Achieving chemical accuracy for strongly correlated molecules is a defining milestone for first-generation, fault-tolerant quantum computers, yet the factorial growth of three, four, and six-index tensor contractions in coupled-cluster CCSD(T), full configuration interaction (FCI), and multireference CI (MRCI) makes current classical and quantum approaches prohibitive. We introduce tensor-factorized Hamiltonian downfolding (TFHD) and its quantum analogue, qubitized downfolding (QD)- a hybrid classical-quantum framework that collapses every high-rank object to rank-2 networks executed in depth-optimal, block-encoded circuits. The complexity of these operations scales exponentially with the system size. We aim to find properties of chemical systems by optimizing this scaling through mathematical transformations on the Hamiltonian and the state space. By defining a bi-partition of the many-body Hilbert space into electronoccupied and electron-unoccupied blocks for a given orbital, we perform a downfolding transformation that decouples the electron-occupied block from its complement. We factorize high-rank electronic integrals and cluster amplitude tensors into low-rank tensor factors of a downfolding transformation, mapping the full many-body Hamiltonian into a smaller dimensional block-Hamiltonians. This reduces the computational complexity of solving the residual equations for Hamiltonian downfolding from O(N7) for CCSD(T) and O(N9) - O(N10) for CI and MRCI to O(N3). This operations can be implemented as a family of tensor networks solely made from two-rank tensors. Additionally, we create block-encoding quantum circuits of the tensor networks, generating circuits of O(N2) depth with O(logN) qubits. We demonstrate super-quadratic speedups of expensive quantum chemistry algorithms on both classical and quantum computers.
Achieving chemical accuracy for strongly correlated molecules is a defining milestone for first-generation, fault-tolerant quantum computers, yet the factorial growth of three, four, and six-index tensor contractions in coupled-cluster CCSD(T), full configuration interaction (FCI), and multireference CI (MRCI) makes current classical and quantum approaches prohibitive. We introduce tensor-factorized Hamiltonian downfolding (TFHD) and its quantum analogue, qubitized downfolding (QD)- a hybrid classical-quantum framework that collapses every high-rank object to rank-2 networks executed in depth-optimal, block-encoded circuits. The complexity of these operations scales exponentially with the system size. We aim to find properties of chemical systems by optimizing this scaling through mathematical transformations on the Hamiltonian and the state space. By defining a bi-partition of the many-body Hilbert space into electronoccupied and electron-unoccupied blocks for a given orbital, we perform a downfolding transformation that decouples the electron-occupied block from its complement. We factorize high-rank electronic integrals and cluster amplitude tensors into low-rank tensor factors of a downfolding transformation, mapping the full many-body Hamiltonian into a smaller dimensional block-Hamiltonians. This reduces the computational complexity of solving the residual equations for Hamiltonian downfolding from O(N7) for CCSD(T) and O(N9) - O(N10) for CI and MRCI to O(N3). This operations can be implemented as a family of tensor networks solely made from two-rank tensors. Additionally, we create block-encoding quantum circuits of the tensor networks, generating circuits of O(N2) depth with O(logN) qubits. We demonstrate super-quadratic speedups of expensive quantum chemistry algorithms on both classical and quantum computers.
The practical application of quantum technologies to chemical problems faces significant challenges, particularly in the treatment of realistic basis sets and the accurate inclusion of electron correlation effects. A direct approach to these problems is currently infeasible due to limitations in the number of logical qubits, their fidelity, and the shallow circuit depths supported by existing hardware, all of which hinder simulations at the required level of accuracy. A promising alternative is hybrid quantum-classical computing, where classical resources are used to construct effective Hamiltonians characterized by dimensions that conform to the constraints of current quantum devices. In this paper, we demonstrate the performance of a hybrid approach: coupled cluster downfolded Hamiltonians are first evaluated in reduced-dimensionality active spaces, and the corresponding ground-state energies are subsequently computed using quantum algorithms. Our comprehensive analysis explores the achievable accuracy in recovering correlation energies when hundreds of orbitals are downfolded into a problem size tractable by today’s quantum hardware. We argue that such flexible hybrid algorithms, where problem size can be tailored to available quantum resources, can serve as a bridge between noisy intermediate-scale quantum devices and future fault-tolerant quantum computers, marking a step toward the early realization of quantum advantage in chemistry.
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%.
Applications of quantum simulation algorithms to obtain electronic energies of molecules on noisy intermediate-scale quantum (NISQ) devices require careful consideration of resources describing the complex electron correlation effects. In modeling second-quantized problems, the biggest challenge confronted is that the number of qubits scales linearly with the size of the molecular basis. This poses a significant limitation on the size of the basis sets and the number of correlated electrons included in quantum simulations of chemical processes. To address this issue and enable more realistic simulations on NISQ computers, we employ the double unitary coupled-cluster (DUCC) method to effectively downfold correlation effects into the reduced-size orbital space, commonly referred to as the active space. Using downfolding techniques, we demonstrate that properly constructed effective Hamiltonians can capture the effect of the whole orbital space in small-size active spaces. Combining the downfolding preprocessing technique with the variational quantum eigensolver, we solve for the ground-state energy of H 2, Li 2, and BeH 2 in the cc-pVTZ basis using the DUCC-reduced active spaces. We compare these results to full configuration-interaction and high-level coupled-cluster reference calculations.
The accurate first-principles description of strongly correlated materials is an important and challenging problem in condensed matter physics. Ab initio downfolding has emerged as a way of deriving compressed many-body Hamiltonians that maintain the essential physics of strongly correlated materials. The solution of these material-specific models is still exponentially difficult to generate on classical computers, but quantum algorithms allow for a significant speed-up in obtaining the ground states of these compressed Hamiltonians. Here, we demonstrate that using quantum algorithms to obtain the properties of downfolded Hamiltonians can indeed yield high-fidelity solutions. By combining ab initio downfolding and variational quantum eigensolvers, we correctly predict the antiferromagnetic state of one-dimensional cuprate Ca2CuO3, the excitonic ground state of monolayer WTe2, and the charge-ordered state of correlated metal SrVO3. Numerical simulations using a classical tensor network implementation of variational quantum eigensolvers allow us to simulate large models with up to 54 qubits and encompassing up to four bands in the correlated subspace, which is indicative of the complexity that our framework can address. Through these methods we demonstrate the potential of classical preoptimization and downfolding techniques for enabling efficient materials simulation using quantum algorithms.
Abstract Electronic ground states are of central importance in chemical simulations, but have remained beyond the reach of efficient classical algorithms except in cases of weak electron correlation or one-dimensional spatial geometry. We introduce a hybrid quantum-classical eigenvalue solver that constructs a wavefunction ansatz from a linear combination of matrix product states in rotated orbital bases, enabling the characterization of strongly correlated ground states with arbitrary spatial geometry. The energy is converged via a gradient-free generalized sweep algorithm based on quantum subspace diagonalization, with a potentially exponential speedup in the off-diagonal matrix element contractions upon translation into compact quantum circuits of linear depth in the number of qubits. Chemical accuracy is attained in numerical experiments for both a stretched water molecule and an octahedral arrangement of hydrogen atoms, achieving substantially better correlation energies compared to a unitary coupled-cluster benchmark, with orders of magnitude reductions in quantum resource estimates and a surprisingly high tolerance to shot noise. This proof-of-concept study suggests a promising new avenue for scaling up simulations of strongly correlated chemical systems on near-term quantum hardware.
Accurate prediction of molecular structures and properties is vital for chemistry; materials science, and drug discovery, yet classical electronic-structure methods often fail for strongly correlated systems, large basis sets, and complex potential-energy landscapes. Quantum technology encompassing quantum computing, quantum machine learning, and hybrid quantum classical strategies offers a fundamentally new paradigm by encoding many-electron wave functions directly on qubits and exploiting superposition and entanglement to explore exponentially large Hilbert spaces. This review synthesizes recent algorithmic and hardware advances relevant to molecular modelling, including the Variational Quantum Eigensolver (VQE), Quantum Phase Estimation (QPE), quantum unitary coupled-cluster (q-UCC) families, equation-of-motion and subspace methods for excited states, tensor-network hybrids, and quantum kernel and variational QML approaches. We examine noise-aware hybrid workflows, error-mitigation techniques, symmetry-preserving ansätze, and operator-factorization methods that reduce measurement and gate overhead. Representative applications are discussed: ground- and excited-state energy prediction, potential-energy surface mapping, geometry and transition-state optimization, spectroscopic property estimation (IR, UV–Vis, NMR, EPR), and reaction-dynamics scenarios where non-adiabatic effects and conical intersections dominate. Resource estimates and scaling analyses clarify current NISQ limitations qubit counts, circuit depth, shot complexity and delineate the roadmap to fault-tolerant QPE for chemical accuracy. We compare quantum approaches with classical baselines (DFT, CCSD (T), multireference methods), identifying domains where quantum methods already show promise (strong correlation, multi-reference dissociation, spin-state ordering) and where classical methods remain competitive. Finally, we highlight near-term industrial opportunities in drug design, catalysis, CO2 capture, and energy materials, and outline critical research directions: algorithmic reductions in measurement/precision cost, hardware improvements in fidelity and connectivity, scalable ansatz design, and integrated software stacks for reproducible hybrid simulations. Together, these developments indicate that while practical, large-scale quantum advantage for general chemistry remains future work, quantum technologies are rapidly maturing into powerful tools for targeted molecular problems that are intractable with existing classical techniques.