Quantum state preparation with polynomial resources: Branched-Subspaces Adiabatic Preparation (B-SAP)

TL;DR

The paper tackles the challenge of efficiently preparing ground, excited, and thermal states in many-body quantum systems under limited resources. It introduces Branched-Subspaces Adiabatic Preparation (B-SAP), a hybrid framework that leverages group-theoretic irreps to explore degenerate subspaces with a polynomially bounded parameter set, followed by adiabatic evolution and classical post-processing via MC-VQE. Applied to the XYZ Heisenberg model, B-SAP achieves accurate low-energy eigenstates with circuit depths that scale polynomially with system size, and demonstrates improved performance for excited states over conventional AP. The work suggests broad implications for practical quantum simulations on near-term devices and provides a pathway toward efficient, robust state preparation beyond standard variational or adiabatic methods.

Abstract

Quantum state preparation lies at the heart of quantum computation and quantum simulations, enabling the investigation of complex manybody systems across physics, chemistry, and data science. While existing methods such as Variational Quantum Algorithms (VQAs) and Adiabatic Preparation (AP) offer viable pathways, both face substantial limitations. Here we introduce a hybrid algorithm that integrates the conceptual strengths of both VQAs and AP, enhanced via the use of group-theoretic structures and classical post-processing to approximate ground and excited states of many-body Hamiltonian models. We validate our approach by applying it to the one-dimensional XYZ Heisenberg model with periodic boundary conditions, evaluating its performance across a broad range of parameters and system sizes. Our results show accurate preparation of low-energy eigenstates, achieved with circuit depths with polynomial scaling versus system size.

Figures (8)

• Figure 1: Illustrations of adiabatic energy spectra for two processes sharing the same final Hamiltonian but differing in their initial ones. The left case \ref{['fig: AP spectrum']}, related to conventional AP, starts with a non-degenerate spectrum, but it features several level crossings. In contrast, the right panel illustrates the evolution performed through B-SAP \ref{['fig: BSAP spectrum']}, which displays a forked structure with degeneracies that never increase throughout the evolution. Notice that the time evolution is plotted from 0 to 1 in \ref{['fig: AP spectrum']}, and mirrored (i.e., from 1 to 0) in \ref{['fig: BSAP spectrum']}, highlighting the differences in the final spectra.
• Figure 2: Schematic representation of the B-SAP protocol implemented on a quantum computer with $L$ qubits. Two copies of the Hilbert space are depicted as gray manifolds. The hybrid algorithm trains a quantum circuit characterized by a polynomial number of tunable parameters $\{\underline{\alpha}\}$, enabling efficient exploration of a polynomially large subspace of the Hilbert space. In contrast, a standard VQA would need to explore the exponentially large Hilbert space. Finally, the adiabatic procedure, represented as a pipeline, bijectively maps the initial subspace onto a new one that includes the desired target state.
• Figure 3: Schematic illustration of the quantum circuit implementing the B-SAP algorithm. The initial quantum state $\ket{\Psi} \in \mathcal{S}(\mathcal{B}_n)$ is first prepared by applying a low-depth circuit to the default $\ket{0}^{\otimes L}$ state. The parametrized unitary $G(\underline{\alpha})$ then allows exploration of the full subspace $\mathcal{S}(\mathcal{B}n)$, by varying the parameters $\underline{\alpha}$. Finally, the state undergoes adiabatic evolution via $U_\tau(1)$, after which the energy of the resulting state is measured to update the variational parameters.
• Figure 4: Error between the target ground state and the actually prepared state in the two parity sectors $\mathcal{X} = \pm 1$. Such an error is quantified by $\mathcal{E}$ (see Eq. \ref{['eq: error B-SAP']}) for all the possible ratios among the coupling constants of the XYZ Heisenberg Hamiltonian $H_T$ in Eq. \ref{['eq: Heisenberg Hamiltonian']}. Both panels refer to the ferromagnetic model with $L = 10$. The adiabatic procedure has been performed for $L/2 = 5$ Trotter steps, equivalent to half of the chain length, each with $0.25\cdot[J^z]^{-1}$ duration.
• Figure 5: Error between the target ground state and the actually prepared state in the $\mathcal{X} = +1$ sector, calculated for different numbers of Trotter steps in the adiabatic protocol of the B-SAP method. The error, $\mathcal{E}$ (see Eq. \ref{['eq: error B-SAP']}), is reported for possible ratios among the coupling constants of the ferromagnetic XYZ Heisenberg Hamiltonian, see $H_T$ in Eq. \ref{['eq: Heisenberg Hamiltonian']}, with $L = 10$. The adiabatic procedures have been performed with (a) 5, (b) 10, and (c) 20 Trotter steps of duration $0.25\cdot [J^z]^{-1}$.

Theorems & Definitions (1)

• Theorem 1