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arXiv:2603.11156

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

Joe Gibbs, Lukasz Cincio, Chandan Sarma, Zoë Holmes, Paul Stevenson

quant-phSubmitted Mar 11, 2026• Announced Mar 13, 2026• 0 citations
v1 · Mar 11, 2026v2 · Mar 26, 2026
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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.

Comments
15 pages, 10 figures

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Low TTT-count preparation of nuclear eigenstates with tensor networks
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+T+T gateset, exploring the benefits of specialized U3U3U3 synthesis techniques. For all nuclear systems tested, on up to 76 qubit Hamiltonians, we consistently find low TTT-count circuits preparing the nuclear eigenstates to high fidelity with ∼2×104\sim 2\times 10^4∼2×104 total TTT 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.
We propose a hybrid quantum-classical algorithm for approximating the ground state of two-dimensional quantum systems using an isometric tensor network ansatz, which maps naturally to quantum circuits. Inspired by the density matrix renormalization group, we optimize tensors sequentially by diagonalizing a series of effective Hamiltonians. These are constructed using a tomography-inspired method on a qubit subset whose size depends only on the bond dimension. Our approach leverages quantum computers to enable accurate solutions without relying on approximate contractions, circumventing the exponential complexity faced by classical techniques. We demonstrate our method on the two-dimensional (2D) transverse-field Ising model, achieving ground-state optimization on up to 25 qubits with modest quantum overhead -- significantly less than standard solutions based on variational quantum eigensolvers. Overall, our results offer a path towards scalable variational quantum algorithms in both noisy and fault-tolerant regimes.
Compiling classically optimized matrix product states (MPSs) into quantum circuits is a promising route to initial-state preparation for strongly correlated molecules, but the accuracy, gate cost, and classical scalability of existing MPS compilation methods have not been systematically assessed. Here, we benchmark representative analytical, variational, and hybrid MPS compilation methods for strongly correlated systems including hydrogen chains and iron-sulfur clusters. We compare state infidelity, CNOT gate count, classical runtime, and parallel scalability across different system sizes and bond dimensions. We also introduce a divide-and-conquer environment-aware compilation algorithm (DCE), which optimizes each local isometry approximation according to its contribution to the prepared-state fidelity. The results reveal complementary operating regimes: direct variational compilation methods are gate-efficient for some smaller molecular systems with O(10)O(10)O(10) qubits and O(10)O(10)O(10) bond dimensions or moderate accuracy targets with infidelity of order 10−110^{-1}10−1, whereas divide-and-conquer methods scale more favorably for larger systems and relatively higher bond dimensions of O(10-102)O(10\text{-}10^2)O(10-102), particularly when target infidelities below 10−110^{-1}10−1 are required. Within the latter family, environment-aware optimization improves accuracy over unitary-distance-based compilation while retaining task-level parallelism. For the FeMo-cofactor (FeMoco), we find the DCE algorithm together with the MPS obtained in the entanglement-minimized orbitals (EMOs) provides a resource-efficient route to initial-state preparation. Our results provide practical guidance for selecting and developing MPS compilation methods for strongly correlated molecular simulations.
Quantum algorithms for ground-state energy estimation require initial states with non-negligible fidelity to the ground state. Guided by classical simulations, we design shallow circuits for the two-dimensional half-filled Hubbard model by first preparing an approximate Heisenberg ground state and then applying charge-fluctuation gates derived directly from the Schrieffer-Wolff (SW) generator. For the Heisenberg model, we demonstrate effective parameter transfer from a 4×44\times44×4 lattice to lattices up to 10×1010\times1010×10 without further optimization. We obtain promising lower bounds on the ground-state fidelity using tensor-network simulations, variational Monte Carlo reference states, and estimates of the ground-state energy and singlet gap. Exact 4×44\times44×4 calculations show that the SW gates substantially improve the Hubbard ground-state fidelity of the embedded Heisenberg state. The successful Heisenberg parameter transfer supports the use of these small-lattice results to design shallow Hubbard circuits on larger lattices beyond the reach of classical simulations. For a 10×1010\times1010×10 lattice, the estimated preparation cost is on the order of 10510^5105 TTT gates, including Clifford+TTT synthesis, which is well within the megaquop regime. These results offer a route to low-cost initial-state preparation for ground-starongly correlated systems.
Hamiltonian simulation on quantum computers is strongly constrained by gate counts, motivating techniques to reduce circuit depths. While tensor networks are natural competitors to quantum computers, we instead leverage them to support circuit design, with datasets of tensor networks enabling a unitary synthesis inspired by quantum machine learning. For a target simulation in the thermodynamic limit, translation invariance is exploited to significantly reduce the optimization complexity, avoiding a scaling with system size. Our approach finds circuits to efficiently prepare ground states, and perform time evolution on both infinite and finite systems with substantially lower gate depths than conventional Trotterized methods. In addition to reducing CNOT depths, we motivate similar utility for fault-tolerant quantum algorithms, with a demonstrated 5.2×5.2\times5.2× reduction in TTT-count to realize e−iHte^{-iHt}e−iHt. The key output of our approach is the optimized unit-cell of a translation invariant circuit. This provides an advantage for Hamiltonian simulation of finite, yet arbitrarily large, systems on real quantum computers.
To maximize the value of fault-tolerant quantum computers, it is essential to develop concrete applications beyond well-established domains such as chemistry and condensed-matter physics. Here we construct and compile quantum algorithms to simulate the structure of atomic nuclei -- a topic that has received relatively little attention from the quantum computing community despite its similarities to the electronic structure problem in chemistry -- via effective shell-model Hamiltonians and no-core-shell-model Hamiltonians with three-body interactions derived from chiral effective field theory. Furthermore, we provide quantum resource estimates, in terms of Toffoli gate and qubit counts, for these algorithms, which, to our knowledge, are the first such estimates for fault-tolerant quantum simulation of atomic nuclei. Notably, the estimates for 32^{32}32Mg and 219^{219}219At shell-model Hamiltonians are comparable to recent estimates of Femoco simulations, a standard benchmark in chemistry. For no-core-shell-model Hamiltonians suitable for light nuclei (up to 40^{40}40Ca or so), we find that resource requirements are significantly higher, suggesting that more bespoke strategies are required to make such simulations practicable. Throughout this work, we draw upon the similarities between nuclear and electronic structure problems, while also highlighting challenges that are specific to the former. We hope this work will spur long-term collaborations between the nuclear and quantum computing community with the ultimate goal of realizing useful nuclear simulations on quantum computers.
The ability to simulate a fermionic system on a quantum computer is expected to revolutionize chemical engineering, materials design, nuclear physics, to name a few. Thus, optimizing the simulation circuits is of significance in harnessing the power of quantum computers. Here, we address this problem in two aspects. In the fault-tolerant regime, we optimize the Rz and T gate counts along with the ancilla qubit counts required, assuming the use of a product-formula algorithm for implementation. We obtain a savings ratio of two in the gate counts and a savings ratio of eleven in the number of ancilla qubits required over the state of the art. In the pre-fault tolerant regime, we optimize the two-qubit gate counts, assuming the use of the variational quantum eigensolver (VQE) approach. Specific to the latter, we present a framework that enables bootstrapping the VQE progression towards the convergence of the ground-state energy of the fermionic system. This framework, based on perturbation theory, is capable of improving the energy estimate at each cycle of the VQE progression, by about a factor of three closer to the known ground-state energy compared to the standard VQE approach in the test-bed, classically-accessible system of the water molecule. The improved energy estimate in turn results in a commensurate level of savings of quantum resources, such as the number of qubits and quantum gates, required to be within a pre-specified tolerance from the known ground-state energy. We also explore a suite of generalized transformations of fermion to qubit operators and show that resource-requirement savings of up to more than 20% , in small instances, is possible.