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arXiv:2610.33834

Initial-State Preparation for Strongly Correlated Molecules from Matrix Product States: Benchmarking and Environment-Aware Divide-and-Conquer Compilation

Haiqi Liu, Longfei Chang, Zhendong Li, Wei-Hai Fang

quant-phSubmitted Oct 24, 2026• Announced Oct 27, 2026• 0 citations
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Abstract

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 $O(10)$ qubits and $O(10)$ bond dimensions or moderate accuracy targets with infidelity of order $10^{-1}$, whereas divide-and-conquer methods scale more favorably for larger systems and relatively higher bond dimensions of $O(10\text{-}10^2)$, particularly when target infidelities below $10^{-1}$ 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.

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Quantum computation offers significant potential for accelerating the simulation of molecules and materials through algorithms such as quantum phase estimation (QPE). However, the expected speedup in ground-state energy estimation depends critically on the ability to efficiently prepare an initial state with high overlap with the true ground state. For strongly correlated molecules such as iron-sulfur clusters, this overlap is demonstrated to decay exponentially with system size. To alleviate this problem, we introduce an efficient classical algorithm to find entanglement-minimized orbitals (EMOs) and their orderings using spin-adapted low-bond-dimension matrix product states (MPS). The EMO basis yields a more compact ground-state representation, significantly easing initial state preparation for challenging systems. Our algorithm improves initial state overlap by nearly an order of magnitude over prior orbital optimization approaches for an iron-sulfur cluster with four irons, and is scalable to larger systems with many unpaired electrons, including the P-cluster and FeMo-cofactor in nitrogenase with eight transition metal centers. For these systems, we achieve substantial enhancements on initial state overlap by factors of O(10^{2}) and O(10^{5}), respectively, compared to results obtained using localized orbitals. Our results show that initial state preparation for these challenging systems requires far fewer resources than prior estimates suggested.

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Scalable Preparation of Matrix Product States with Sequential and Brick Wall Quantum Circuits

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Preparing arbitrary quantum states requires exponential resources. Matrix Product States (MPS) admit more efficient constructions, particularly when accuracy is traded for circuit complexity. Existing approaches to MPS preparation mostly rely on heuristic circuits that are deterministic but quickly saturate in accuracy, or on variational optimization methods that reach high fidelities but scale poorly. This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline. Heuristic staircase-like and brick wall disentangler circuits provide warm-start initializations for variational optimization, enabling high-fidelity state preparation for large systems. Target MPSs are either specified as physical quantum states or constructed from classical datasets via amplitude encoding, using step-by-step singular value decompositions or tensor cross interpolation. The framework incorporates entanglement-based qubit reordering, reformulated as a quadratic assignment problem, and low-level optimizations that reduce depths by up to 50% and CNOT counts by 33%. We evaluate the full pipeline on datasets of varying complexity across systems of 19-50 qubits and identify trade-offs between fidelity, gate count, and circuit depth. Optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts. Overall, our results provide principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.

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Rapid Initial-State Preparation for the Quantum Simulation of Strongly Correlated Molecules

Dominic William Berry, Yu Tong, Tanuj Khattar +10 more

Studies on quantum algorithms for ground-state energy estimation often assume perfect ground-state preparation; however, in reality the initial state will have imperfect overlap with the true ground state. Here, we address that problem in two ways: by faster preparation of matrix-product-state (MPS) approximations and by more efficient filtering of the prepared state to find the ground-state energy. We show how to achieve unitary synthesis with a Toffoli complexity about 7 × lower than that in prior work and use that to derive a more efficient MPS-preparation method. For filtering, we present two different approaches: sampling and binary search. For both, we use the theory of window functions to avoid large phase errors and minimize the complexity. We find that the binary-search approach provides better scaling with the overlap at the cost of a larger constant factor, such that it will be preferred for overlaps less than about 0.003 . Finally, we estimate the total resources to perform ground-state energy estimation of Fe - S cluster systems, including the Fe Mo cofactor by estimating the overlap of different MPS initial states with potential ground states of the Fe Mo cofactor using an extrapolation procedure. With a modest MPS bond dimension of 4000 , our procedure produces an estimate of approximately 0.9 overlap squared with a candidate ground state of the Fe Mo cofactor, producing a total resource estimate of 7.3 × 10 10 Toffoli gates; neglecting the search over candidates and assuming the accuracy of the extrapolation, this validates prior estimates that have used perfect ground-state overlap. This presents an example of a practical path to prepare states of high overlap in a challenging-to-compute chemical system.

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Abstract Preparing arbitrary quantum states requires exponential resources. Matrix product states (MPS) admit more efficient constructions, particularly when accuracy is traded for circuit complexity. Existing approaches to MPS preparation mostly rely on heuristic circuits that are deterministic but quickly saturate in accuracy, or on variational optimization methods that reach high fidelities but scale poorly. This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline. Heuristic staircase-like and brick wall disentangler circuits provide warm-start initializations for variational optimization, enabling high-fidelity state preparation for large systems. Target MPSs are either specified as physical quantum states or constructed from classical datasets via amplitude encoding, using step-by-step singular value decompositions or tensor cross interpolation. The framework incorporates entanglement-based qubit reordering, reformulated as a quadratic assignment problem, and low-level optimizations that reduce depths by up to 50 % and CNOT counts by 33 % . We evaluate the full pipeline on datasets of varying complexity across systems of 19–50 qubits and identify trade-offs between fidelity, gate count, and circuit depth. Optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts. Overall, our results provide principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.

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Quantum Circuit Matrix Product State Ansatz for Large-Scale Simulations of Molecules

Yi Fan, Jie Liu, Zhenyu Li +1 more

As demonstrated in the density matrix renormalization group (DMRG) method, approximating many-body wave function of electrons using a matrix product state (MPS) is a promising way to solve electronic structure problems. The expressibility of an MPS is determined by the size of the matrices or, in other words, the bond dimension, which unfortunately may be required to be very large in quantum chemistry simulations. In this study, we propose to calculate the ground state energies of molecular systems by variationally optimizing quantum circuit MPS (QCMPS) with a relatively small number of qubits. It is demonstrated that with carefully chosen circuit structure and orbital localization scheme, QCMPS can reach a similar accuracy as that achieved in DMRG with an exponentially large bond dimension. QCMPS simulation of a linear hydrogen molecular chain with 50 orbitals can reach the chemical accuracy using only 6 qubits at a moderate circuit depth. These results suggest that QCMPS is a promising wave function ansatz in the variational quantum eigensolver algorithm for molecular systems.

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Explicitly correlated ground states with a feedback-free variational hybrid algorithm

Benjamin Corbett, Akimasa Miyake

Explicitly correlated methods, which represent dominant short-range correlations by a diagonal Jastrow-type correlator, are designed to accurately describe systems of strongly correlated electrons, such as the fully periodic 2D Fermi--Hubbard model. However, as the reference state is compact in momentum space whereas the correlator acts naturally in real space, there is a fundamental mismatch of bases that classical algorithms resolve only by restricting the reference or the correlator. Here we propose a hybrid algorithm in which a quantum processor prepares a low-entanglement momentum-space matrix product state (MPS), precomputed classically by the transcorrelated method at the cost of a nonvariational energy, rotates it to real space, and measures in bases fixed by the Hamiltonian. Because the measurement bases are independent of the correlator, the correlator can be variationally optimized entirely in classical post-processing with no quantum-classical feedback loop. We demonstrate our algorithm on 32 qubits of Quantinuum's H2-2 with a bond-dimension-4 MPS for the half-filled $4 \times 4$ lattice and achieve a relative ground-state energy error of $16 \pm 4\%$ after postselection, reduced to $3.5 \pm 0.5\%$ by calibration using low-order moments from the MPS. Compared with other feedback-free hybrid algorithms, even under generous assumptions, our method requires fewer samples than quantum-selected configuration interaction and fewer two-qubit gates than quantum Krylov methods.

quant-pharXiv:2610.18681•Oct 14, 2026•0 citations
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 computation offers significant potential for accelerating the simulation of molecules and materials through algorithms such as quantum phase estimation (QPE). However, the expected speedup in ground-state energy estimation depends critically on the ability to efficiently prepare an initial state with high overlap with the true ground state. For strongly correlated molecules such as iron-sulfur clusters, this overlap is demonstrated to decay exponentially with system size. To alleviate this problem, we introduce an efficient classical algorithm to find entanglement-minimized orbitals (EMOs) and their orderings using spin-adapted low-bond-dimension matrix product states (MPS). The EMO basis yields a more compact ground-state representation, significantly easing initial state preparation for challenging systems. Our algorithm improves initial state overlap by nearly an order of magnitude over prior orbital optimization approaches for an iron-sulfur cluster with four irons, and is scalable to larger systems with many unpaired electrons, including the P-cluster and FeMo-cofactor in nitrogenase with eight transition metal centers. For these systems, we achieve substantial enhancements on initial state overlap by factors of O(10^{2}) and O(10^{5}), respectively, compared to results obtained using localized orbitals. Our results show that initial state preparation for these challenging systems requires far fewer resources than prior estimates suggested.
Preparing arbitrary quantum states requires exponential resources. Matrix Product States (MPS) admit more efficient constructions, particularly when accuracy is traded for circuit complexity. Existing approaches to MPS preparation mostly rely on heuristic circuits that are deterministic but quickly saturate in accuracy, or on variational optimization methods that reach high fidelities but scale poorly. This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline. Heuristic staircase-like and brick wall disentangler circuits provide warm-start initializations for variational optimization, enabling high-fidelity state preparation for large systems. Target MPSs are either specified as physical quantum states or constructed from classical datasets via amplitude encoding, using step-by-step singular value decompositions or tensor cross interpolation. The framework incorporates entanglement-based qubit reordering, reformulated as a quadratic assignment problem, and low-level optimizations that reduce depths by up to 50% and CNOT counts by 33%. We evaluate the full pipeline on datasets of varying complexity across systems of 19-50 qubits and identify trade-offs between fidelity, gate count, and circuit depth. Optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts. Overall, our results provide principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.
Studies on quantum algorithms for ground-state energy estimation often assume perfect ground-state preparation; however, in reality the initial state will have imperfect overlap with the true ground state. Here, we address that problem in two ways: by faster preparation of matrix-product-state (MPS) approximations and by more efficient filtering of the prepared state to find the ground-state energy. We show how to achieve unitary synthesis with a Toffoli complexity about 7 × lower than that in prior work and use that to derive a more efficient MPS-preparation method. For filtering, we present two different approaches: sampling and binary search. For both, we use the theory of window functions to avoid large phase errors and minimize the complexity. We find that the binary-search approach provides better scaling with the overlap at the cost of a larger constant factor, such that it will be preferred for overlaps less than about 0.003 . Finally, we estimate the total resources to perform ground-state energy estimation of Fe - S cluster systems, including the Fe Mo cofactor by estimating the overlap of different MPS initial states with potential ground states of the Fe Mo cofactor using an extrapolation procedure. With a modest MPS bond dimension of 4000 , our procedure produces an estimate of approximately 0.9 overlap squared with a candidate ground state of the Fe Mo cofactor, producing a total resource estimate of 7.3 × 10 10 Toffoli gates; neglecting the search over candidates and assuming the accuracy of the extrapolation, this validates prior estimates that have used perfect ground-state overlap. This presents an example of a practical path to prepare states of high overlap in a challenging-to-compute chemical system.
Abstract Preparing arbitrary quantum states requires exponential resources. Matrix product states (MPS) admit more efficient constructions, particularly when accuracy is traded for circuit complexity. Existing approaches to MPS preparation mostly rely on heuristic circuits that are deterministic but quickly saturate in accuracy, or on variational optimization methods that reach high fidelities but scale poorly. This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline. Heuristic staircase-like and brick wall disentangler circuits provide warm-start initializations for variational optimization, enabling high-fidelity state preparation for large systems. Target MPSs are either specified as physical quantum states or constructed from classical datasets via amplitude encoding, using step-by-step singular value decompositions or tensor cross interpolation. The framework incorporates entanglement-based qubit reordering, reformulated as a quadratic assignment problem, and low-level optimizations that reduce depths by up to 50 % and CNOT counts by 33 % . We evaluate the full pipeline on datasets of varying complexity across systems of 19–50 qubits and identify trade-offs between fidelity, gate count, and circuit depth. Optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts. Overall, our results provide principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.
As demonstrated in the density matrix renormalization group (DMRG) method, approximating many-body wave function of electrons using a matrix product state (MPS) is a promising way to solve electronic structure problems. The expressibility of an MPS is determined by the size of the matrices or, in other words, the bond dimension, which unfortunately may be required to be very large in quantum chemistry simulations. In this study, we propose to calculate the ground state energies of molecular systems by variationally optimizing quantum circuit MPS (QCMPS) with a relatively small number of qubits. It is demonstrated that with carefully chosen circuit structure and orbital localization scheme, QCMPS can reach a similar accuracy as that achieved in DMRG with an exponentially large bond dimension. QCMPS simulation of a linear hydrogen molecular chain with 50 orbitals can reach the chemical accuracy using only 6 qubits at a moderate circuit depth. These results suggest that QCMPS is a promising wave function ansatz in the variational quantum eigensolver algorithm for molecular systems.
Explicitly correlated methods, which represent dominant short-range correlations by a diagonal Jastrow-type correlator, are designed to accurately describe systems of strongly correlated electrons, such as the fully periodic 2D Fermi--Hubbard model. However, as the reference state is compact in momentum space whereas the correlator acts naturally in real space, there is a fundamental mismatch of bases that classical algorithms resolve only by restricting the reference or the correlator. Here we propose a hybrid algorithm in which a quantum processor prepares a low-entanglement momentum-space matrix product state (MPS), precomputed classically by the transcorrelated method at the cost of a nonvariational energy, rotates it to real space, and measures in bases fixed by the Hamiltonian. Because the measurement bases are independent of the correlator, the correlator can be variationally optimized entirely in classical post-processing with no quantum-classical feedback loop. We demonstrate our algorithm on 32 qubits of Quantinuum's H2-2 with a bond-dimension-4 MPS for the half-filled 4×44 \times 44×4 lattice and achieve a relative ground-state energy error of 16±4%16 \pm 4\%16±4% after postselection, reduced to 3.5±0.5%3.5 \pm 0.5\%3.5±0.5% by calibration using low-order moments from the MPS. Compared with other feedback-free hybrid algorithms, even under generous assumptions, our method requires fewer samples than quantum-selected configuration interaction and fewer two-qubit gates than quantum Krylov methods.