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arXiv:2610.33800

Efficient Block Encodings of Vibrational Hamiltonians

Shashank G. Mehendale, Ignacio Loaiza, Shreyas Malpathak, Stepan Fomichev, Danial Motlagh, Artur F. Izmaylov

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

Simulation of molecular vibrations is emerging as a promising application of quantum computing in quantum chemistry. In this work, we develop new strategies for block-encoding the second-quantized $n$-mode representation of the vibrational Hamiltonian, whose algebraic structure closely parallels that of the second-quantized electronic Hamiltonian. We derive linear-combination-of-unitaries (LCU) representations of this Hamiltonian based on Pauli decomposition and tensor hypercontraction (THC) and construct efficient circuits for the corresponding block encodings. We also extend the Block-Invariant Symmetry Shift (BLISS) technique to vibrational Hamiltonians, yielding a lower LCU 1-norm and, consequently, a lower overall circuit cost. For the molecules considered here, BLISS-THC reduces the Toffoli cost of quantum phase estimation by more than two orders of magnitude relative to existing block-encoding approaches. In comparison with recent Trotter-based methods, it shows up to a ninefold reduction in the T-gate count, with the relative savings tending to increase with system size. Asymptotically, and up to polylogarithmic factors, both the Pauli and THC LCUs trade additional ancillas for a quadratic improvement in scaling with the number of modes, while THC provides an additional quadratic improvement in scaling with the number of modals. These results highlight the importance of Hamiltonian factorizations and Toffoli-qubit tradeoffs for reducing the computational cost of quantum simulations of molecular vibrations.

Comments
42 pages, 10 figures, 4 tables

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Simulation of molecular vibrations is emerging as a promising application of quantum computing in quantum chemistry. In this work, we develop new strategies for block-encoding the second-quantized nnn-mode representation of the vibrational Hamiltonian, whose algebraic structure closely parallels that of the second-quantized electronic Hamiltonian. We derive linear-combination-of-unitaries (LCU) representations of this Hamiltonian based on Pauli decomposition and tensor hypercontraction (THC) and construct efficient circuits for the corresponding block encodings. We also extend the Block-Invariant Symmetry Shift (BLISS) technique to vibrational Hamiltonians, yielding a lower LCU 1-norm and, consequently, a lower overall circuit cost. For the molecules considered here, BLISS-THC reduces the Toffoli cost of quantum phase estimation by more than two orders of magnitude relative to existing block-encoding approaches. In comparison with recent Trotter-based methods, it shows up to a ninefold reduction in the T-gate count, with the relative savings tending to increase with system size. Asymptotically, and up to polylogarithmic factors, both the Pauli and THC LCUs trade additional ancillas for a quadratic improvement in scaling with the number of modes, while THC provides an additional quadratic improvement in scaling with the number of modals. These results highlight the importance of Hamiltonian factorizations and Toffoli-qubit tradeoffs for reducing the computational cost of quantum simulations of molecular vibrations.
Computational cost of energy estimation for molecular electronic Hamiltonians via quantum phase estimation (QPE) grows with the difference between the largest and smallest eigenvalues of the Hamiltonian. In this work, we propose a preprocessing procedure that reduces the norm of the Hamiltonian without changing its eigenspectrum for the target states of a particular symmetry. The new procedure, block-invariant symmetry shift (BLISS), builds an operator T̂ such that the cost of implementing H ^ − T ^ is reduced compared to that of Ĥ, yet H ^ − T ^ acts on the subspaces of interest the same way as Ĥ does. BLISS performance is demonstrated for a linear combination of unitaries (LCU)-based QPE approaches on a set of small molecules. Using the number of electrons as the symmetry specifying the target set of states, BLISS provided a factor of 2 reduction of 1-norm for several LCU decompositions compared to their unshifted versions.
Accurately characterizing molecular properties and reactivity often requires electronic-structure calculations across an ensemble of molecular geometries. Quantum algorithms that process such ensembles coherently could address this challenge, but they require Hamiltonian block encoding circuits that combine advanced factorization techniques with sublinear scaling in ensemble size. Here, we introduce SVD-THC, which extends tensor hypercontraction to sets of Hamiltonians defined at different nuclear geometries through a singular value decomposition of the ensemble. We use SVD-THC to construct a single multiplexed block encoding circuit for all geometries in an ensemble. We then evaluate the approach on ensembles generated by varying internal rotational coordinates in 1-bromo-2-chloroethane as well as in glycine and alanine dipeptides, corresponding to established model systems for peptide conformational landscapes, with up to 5,184 geometries. Across several resolutions of the internal-coordinate grids, the Toffoli complexity of the resulting circuits grows only weakly with the number of geometries, changing by at most 12\% over grids that differ in size by up to 576×\times×. In an energy-thresholding circuit, this yields 35--200×\times× fewer Toffoli gates than slicing together single-geometry block encodings. This approach could help make coherent processing of molecular geometry ensembles a useful primitive for fault-tolerant quantum computing.
We describe quantum circuits with only O(N ) Toffoli complexity that block encode the spectra of quantum chemistry Hamiltonians in a basis of N arbitrary (e.g., molecular) orbitals.With O(λ/ ) repetitions of these circuits one can use phase estimation to sample in the molecular eigenbasis, where λ is the 1-norm of Hamiltonian coefficients and is the target precision.This is the lowest complexity shown for quantum computations of chemistry within an arbitrary basis.Furthermore, up to logarithmic factors, this matches the scaling of the most efficient prior block encodings that can work only with orthogonal-basis functions diagonalizing the Coloumb operator (e.g., the plane-wave dual basis).Our key insight is to factorize the Hamiltonian using a method known as tensor hypercontraction (THC) and then to transform the Coulomb operator into an isospectral diagonal form with a nonorthogonal basis defined by the THC factors.We then use qubitization to simulate the nonorthogonal THC Hamiltonian, in a fashion that avoids most complications of the nonorthogonal basis.We also reanalyze and reduce the cost of several of the best prior algorithms for these simulations in order to facilitate a clear comparison to the present work.In addition to having lower asymptotic scaling space-time volume, compilation of our algorithm for challenging finitesized molecules such as FeMoCo reveals that our method requires the least fault-tolerant resources of any known approach.By laying out and optimizing the surface-code resources required of our approach we show that FeMoCo can be simulated using about four million physical qubits and under 4 days of runtime, assuming 1-μs cycle times and physical gate-error rates no worse than 0.1%.
Hamiltonian simulation is a fundamental application of quantum computing, particularly in quantum chemistry, where solving the electronic structure problem requires efficiently representing electron repulsion integrals (ERI). This thesis analyzes the computational complexity of Hamiltonian simulation using the Linear Combination ofUnitaries (LCU) algorithm and Tensor Hypercontraction (THC), a factorization method that reduces the cost of ERI representation. By decomposing the Coulomb tensor via Interpolative Separable Density Fitting (ISDF) and mapping the THC representation to an LCU form using the Jordan-Wigner transformation, we achieve a more efficient block encoding of the Hamiltonian in a basis of N molecular orbitals. This enables quantum simulation with an asymptotic Toffoli complexity of O(Nl/e) and a space complexity of O(N), where e = 0.0016 Hartree is the target accuracy and l denotes the 1-norm of the coefficients of the electronic Hamiltonian in the THC representation. Beyond Hamiltonian simulation, we explore how Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) leverage block encoding for eigenvalue transformations and time evolution, extending its utility to a broader class of quantum algorithms. Despite these advancements, large-scale quantum chemistry simulations are far from experimental feasibility, for instance, simulating the FeMoCo molecule in an active space model under surface code constraints would require billions of Toffoli gates, millions of physical qubits and several days of runtime using error-corrected qubits; and that is assuming it ever becomes feasible. This work contributes to the effort of making quantum chemistry simulations more computationally viable by investigating how LCU and THC can be integrated to reduce costs and maintain chemical accuracy, while exploring different methods and their combinations to identify key challenges in quantum chemistry simulations.
Fault tolerant quantum simulation via the phase estimation algorithm and qubitization has a T-gate count that scales proportionally to the 1-norm of the Hamiltonian, the cost of block encoding the Hamiltonian, and inversely proportionally to the desired accuracy. Tensor factorization methods have been successfully used to reduce T-gate counts in the ground state electronic structure problem. Here we introduce the use of tensor factorization methods to reduce the T-gate count of quantum phase estimation. In particular, we show how Canonical Polyadic and Tucker decompositions of the tensors representing the vibrational and vibronic Hamiltonians can be utilized to rewrite the Hamiltonian in terms of linear combination of bosonic position operators representing nuclear vibrations. We demonstrate the use of these factorization methods on the water and monodeutered methane molecules.
In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous best techniques for industrially and biologically relevant molecules, including the nitrogenase cofactor (FeMoCo) and cytochrome P450. Benchmark calculations comparing our method with the original double-factorization method show a 75% reduction in the block-encoding scaling constant. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the journey to the feasibility of practical Hamiltonian simulations.