground states in time that scales polynomially with system size. We analytically show the performance our protocol for the dynamics of a single-particle; we argue the same mechanism generalizes to many-body systems based on the physics of quasiparticles. Our theory predicts that a transient cooling dynamics directly reveals the system's universal critical properties. In particular, the state preparation time is linear in the inverse of the finite-size gap (up to log correction) when the system's dynamical critical exponent is larger or equal the effective spatial dimension explored by the quasiparticles. We verify these predictions in numerical simulations of ferromagnetic Heisenberg models in one- and two dimensions, a Fredkin spin chain, and a two-dimensional model of resonating valence bond states. Our protocol stabilizes gapless many-body ground states fully digitally without requiring analog rotations, enabling access to high-fidelity states beyond conventional adiabatic approaches in near-term experiments.

Figures (12)

• Figure 1: Protocol.a) Our protocol consists of sequentially measuring the projectors of the Hamiltonian Eq.(\ref{['eq:frustration_free_H']}) and applying local corrections $U_C$ for outcomes $P_i=1$. b) For a variety of gapless many-body systems, the average time to prepare the ground state up to a fixed many-body infidelity, $\epsilon$, scales inverse linearly with finite size energy gap $\Delta\sim N^{-z/d}$, as shown by our numerical simulation with $\epsilon =0.2$. c) This performance scaling can be exactly shown for a solvable model of effective single particles dynamics: A localized excitation undergoes imaginary time evolution with stochastic resets to the initial state (gray). The solution reveals the universal scaling of the average dynamics of the energy (red).
• Figure 2: Single particle dynamics.a) The trajectory-averaged energy $\overline{E}(t)$ in the single-particle Heisenberg chain. $\overline{E}(t)$ decays with rate linear in the energy gap $\Delta(N)$, following the prediction of Eq.(\ref{['eq:avg_energy']}) (with $\beta = 1/2$). $\lambda \approx 1.0$ is a system-size independent constant. b) At early times, $\overline{E(t)} \sim 1/\sqrt{t}$ decays algebraically, in agreement with Eq.(\ref{['eq:avg_energy']}) for $\beta=1/2$. In contrast, dynamics in 2D, where $\beta=1$, shows no clear regime with algebraic decay (discernible from a predicted slower logarithmic decay at early times for $\beta=1$, see SM SM). c) Average ground state infidelity $\overline{\epsilon(t)}$, decaying with the same rate $\lambda \Delta$ as the energy.
• Figure 3: Many-body dynamics.a)Left: Average energy in the Heisenberg chain starting from a Néel product state. $\overline{E(t)}$ decays with rate linear in $\Delta(N)$, following our expectation for $\beta = 1/2$ (black dashed lines with $\lambda \approx 1.0$). Upper right: At early times, $\overline{E(t)} / N \sim 1/\sqrt{t}$ decays algebraically. Lower right: Dynamics of the average infidelity. The asymptotic decay $\overline{\epsilon(t)} \sim f(N) \, e^{-\lambda\Delta t}$, with $f(N) \lesssim N / \sqrt{\Delta}$, implies an efficient preparation time. Black dashed lines show $f(N)\sim \sqrt{N}$. b) Average energy in the 2D Heisenberg model starting from a Néel state, consistent with a log-correction to the decay rate as expected for $\beta = 1$ (black dashed lines with $\lambda \approx 3.1$). There is no discernible early time algebraic decay.
• Figure 4: Algebraically correlated systems.a) Average energy starting from a Néel state in the Fredkin spin chain with $z\approx 8/3$. The dynamics is consistent with the scaling form expected for $\beta = d/z = 3/8$ (black dashed lines with $\lambda \approx 1.9$). b) Average energy starting from a columnar product state of a 2D square lattice quantum dimer model (first state depicted in the RVB superposition). $\overline{E(t)}$ decays with rate linear in the gap $\Delta(N)$ and follows our prediction for $\beta = 1/2$ (black dashed lines with $\lambda \approx 2.1$). Dynamics was simulated with MPS at bond dimension $\chi=512$, averaging over at least $10^3$ trajectories for each system size; gaps $\Delta(N)$ were obtained using exact diagonalization. We further found a decay of the infidelity consistent with $\overline{\epsilon(t)} \sim \sqrt{N} e^{-\lambda\Delta t}$ (see SM).
• Figure S1: Numerical verification of $\mathcal{P}^\tau \approx \tilde{\mathcal{P}}^{4\tau/3}$.a) Numerical evaluation of the norm $\|\mathcal{P}^\tau \ket{\tilde{\psi}_j}\|$ for selected eigenstates $\ket{\tilde{\psi}_j}$ of the operator $\tilde{\mathcal{P}}$ in the Heisenberg chain with $N=20$ and magnetization $\frac{1}{2}\sum_i (1-Z_i)=3$. The eigenstates with $\tilde{\mathcal{P}}\ket{\tilde{\psi}_j}=\tilde{\lambda}_{\tilde{\psi}_j} \ket{\tilde{\psi}_j}$ are organized such that $\tilde{\lambda}_{\tilde{\psi}_1}=1>\tilde{\lambda}_{\tilde{\psi}_2}\geq ...$. The exponential decay of the norm follows the leading order behavior $\sim \tilde{\lambda}_{\tilde{\psi}_j}^{4\tau/3}$ predicted by Eq.(\ref{['eq:EVinverse']}). For the smallest eigenvalue $\tilde{\lambda}_{\tilde{\psi}_{14}}\approx 0.71$ displayed here, a very small deviation from the leading behavior becomes visible at late times. b) Upon dividing out the leading order scaling behavior, we find that these corrections are in excellent agreement with the predicted form of Eq.(\ref{['eq:EVinverse']}). c) Overlap of the evolved state $\mathcal{P}^\tau \ket{\tilde{\psi}_j}$ with the initial eigenstate $\ket{\tilde{\psi}_j}$ of $\tilde{\mathcal{P}}$, relative to its total norm. This relative overlap takes on $\mathcal{O}(1)$ value. Inset: Deviations from unity scale approximately linearly in $|\log \tilde{\lambda}_{\tilde{\psi}_j}|$ as predicted in Eq.(\ref{['eq:stateOverlap']}).