Variational ground-state quantum adiabatic theorem

TL;DR

The paper tackles whether adiabatic quantum evolution can be faithfully simulated on low-entanglement variational manifolds when the target ground state is classical. It derives a variational ground-state quantum adiabatic theorem for Hamiltonians $H(s)=(1-s)H_0+sH_1$ and validates it across two-qubit, Lipkin-Meshkov-Glick, and Ising spin-glass models using time-dependent variational principles on restricted manifolds. The results show that the variational state remains within $\mathcal{O}(1/T)$ of the instantaneous variational ground state and can converge to the target state even when exact dynamics traverse highly entangled regions, with distinct scaling regimes tied to phase transitions. This work proposes a quantum-inspired classical framework for ground-state search and benchmarking of quantum annealing, with avenues for extending to imaginary time, alternative variational manifolds, and excited states, potentially enabling efficient problem solving and benchmarking strategies.

Abstract

We present a variational quantum adiabatic theorem, which states that, under certain assumptions, the adiabatic dynamics projected onto a variational manifold follow the instantaneous variational ground state. We focus on low-entanglement variational manifolds and target Hamiltonians with classical ground states. Despite the presence of highly entangled intermediate states along the exact quantum annealing path, the variational evolution converges to the target ground state. We demonstrate this approach with several examples that align with our theoretical analysis.

Figures (10)

• Figure 1: Schematic representation of the variational ground-state quantum adiabatic theorem. The dashed green line represents the exact quantum adiabatic evolution $\ket{\psi(s)} \approx \ket{\psi_0(s)}$, visiting high-entangled regions. Solid black line corresponds to the instantaneous variational ground state $\ket{\psi^{\rm VS}_0(s)}$, with the small red arrows indicating that it changes slowly with time. The time-dependent variational state $\ket{\psi^{\rm VS}(s)}$ is separated into $\ket{\psi^{\rm VS}_0(s)}$ and their difference $\ket{\delta\psi(s)}$. The latter can be decomposed in the eigenbasis of the linearized evolution map, whose eigenvalues $\omega_l$ determine the frequencies of the elliptic trajectories around the instantaneous variational ground state. Since $1/T \ll \omega_l$ (denoted by long, dark-red arrows), the effect of $\ket{\delta\psi(s)}$ averages to zero, and the time-dependent variational state $\ket{\psi^{\rm VS}(s)}$ follows the instantaneous variational ground state $\ket{\psi^{\rm VS}_0(s)}.$ When the target ground state of the final Hamiltonian lies within the variational manifold, the time-dependent variational state will converge to it as the annealing time $T$ becomes large.
• Figure 2: a) The final norm $\lVert\delta\psi(s)\rVert_2$ at $s=1$, a measure of the distance between $\ket{\psi_0(1)}$ (the target ground state of $H_1$) and $\ket{\psi^{\rm VS}(1)}$ (the variational state at the end of the annealing). We plot it as a function of the annealing time $T$ and parameter $A$. b) Half-system entanglement entropy at intermediate time $s=0.5$ (grey line) as a function of the parameter $A$, and the norm $\lVert \delta\psi(1)\rVert_2$ for $T=0.5$ --- dashed orange, and $T=1.5$ --- dash-dotted black. Higher entanglement entropy corresponds to smaller values of $\lVert \delta\psi (1)\rVert_2$. c) Scaling of the norm $\lVert \delta\psi(1)\rVert_2$ with annealing time $T$. The thin line corresponds to $T^{-1}$ scaling, the dashed orange line to $A=0$, and the grey line to $A=5$.
• Figure 3: a) An example evolution of the vector $S^{\rm\, VS}(s)$ (black line) for $N=4$ and $T=500$, $s \in [0,1]$. The red dot corresponds to the initial condition, and the yellow dot corresponds to the final variational state $S^{\rm\, VS}(1)$. The dashed orange line shows the variational ground state vector $S_0^{\rm VS}(s)$. b) Three examples of the evolution of the rescaled norm $\sqrt{T}\lVert \delta S (s)\rVert_2$ for $T=10^2$ (red), $T=10^3$ (green), and $T=10^4$ (blue). c) Convergence of $\delta S (s)$ during the variational time evolution --- same data as in panel b). d) Scaling of $\lVert\delta S (1)\rVert_2$: the solid line represent the predicted $1/\sqrt{T}$, the dots correspond to numerical results.
• Figure 4: a) Convergence of the time-dependent variational state towards the instantaneous variational ground state upon increasing the annealing time $T$ for a particular realization of the Ising spin glass model with $N=8$. We fix the bond dimension to $D=1$ and plot $\lVert\delta\psi(s)\rVert_2$. The black dotted line shows the norm of the difference between the exact instantaneous ground state and the instantaneous variational ground state. b) Norm of the difference between the final variational state at $s=1$, which is $\ket{\psi^{\rm VS}(1)}$, and the exact target ground state of $H_1$, i.e. $\ket{\psi_0(1)}$, for increasing values of $D$. Notice that the target state belongs to the MPS manifold with $D=1$ (dotted black line in panel a, at $s=1$). The bond dimension $D=16$ corresponds to the exact evolution, and the dashed line highlights the predicted scaling $1/T$. c) The spectral gap calculated from the effective Hamiltonian in the MPS-TDVP evolution for various bond dimensions $D$. The dashed line corresponds to the exact spectral gap. We set $T=3200$. d) Entanglement entropy during the protocol with $T=3200$ and different bond dimensions.
• Figure 5: Histogram of the norm $\lVert\delta\psi(1)\rVert_2$ at the end of the annealing protocol with $D=1,2,4,8$ and $T=1600$. The histogram is computed from 100 random realizations of the final Hamiltonian $H_1$. We fixed $N=8$. The red bars correspond to instances with values larger than $0.1$.

Theorems & Definitions (2)

• Theorem 1
• proof