Low $T$-count preparation of nuclear eigenstates with tensor networks

Abstract

We present an efficient protocol leveraging classical computation to support Initial State Preparation for strongly correlated fermionic systems, a critical bottleneck for fault-tolerant quantum simulation. Focusing on nuclear shell model eigenstates, we first demonstrate that the Density Matrix Renormalization Group algorithm can efficiently approximate target states as Matrix Product States, capitalizing on the favourable entanglement structure of these fermionic systems. These high-fidelity approximations are then leveraged as a classical resource in a variational circuit optimization scheme to compile shallow quantum circuits. We establish concrete resource estimates by decomposing the resulting circuits into the industry-standard Clifford$+T$ gateset, exploring the benefits of specialized $U3$ synthesis techniques. For all nuclear systems tested, on up to 76 qubit Hamiltonians, we consistently find low $T$-count circuits preparing the nuclear eigenstates to high fidelity with $\sim 2\times 10^4$ total $T$ gates. This low number gives confidence these eigenstates can be prepared on early fault-tolerant quantum computers. Our work establishes a viable path toward practical ground state preparation for nuclear structure and other fermionic applications.

Figures (9)

• Figure 1: Overview. Sketch of method to approximately prepare nuclear shell model eigenstates on fault-tolerant quantum computers, aided by tensor networks. a) Nuclear shell model Hamiltonians are defined on a valence space of proton and neutron orbitals. Here this is visualized for a 24 qubit example. b) We approximately represent a target nuclear eigenstate, $|E_\lambda\rangle$, as an MPS using DMRG, exploiting known entanglement structures in nuclei. c) These MPS are used as a resource in a variational circuit optimization, maximizing fidelity with the target MPS. d) These circuits have a simple representation in Clifford$+R_z$ gates. A further decomposition by unitary synthesis finally gives circuits described in the Clifford$+T$ gateset. Across the nuclei studied, on up to 76 qubit Hamiltonians, we consistently find circuits preparing eigenstates to high fidelities with low $T$-counts of $\sim2\times 10^4$ total $T$ gates. e) These circuits are ready to prepare the initial state in future fault-tolerant quantum algorithms.
• Figure 2: Neon and Sodium - MPS Bond Dimension Distribution. For the three neon isotopes $^{(20-22)}$Ne and three sodium isotopes $^{(22-24)}$Na, we apply perform a high-accuracy DMRG calculation to determine the three lowest-lying eigenstates (enumerated by $\lambda\in\{0,1,2\}$). The x-axis indicates the MPS site tensors, labelled by the quantum numbers of single-particle orbital they represent. The y-axis measures the virtual bond dimension between these site tensors in the outputted MPS, after DMRG is performed with a minimum singular value of $10^{-8}$ retained after tensor decompositions.
• Figure 3: Restricted bond dimension DMRG. Convergence of variational energy computed by DMRG with respect to the exact value, as the maximum allowed bond dimension of the outputted MPS increases.
• Figure 4: MPS Compression. Here we variationally compress the set of MPS shown in Fig. \ref{['fig:usdb_dmrg_chis']} representing the states $|E_\lambda\rangle$ for different nuclei, producing the MPS $|\Phi(\chi)\rangle$ compressed to a maximum bond dimension of $\chi$. After the compression we compute the overlap error $1-|\langle\Phi(\chi)|E_\lambda\rangle|$, for increasing $\chi$, and find a rapid decrease in error to the exact eigenstate.
• Figure 5: Approximate state preparation with shallow circuits. Short depth circuits, following the construction described in Sec. \ref{['sec:circuit_opt']}, are optimized to maximize the overlap with the target eigenstate $|E_\lambda\rangle$. Here the overlap magnitude is computed by $|\langle E_\lambda|U|\mathbf{0}\rangle|$ where $U$ is the optimized circuit.