Preparation of many-body ground states by time evolution with variational microscopic magnetic fields and incomplete interactions

State preparation is of fundamental importance in quantum physics, which can be realized by constructing the quantum circuit as a unitary that transforms the initial state to the target, or implementing a quantum control protocol to evolve to the target state with a designed Hamiltonian. In this article, we study the latter on quantum many-body systems by time evolution with fixed couplings and variational magnetic fields. Specifically, we consider preparing the ground states of the Hamiltonians containing certain interactions that are missing in the Hamiltonians for the time evolution. An optimization method is proposed to optimize the magnetic fields by “fine graining” the discretization of time, in order to gain high precision and stability. The automatic differentiation technique is utilized to obtain the gradients of the fields against the logarithmic fidelity. Our method is tested on preparing the ground state of the Heisenberg chain with the time evolution by the $XY$ and Ising interactions, and its performance surpasses two baseline methods that use local and global optimization strategies, respectively. Our work can be applied and generalized to other quantum models such as those defined on higher-dimensional lattices. It enlightens to reduce the complexity of the required interactions for implementing quantum control or other tasks in quantum information and computation by means of optimizing the magnetic fields.

Figure 1

The illustration of state preparation by time evolution with variational magnetic fields. Here, we take the target state as the ground state of the Heisenberg chain as an example. For a given Hamiltonian, the ground state can be obtained in an annealing process. We propose to tune the magnetic fields to evolve from a product state to the target state, while the Hamiltonian for time evolution contains less interaction terms (such as the $XY$ or Ising interactions). The magnetic fields are optimized using the automatic differentiation technique that was originally developed in machine learning.

Figure 2

The flowchart of fine-grained time optimization. The letter $K$ represents the total number of time slices in a time evolution. We have $T=K\tau$, with $T$ the total evolution time and $\tau$ the time interval for each slice. The letter $\tilde{\kappa }$ represents the total number of fine-grained steps. For the $\kappa \mathrm{th}$ fine-graining step with $\kappa =0,...,\tilde{\kappa }$, we have $K=2^{\kappa }$. The letter $k$ labels the time slice in a specific time evolution and takes $k=1,...,K$. The iteration shown in the upper half of the figure is the optimization of magnetic fields for a fixed $K$. The iteration shown by the lower half is to gradually fine grain time by increasing $K$ to $2K$.

Figure 3

The fidelity $f$ in Eq. (4) vs the number of time slices $K$. The target state is the ground state of the Heisenberg model and the evolution Hamiltonian is taken as the $XY$ model. We employ the FGTO method, and take the number of spins $N=10$ and the total evolution time $T=0.5$, 1, 2, 3, and 5.

Figure 4

The lines with symbols show the fidelity $f$ in Eq. (4) vs the total evolution time $T$ for the STO, GTO, and FGTO methods. The target state is the ground state of the Heisenberg model and the evolution Hamiltonian is taken as the $XY$ model. The lines without symbols show $f$ at the evolving time $t$ [see Eq. (10)] varying from $t=0$ to $t=6$. We fix the number of spins $N=10$ and the total number of time slices $K=16$ as an example.

Figure 5

The fidelity $f$ in Eq. (4) optimized by FGTO vs the number of spins $N$. The data shown by the red squares are obtained using $F$ [Eq. (3)] as the loss function, and those by the blue circles use $F^{\prime }$ [Eq. (11)] as the loss. Our results imply the exponential decrease of $f$ against $N$ as $f=1-0.0018e^{0.32N}$. The target state is the ground state of the Heisenberg chain, and the evolution Hamiltonian possesses the $XY$ interactions. We take the total time $T=6$ and $K=16$ time slices for evolution.

Figure 6

The fidelity $f$ in Eq. (4) vs the optimization epoch. We take the target state as the ground state of the Heisenberg model, and test the Ising, $XY$, and Heisenberg models as the evolution Hamiltonians. We fix the total evolution time $T=4.5$ and increase the number of time slices from $K=1$ to 16 as the optimization proceeds. We take the number of spins as $N=10$. See details in the main text.

Figure 7

The space-and-time “landscapes” of the magnetic fields in the (a) $x$, (b) $y$, and (c) $z$ directions for preparing the ground state of the Heisenberg chain by time evolution with $XY$ interactions. We have $t=k\tau$ that denotes the time in the evolution and use $n$ to denote the position in the chain. The colors illustrate the strength of the magnetic fields. See details in the main text.

Figure 8

The difference between the time-evolution gate and the target gate with different evolution times $T$. The swap, $\sqrt{\mathrm{SWAP}}$, and cnot gates are taken as the target as examples. The minimal time $t^{*}$ given by the analytical solution for realizing each gate is indicated by the vertical dashed lines.