We propose a practical method for accurately evaluating molecular energies using a hybrid approach that integrates fault-tolerant quantum computers with classical computing. Our scheme comprises two complementary methods: quantum dominant orbital selection (QDOS) and subspace dynamical correlation (SDC). The QDOS method extracts only the relevant active orbitals from the complete active space (CAS) configuration interaction (CI) state on a quantum computer, thereby defining a more compact active space suitable for subsequent classical CASCI calculations. The SDC method evaluate correction of dynamical correlation of the CASCI obtained by quantum computing by using the compact CASCI state, which can be handled by classical computing. To demonstrate that the CAS energy resulting from the quantum computation is post-corrected by the SDC method, we examine the two frameworks, multi-reference perturbation theory and tailored coupled-cluster theory, for the SDC method. Our scheme does not suffer from massive task to read out quantum data readout and demonstrates the potential to efficiently compute large, complex molecular systems by leveraging quantum-classical hybrid computation with reasonable computational resources.
Accurate computation of non-covalent, intermolecular interaction energies is important to understand various chemical phenomena, and quantum computers are anticipated to accelerate it. Although the state-of-the-art quantum computers are still noisy and intermediate-scale ones, development of theoretical frameworks those are expected to work on a fault-tolerant quantum computer is an urgent issue. In this work, we explore resource-efficient implementation of the quantum phase estimation-based complete active space configuration interaction (QPE-CASCI) calculations, with the aid of the second-order Møller--Plesset perturbation theory (MP2)-based active space selection with Boys localized orbitals. We performed numerical simulations of QPE for the supramolecular approach-based intermolecular interaction energy calculations of the hydrogen-bonded water dimer, using 6 system and 6 ancilla qubits. With the aid of algorithmic error mitigation, the QPE-CASCI simulations achieved interaction energy predictions with an error of 0.02 kcal mol
−1 relative to the CASCI result, demonstrating the accuracy and efficiency of the proposed methodology. Preliminary results on quantum circuit compression for QPE are also presented to reduce the number of two-qubit gates and depth.
We present the quantum-selected configuration interaction-tailored coupled-cluster (QSCI-TCC) method, a hybrid quantum-classical scheme that tailors coupled-cluster (CC) theory with a quantum-selected configuration interaction (QSCI) wave function. QSCI provides a scalable, shot-efficient approach to reconstructing the many-electron state prepared on quantum hardware on a classical computer. The resulting active-space CI coefficients, which are free from additive shot noise, are mapped to fixed cluster amplitudes within the tailored coupled-cluster framework, after which a conventional CC calculation optimizes the remaining amplitudes. This workflow embeds static (strong) correlation from the quantum device and subsequently recovers dynamical (weak) correlation, yielding a balanced description of both. The method is classically simulated and applied to the simultaneous O-H bond dissociation in H
2O and the triple-bond dissociation in N
2. QSCI-TCC and its perturbative-triples variant, QSCI-TCC(T), provide accurate results even where CCSD or CCSD(T) begin to break down. Shot-count tests for the N
2 (6e, 6o) active space demonstrate that, with the (c) correction, chemically sufficient precision (
≤1 kcal/mol) is achieved with only
1.0×105 shots in the strongly correlated regime (
r=2.2 Å) -- an order of magnitude fewer than required by an earlier matchgate-shadows implementation [J. Chem. Theory Comput., 20, 5068 (2024)]. By pairing resource-efficient quantum sampling with the CC theory, QSCI-TCC provides a promising pathway to quantum-chemical calculations of classically intractable systems.
We present a multilevel embedding framework for quantum chemistry calculations on a quantum computer. In our framework, a quantum algorithm treats the strongly correlated active space, while a high-level wave-function method such as coupled cluster theory or multireference perturbation theory recovers the remaining correlation in the surrounding region. A sampling-based quantum algorithm, quantum-selected configuration interaction, bridges the quantum and classical treatments. The entire calculation is embedded in a low-cost density functional theory description of the surrounding environment using Manby's projection technique. We apply the framework to organic, metal-organic, and metallic systems, computing bond dissociation energies, adsorption energies, and reaction barriers using only the subset of qubits of a 144-qubit superconducting quantum computer at the University of Osaka and achieving $\sim$1 kcal/mol agreement with classical references for a Menshutkin
SN2 reaction inside a carbon nanotube. Our results may open the way to quantitatively reliable quantum-classical hybrid calculations for large-scale chemical systems.
Quantum simulation of molecular electronic structure is one of the most promising applications of quantum computing. However, achieving chemically accurate predictions for strongly correlated systems requires quantum phase estimation (QPE) on fault-tolerant quantum computing (FTQC) devices. Existing resource estimates for typical FTQC architectures suggest that such calculations demand millions of physical qubits, thereby placing them beyond the reach of near-term devices. Here, we investigate the feasibility of performing QPE for chemically relevant molecular systems in an early-FTQC regime, characterized by partial fault tolerance, constrained qubit budgets, and limited circuit depth. Our framework is based on single-ancilla, Trotter-based QPE implementations combined with partially randomized time evolution. Within this framework, we develop a novel Hamiltonian optimization strategy, termed unitary weight concentration, that reduces algorithmic cost by reshaping linear-combination-of-unitaries representations. Applying this framework to active-space models of iron-sulfur clusters, cytochrome P450 active sites, and CO
2-utilization catalysts, we perform end-to-end resource estimation using the latest version of the space-time efficient analog rotation (STAR) architecture. Our results indicate that ground-state energy estimation for active spaces of approximately 20 to 50 spatial orbitals, well beyond the reach of classical full configuration interaction, is achievable using
∼105 physical qubits, with runtimes on the order of days to weeks. These findings demonstrate that while full-fledged fault-tolerant quantum computers will be required for even larger molecular simulations, chemically meaningful quantum chemistry problems are already within reach in an experimentally relevant, early-FTQC regime.
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.
The wave function prepared by a variational quantum eigensolver (VQE) can contain a substantial classically tractable component, yet its energy contribution is sampled on the quantum device. Representing the wave function as a combination of a classical reference and separately prepared quantum states allows the reference energy to be evaluated without quantum sampling. However, near-linear dependence among these components can amplify measurement errors and destabilize the resulting energy estimate. We introduce Classical Active-Space Hybrid Quantum Subspace Expansion (CASH-QSE), which retains a complete-active-space self-consistent field (CASSCF) reference classically and uses occupation structure to construct quantum states exactly orthogonal to the reference and to one another. The energy follows from an ordinary Hermitian eigenvalue problem without overlap measurements, while the same occupation constraints simplify the measured operators. Using full configuration interaction to guide component selection, we test CASH-QSE along H
2O and N
2 bond-stretching coordinates spanning weakly to strongly correlated regimes. The benchmarks use STO--3G for both molecules and a restricted cc-pVDZ orbital space for H
2O. CASH-QSE reaches chemical accuracy while limiting the largest complete measurement circuits to a few hundred all-to-all logical controlled-NOT gates. In favorable cases, it also reduces the idealized final-energy sampling cost by several orders of magnitude relative to VQE with an adaptive derivative-assembled pseudo-Trotter ansatz (ADAPT-VQE).