An alternating-minimization method for preparing low-energy states

Abstract

Preparing low energy states is a central challenge in quantum computing and quantum complexity theory. Several known approaches to prepare low energy states often get stuck in suboptimal states, such as high energy eigenstates (or low variance high energy states). We develop a heuristic method to go past this barrier for local Hamiltonian systems with relatively low frustration, by taking advantage of the fact that such systems come with multiple Hamiltonians that agree on the low-energy subspaces. We establish an energy-based uncertainty principle, which shows that these Hamiltonians in fact do not have common eigenstates in the high energy regime. This allows us to run energy lowering steps in an alternating manner over the Hamiltonians. We run numerical simulations to check the performance of the `alternating' algorithm on small system sizes, for the 1D AKLT model and instances of Heisenberg model on general graphs. We also formulate a version of the energy-based uncertainty principle using sparse Hamiltonians, which shows a quadratically larger variance at higher energies and hence leads to a larger energy change. We use this version to simulate the method on energy profiles with high energy barriers.

Figures (8)

• Figure 1: Figure (a) plots the mean energy (black) and quartile energy (blue) for every $50$th eigenvector of $H_{QMC}$ in the eigenbasis of a randomly sampled $H_{QMC, \vec{\phi}}$ from Equation \ref{['eq:alteredHloc']}. Figure (b) plots the mean energy (black) and quartile energy (blue) for every $50$th eigenvector of a randomly sampled $H_{QMC, \vec{\phi}}$ in the eigenbasis of an independently sampled $H_{QMC, \vec{\phi'}}$. The gap between the black and the blue lines persists throughout the spectrum and reduces near the ground energy. Recall that $H_{QMC}$ is a frustrated model. These simulations are done on $12$ qubits.
• Figure 2: Figure (a) plots the mean energy (black) and quartile energy (blue) for every $50$th eigenvector of $H_{AKLT}$ in the eigenbasis of a randomly sampled $H_{AKLT, \vec{\phi}}$ from Equation \ref{['eq:alteredHloc']}. Figure (b) plots the mean energy (black) and quartile energy (blue) for every $50$th eigenvector of a randomly sampled $H_{AKLT, \vec{\phi_1}}$ in the eigenbasis of an independently sampled $H_{AKLT, \vec{\phi_2}}$. The gap between black and blue lines closes near the ground energy, which is expected from a frustration-free model. These simulations are done on $8$ qutrits.
• Figure 3: Figure (a) plots the mean energy (black) and quartile energy (blue) for every $50$th eigenvector of $H_{MC}$ in the eigenbasis of a randomly sampled $H_{MC, \vec{\phi}}$ from Equation \ref{['eq:alteredHloc']}. The gap between mean and quartile is significantly smaller than what we see in Figures \ref{['fig:qmcmeanquantile']} and \ref{['fig:AKLTmeanquantile']}. Figure (b) plots the mean energy (black) and quartile energy (blue) for every $50$th eigenvector of a randomly sampled $H_{MC, \vec{\phi}}$ in the eigenbasis of an independently sampled $H_{MC, \vec{\phi'}}$. We find that the gap between mean and quartile is now larger. These simulations are done on $12$ qubits.
• Figure 4: Figure (a) plots the mean energy (black) and quartile energy (blue) for every $50$th eigenvector of $H_{MC}$ in the eigenbasis of a randomly sampled $H_{MC, T,f}$ from Equation \ref{['eq:Tchoiceham']}. Figure (b) plots the mean energy (black) and quartile energy (blue) for every $50$th eigenvector of a randomly sampled $H_{MC}$ in the eigenbasis of an independently sampled $H_{MC, T,f}$ from Equation \ref{['eq:Tchoicet']}. We see a significantly larger gap in comparison to Figure \ref{['fig:MCmeanquartilelocal']}, as suggested by Theorem \ref{['theo:sparsevar']}. These simulations are done on $12$ qubits.
• Figure 5: Figure (a) Algorithm \ref{['alg:alt-min-energymeas']} is run on a sparse $H$ with many local minima (see text), setting $n=12, K=3$, on the initial state $\ket{+}^{\otimes n}$. The altered Hamiltonians are as defined in Equation \ref{['eq:Tchoiceham']}. Figure shows that at $L=8$, the energy is nearly $0$ and hence the state of the system is very close to the ground space $\text{span}\{\ket{1},\ket{2^n}\}$. The total number of energy measurements is $K^8+K^7+\ldots+1 = 9841$. In Figure (b), we plot the run of simulated annealing for 50000 steps to highlight that this landscape indeed has a strong energy well. We start from the uniform distribution, then perform jumps with temperature falling from $\beta=0.1$ to $\beta=50$. The minimum energy recorded is about $0.46$, which does not change for a very long time in the simulation. Figure (c) depicts a slightly different energy profile for $n=12$, where ground states have no energy barrier with adjacent energy $1$ states. Figure (d) shows that running Algorithm \ref{['alg:alt-min-energymeas']} with $K=3$ using altered Hamiltonian from Equation \ref{['eq:Tchoicet']}, ground state is found in $L=4$ iterations, which corresponds to $121$ energy measurements. Simulated annealing (not shown) on the same landscape reaches energy about $0.8$ in 40000 runs.

Theorems & Definitions (8)

• Theorem 2.1
• proof
• Claim 2.2
• proof
• Theorem 2.3
• Claim 3.1
• proof
• proof : Proof of Theorem \ref{['theo:sparsevar']}