Preparation of initial states with open and periodic boundary conditions on quantum devices using matrix product states

TL;DR

This work develops a scalable framework to prepare quantum states on circuits directly from matrix product states (MPS) with open or periodic boundary conditions. It maps MPS tensors to unitary gates, decomposes them into hardware-native operations using an autodiff-driven SO(4) ansatz, and handles periodic boundaries via ancilla-assisted post-selection with an exact expression for the success probability. The authors demonstrate two applications: ground-state preparation and quench dynamics for the Heisenberg model with PBC, and high-fidelity excited-state preparation for the Schwinger model, illustrating potential quantum advantages for strongly correlated systems. The approach is modular, extensible to general bond dimensions through disentangling gates, and applicable to near-term devices for dynamical simulations and improved state initialization in variational protocols.

Abstract

We present a framework for preparing quantum states from matrix product states (MPS) with open and periodic boundary conditions on quantum devices. The MPS tensors are mapped to unitary gates, which are subsequently decomposed into native gates on quantum hardware. States with periodic boundary conditions (pbc) can be represented efficiently as quantum circuits using ancilla qubits and post-selection after measurement. We derive an exact expression for the success rate of this probabilistic approach, which can be evaluated a priori. The applicability of the method is demonstrated in two examples. First, we prepare the ground state of the Heisenberg model with pbc and simulate dynamics under a quenched Hamiltonian. The volume-law entanglement growth in the time evolution challenges classical algorithms but can potentially be overcome on quantum hardware. Second, we construct quantum circuits that generate excited states of the Schwinger model with high fidelities. Our approach provides a scalable method for preparing states on a quantum device, enabling efficient simulations of strongly correlated systems on near-term quantum computers.

Figures (15)

• Figure 1: Matrix product states with (a) a general form and (b) a special form obtained after sequential $LQ$ decompositions from site $N$ to site 1. This form is analogous to the right-canonical form, except for the first diagonal bond matrix $\Lambda$. For OBC, setting the boundary index to $1$ reduces (b) to the right-canonical form.
• Figure 2: Divide-and-conquer framework for the initialization of quantum circuits using matrix product states. Taking a four-qubit system as an example, the workflow involves three main steps: determining tensor-network states, mapping local tensors to quantum gates, and operating on the prepared states.
• Figure 3: Schematic divide-and-conquer diagram mapping local tensors to quantum gates, using a $D=4$ MPS as an example. (a) Local tensors after sequential $LQ$ decompositions are unitary or isometric matrices after reshaping. The unitary matrices can be transformed into multi-qubit gates by reshaping them into tensors, where each index corresponds to a qubit index. For isometric matrices, a similar reshaping is applied. However, the tensors have to be extended to obtain unitary gates, and the target state is recovered by applying the initial state $\Ket{0}$ on the corresponding index. (b) Mapped universal multi-qubit gates are decomposed into universal $\mathrm{SO}(4)$ gates by minimizing the squared Frobenius distance between the decomposed and target gates. (c) Each universal $\mathrm{SO}(4)$ gate is further decomposed into two CNOT gates and a set of single-qubit gates.
• Figure 4: The four-site examples for disentangling MPS with OBC (a) and PBC (b) into MPS with bond dimension constrained to powers of $2$, where these MPS with smaller bond dimensions are directly obtained through compression. The disentangling process is implemented using sequential universal $\mathrm{SO}(4)$ gates arranged in a ladder structure, which can be alternatively rearranged into another structure. The universal $\mathrm{SO}(4)$ gates are optimized by minimizing infidelity to achieve effective disentanglement.
• Figure 5: Encoding of the real-valued diagonal bond matrix $\Lambda$ for MPS with PBC. The matrix $\Lambda$ is rescaled by the $C_{\alpha}C_{1-\alpha}$ and then embedded into two gates, $V_{\alpha}$ and $V_{1-\alpha}^{\dagger}$, with a final post-selection step. These two gates $V_\alpha$ and $V^\dagger_{1-\alpha}$ correspond to the first and last four-qubit gates (in pink) in the second step of the PBC case in \ref{['fig:workflow']}.