Method and system for chemically accurate quantum computing

Abstract

The invention relates to a computer implemented method (100) for computing quantum-enhanced chemical property of a chemical system comprising the following steps: - Providing a Hamiltonian (120) of the chemical system subject of the quantum-enhanced chemical computing; - Providing a basis set (130) for the chemical system; - Preparing an initial quantum state (140), on the quantum computation means (10), to represent the chemical system according to the Hamiltonian and using the provided basis set; - Applying, on the quantum computation means (10), a quantum solver (150) on the prepared quantum state in the basis set; and - Computing, on classical computation means (40), a density-based basis-set correction (160) to modify the density-dependent terms in the Hamiltonian

Description

Method and system for chemically accurate quantum computing Field of the invention

[0001] The present invention relates to the field of quantum computing and quantum- computing methods for electronic-structure calculations. In particular, the invention relates to a method and system for chemically accurate quantum computing. Description of Related Art

[0002] In quantum chemistry, computing ground-state energies and molecular properties within chemical accuracy or, in other words, computing ground-state energies and properties in the aim to be comparable to experiments usually requires using very large atomic basis sets on top of an effective wave-function approximation.

[0003] From this requirement, we define three key concepts : (1) the atomic orbitals basis set (AO in this document) which provides the exact ground-state energies and molecular properties is the complete-basis-set (CBS), (2) the molecular orbitals basis set (MO in this document) is a set of functions (in this document, same amount as the amount of AOs) which describes each electron in the molecular environment, it is the set of functions that is actually used in electronic structure calculations, and (3) the wave-function method which leads to the exact solution (by using the CBS) is the full configuration interaction (FCI).

[0004] There exists several mappings of the MO. In the most straightforward quantum computing implementation of quantum chemistry methods, the electronic structure problem is translated from classical to quantum computation by mapping each MO on two qubits (one for each spin). Therefore, increasing the number of MO leads to linear increase of the number of qubits. Current Noisy Intermediate-Scale Quantum (NISQ) devices can only handle a few tens of qubits, so quantum simulations based on the above-mentioned straightforward quantum algorithms have so far been limited to small molecules in small basis sets which forbids to use such algorithm to compute ground state energies and molecular properties with good enough accuracy. This creates a dilemma: achieving chemical accuracy demands large basis sets, but large basis are intractable on near-term quantum hardware (e.g. NISQ) due to qubit limits.

[0005] Quantum algorithms like Quantum Phase Estimation (QPE) and VQE have shown promise for solving molecular electronic structure by finding ground-state wavefunctions.They encode the second-quantized electronic Hamiltonian with one qubit per spin-orbital (one MO = one spin-orbital), allowing representation of an exponentially large FCI (Full Configuration Interaction) Hilbert space with only linearly many qubits. In principle, a fault- tolerant quantum computer could achieve chemically accurate results by applying FCI on complex molecules with the knowledge that it requires a number of orbital, and therefore number of qubit, which increases exponentially with the size of the molecular system. In practice, however, limited qubit counts and circuit depths on NISQ devices prevent using sufficiently extensive bases. Thus, there is a need for further innovation for the quantum chemistry calculations on quantum processors to achieve the accuracy needed for real- world applications.

[0006] Various quantum-algorithm refinements have been proposed to reduce circuit depth or improve ansatz compactness, including ADAPT-VQE and transcorrelated Hamiltonian formulations. These techniques, however, do not directly address the slow convergence of molecular properties with respect to one-electron basis-set size.

[0007] There is therefore a need for quantum-computing procedures that allow the obtention of chemically accurate energies and first-order properties on hardware with limited logical-qubit resources. The present disclosure addresses this need. Summary of the invention

[0008] The following sets forth a simplified summary of selected aspects, embodiments and examples of the present invention for the purpose of providing a basic understanding of the invention. However, the summary does not constitute an extensive overview of all the aspects, embodiments and examples of the invention. The sole purpose of the summary is to present selected aspects, embodiments and examples of the invention in a concise form as an introduction to the more detailed description of the aspects, embodiments and examples of the invention that follow the summary.

[0009] According to a first aspect, a computer-implemented method is provided for computing quantum-enhanced chemical properties of a molecular or periodic system. The method comprises: (a) providing an electronic Hamiltonian, preferably expressed in second-quantized form; (b) providing a finite orbital basis; (c) preparing, on quantum computation means, an initial many-body state thatencodes the electronic Hamiltonian in qubits mapped one-to-one with spin- orbitals of the basis; (d) applying, on the quantum computation means, a quantum solver configured to evaluate expectation values of the Hamiltonian; and (e) applying, on classical computation means, a density-based basis-set correction that, when added to the quantum expectation value, yield a corrected chemical property.

[0010] In certain embodiments, steps (d) and (e) are performed iteratively. At each iteration the functional derivative of the correlation functional of the electron density defines one- and two-electron potential that is added to the Hamiltonian; the modified Hamiltonian is then re-evaluated on the quantum processor until the corrected property converges.

[0011] The invention also relates to a computer implemented method for computing quantum-enhanced chemical properties of a chemical system comprising the following steps: - Providing a Hamiltonian of the chemical system subject of the quantum-enhanced chemical computing; - Providing a basis set for the chemical system; - Preparing an initial quantum state on the quantum computation means to represent the chemical system according to the Hamiltonian and using the provided basis set; - Applying, on the quantum computation means, a quantum solver on the initial quantum state; and - Applying, on classical computation means, a density-based basis-set correction to modify the density-dependent terms in the Hamiltonian, preferably combined with a Hartree-Fock basis-set correction, and computing quantum-enhanced chemical properties of the chemical system.

[0012] The present invention also relates to a computer implemented method for computing quantum-enhanced chemical properties of a chemical system comprising the following steps: - Defining a given number of qubits that will be involved in the computer implemented method; - Providing a basis set having a number of sets of functions for each valence atomic orbital which is defined according to the given number of qubits;- Providing a Hamiltonian of the chemical system subject of the quantum-enhanced chemical computing; - Preparing an initial quantum state on the quantum computation means to represent the chemical system according to the Hamiltonian and using the provided basis set; - Applying, on the quantum computation means, a quantum solver on the initial quantum state; and - Applying, on classical computation means, a density-based basis-set correction to modify the density-dependent terms in the Hamiltonian, preferably combined with a Hartree-Fock basis-set correction, and computing quantum-enhanced chemical properties of the chemical system.

[0013] According to an aspect of the present invention, the invention relates to a computer-implemented method for computing quantum-enhanced chemical property of a chemical system, comprising the following steps: - Providing a Hamiltonian of the chemical system subject of the quantum-enhanced chemical computing, preferably said Hamiltonian being expressed in second- quantised form; - Providing a basis set for the chemical system, preferably a finite basis set; - Preparing an initial quantum state, on the quantum computation means, to represent the chemical system according to the Hamiltonian and using the provided basis set; - Applying, on the quantum computation means, a quantum solver on the prepared quantum state in the basis set; and - Computing, on classical computation means, a density-based basis-set correction to modify the density-dependent terms in the Hamiltonian ; and preferably computing at least one quantum-enhanced chemical properties of the chemical system.

[0014] The computer-implemented methods according to the invention achieve high- precision electronic structure results with dramatically reduced quantum hardware requirements. For example, the computer-implemented methods disclosed herein can provide ground-state electronic energies within ±1.6 mHa (≈1 kcal mol⁻¹). This is demonstrated here after while operating a variational quantum eigensolver on 20 – 28 logical qubits for di-atomic and tri-atomic benchmark systems (H₂, He, LiH). A full-configuration-interaction (FCI) computation in a quintuple-ζ Gaussian basis of corresponding quality would require more than 100 logical qubits and circuit depths exceeding 2000 two-qubit gates. Consequently, the disclosed workflow yields an approximate five-fold qubit reduction and a ten-fold gate-depth reduction relative to a direct FCI realisation on identical hardware.

[0015] The resource savings arise from a hybrid division of labour: (a) a quantum processor evaluates expectation values of the Hamiltonian in a reduced orbital basis selected under a predefined qubit budget; (b) a density-based basis-set correction (DBBSC) is computed on classical processors and applied additively to the quantum result to approximate the complete-basis-set (CBS) limit. Because step (b) executes entirely on CPUs or GPUs, no further qubits or quantum gates are introduced after the initial solver execution. By doing so, one can run a quantum algorithm with a much smaller basis (fewer qubits) and then apply a density-driven correction that lifts the result closer to the CBS (infinite-basis) answer. This hybrid strategy provides a shortcut to chemical accuracy by minimizing the quantum resources needed for a given accuracy target.

[0016] Hence, the present invention allows an offloading of as much computation as possible to classical post-processing without sacrificing accuracy. In practice, one can select a subset of orbitals under a size constraint and use that basis in a variational quantum eigensolver (VQE), then apply the DBBSC to the VQE result. Because the density-based basis-set correction (e.g. DBBSC functional and the associated Hartree– Fock CBS offset) is evaluated on CPUs / GPUs, the correction stage introduces no extra qubits and minimal circuit depth. As it is described in the detailed description, for H₂O, a 24-qubit V5Z-10 basis combined with DBBSC lowers the CBS error from >0.3 Ha to ≈0.04 Ha.

[0017] By splitting the computation, the method achieves accuracy of a large-basis calculation using a small-basis quantum calculation plus a classical functional correction (on classical computers such as GPU or CPU). This hybrid workflow preserves qubit resources (no extra qubits needed for the correction) and improves accuracy, which is a clear problem-solution benefit in the NISQ context. Thus, the method of the invention achieves CBS-level precision at a qubit count and depth compatible with near-term noisy intermediate-scale quantum (NISQ) devices.

[0018] As it is described hereafter, the invention couples a quantum chemistry calculation with a basis set correction which can be completed with a posteriori additive correction(strategy 1) for example with the DBBSC applied once to a converged VQE energy or a self-consistent scheme that iteratively updates an effective one-particle potential during the wavefunction optimization (strategy 2). Both approaches can be implemented without increasing qubit requirements. Properties obtainable with the method include ground-state energies, vertical excitation energies, dipole moments, and potential-energy curves along nuclear coordinate displacements.

[0019] Because the basis-set correction is executed on classical hardware and no additional qubits are introduced, the method attains accuracy comparable to calculations performed in extended Gaussian bases while maintaining reduced qubit counts and circuit depths. Observable quantities accessible through the workflow include ground-state energies, vertical excitation energies, dipole moments and potential-energy profiles along nuclear coordinates. Each quantity benefits from the same qubit-sparing correction mechanism because the DBBSC modifies only density-dependent Hamiltonian terms and leaves the ansatz structure untouched.

[0020] The density-based basis-set correction reduces residual basis-set error without enlarging quantum circuit depth, thereby lowering total quantum gate count relative to an uncorrected computation and enabling execution within a coherence window of the quantum processor. In summary, the density-based correction removes residual basis-set error while preserving the original circuit depth and gate count. This reduction in quantum operations lowers accumulated gate error and enables completion of the computation inside the coherence envelope of current superconducting and trapped-ion processors

[0021] According to other optional features of the method according to the invention, it can optionally include one or more of the following characteristics alone or in combination: - computing the density-based basis-set correction comprises computing a- posteriori correction based only on energy. - computing at least one quantum-enhanced chemical properties comprise the addition of the density-based basis-set correction to an approximation of the full- configuration interaction energy calculated when applying the quantum solver on the prepared quantum state in the basis-set. - the density-based basis-set correction comprises (i) a correlation energy functional of the electron density and (ii) an energy correction term (for example a Hartree–Fock energy correction term that compensates for basis-set in the Hartree–Fock energy). - computing the density-based basis-set correction comprises computing a self- consistent correction based on energy and density. - applying the quantum solver and computing the density-based basis-set correction are repeated iteratively, with an updated (density-based basis-set corrected) Hamiltonian communicated from the classical computation means to the quantum computation means after each iteration. - expectation values are computed when applying the quantum solver on the prepared quantum state in the basis-set; and the computed density-based basis- set correction comprises a functional of the electron density obtained from the expectation values; and the quantum solver being re-executed with an upgraded Hamiltonian with the computed density-based basis-set correction. - the step of computing the density-based basis-set correction and computing quantum-enhanced chemical properties of the chemical system comprises: computing, on the classical computation means, a density‑based basis‑set correction; and combining an approximate expectation value with the density- based basis-set correction to output a quantum-enhanced chemical properties of the chemical system that approaches a complete‑basis‑set value. - the step of computing a density-based basis-set correction and computing quantum-enhanced chemical properties of the chemical system comprises updating the quantum computation means using the computed basis-set correction and re-evaluating the energy and density on the quantum computation means. - the steps of computing the quantum solver and applying the density-based basis- set correction and computing quantum-enhanced chemical properties of the chemical system are repeated iteratively; preferably each iteration comprising: computing a correction potential (e.g. a one-electron correction potential or a two- electron correction potential) as the functional derivative of the density-based basis-set correction with respect to the electron density, adding said electron correction potential to the Hamiltonian, and repeating the quantum solver with the updated Hamiltonian until convergence of the computed energy is reached. - computing quantum-enhanced chemical properties of the chemical system comprises the ground-state, dissociation curves, wavefunctions, and / or dipole moment. - the at least one quantum-enhanced chemical property comprises a corrected ground-state electronic energy, the method further including:o (a) receiving, from the quantum computation means, a numerical ground- state energy obtained for the Hamiltonian in the provided basis set; o (b) adding, on the classical computation means, a density-based basis- set correction to the a numerical ground-state energy to obtain a corrected ground-state energy; and o (c) storing the resulting ground-state energy and wave-function ansatz in computer-readable memory in computer-readable memory. - the at least one quantum-enhanced chemical property comprises a corrected dipole moment, wherein it comprises the following steps: - obtaining, from quantum computing and classical computing means, a corrected wave-function and a one-electron density; - computing, on classical computing means, an expectation value of the dipole moment operator; - computing, on classical computing means, an Hartree-Fock correction to the dipole moment, for example by computing the difference between the Hartree-Fock dipole moment computing on a large basis set and the Hartree-Fock dipole moment computing on the basis-set of interest; - adding to the correlation corrected dipole moment, the Hartree-Fock correction to the dipole moment. - the step of applying the quantum solver on the initial quantum state is done on quantum computation means coupled to classical computation means. - the steps of applying the quantum solver and applying the density-based basis-set correction are done iteratively and a density-corrected Hamiltonian is transferred from the classical computation means to the quantum computation means for further measurements. - the step of applying the density-based basis-set correction is done on an optimized Hamiltonian generated by the quantum solver; in particular, the Hartree-Fock basis-set correction and non-self-consistent correlation basis-set correction (or the first order correlation), are computed on classical computing units and added to the expectation value out of the quantum solver. - the non- self-consistent correlation basis-set correction is functional of the one- electron density and preferably, it is built to correct the basis-set due to short- range electron correlation in such a way that the correction correctly vanishes in the CBS limit. - it comprises a step of defining compact adaptative basis-sets, said step comprising a step of generating atomic-orbital (AO) basis sets under a basis size budget.- it comprises a step of defining compact adaptative basis-sets, said step comprising defining truncated versions of the Dunning basis sets by applying - procedure, such as a greedy, for discarding elements of the full AO-product set that spans the space containing the reference density , which admit the expansion:with D = CC⊤the density matrix and C the orbital coefficients in the AO basis. - it comprises a step of defining compact adaptative basis-sets, said step comprising extracting subsets of a given “large” basis-set (typically comprising more than three sets of functions for each valence atomic orbital) for any index subset achieving a target size and minimal accuracy loss using a step of solving an optimization problem under constraints. - the initial quantum state refers to: an initial wavefunction estimate consistent with the Hamiltonian, an approximation of the ground state. - the basis set is selected among: sto-3G, 6-31g, pc-seg0, VDZ, VTZ, VQZ, V5Z, or V6Z - the quantum solver is configured to solve the Schrödinger equation to determine the fundamental properties of the quantum system of the molecular target. - the quantum solver is selected among: quantum diagonalization solvers such as Quantum Phase Estimation solver (QPE Solver) or Variational Quantum Eigensolver (VQE Solver); quantum stochastic solvers such as quantum Monte Carlo solver or stochastic variational solver; quantum coupled-cluster solvers such as coupled-cluster singles and doubles solver or UCCSDT solver, or non-iterative dUCCD or tensor networks aided solvers. - computing quantum-enhanced chemical properties of the chemical system comprises the ground-state and excited-state energies, dissociation curves, wavefunctions, and / or dipole moment. - it is performed on a hybrid quantum-classical computer, the computer comprising a quantum computing component having a plurality of qubits (at least the given number of qubits), and a classical computing component comprising at least one processor and a non-transitory computer-readable memory, the non-transitory computer-readable memory storing computer instructions. - preparing the initial quantum state comprises configuring at least one qubit register and at least one ancilla qubit.- it comprises applying a set of quantum gates to the involved qubits and measuring an outcome state.

[0022] According to another aspect, the present invention relates to a computer- implemented method to compute a dissociation energy in a molecular system, said method comprising the steps of: i) executing the steps of the method according to the invention for a first nuclear geometry of the molecular system corresponding to an equilibrium bond length; ii) re-executing the steps of the method according to the invention for at least one additional nuclear geometry of the molecular system in which the bonded atoms are separated by a distance sufficient to represent dissociation; and iii) computing, on the classical computation means, a numerical difference between the corrected ground-state energies generated in steps (i) and (ii).

[0023] According to another aspect, the present invention relates to a computer system specifically configured to implement the method according to the invention. For example, the present invention relates to a computing apparatus is provided comprising quantum computation means, classical computation means, and memory storing instructions which, when executed, configure the apparatus to perform any embodiment of the methods of the invention.

[0024] In particular, according to another aspect, the present invention relates to a computer system comprising : - one or more quantum computation means; and - one or more classical computation means, the system being further configured to implement a computer-implemented method according to the present invention.

[0025] According to another aspect, the present invention relates to a computer implemented method for performing a system-adapted basis-set generation as described in the attached description.

[0026] According to another aspect, the present invention relates to a non-transitory computer-readable medium is provided storing program code that, when executed by processing circuitry comprising quantum and classical computation means, causes execution of any embodiment of the method set forth above. The present invention relates to a computer-readable medium with instructions that, when executed by a classicalcomputation means in operative communication with a quantum computation means, perform the methods of the invention.

[0027] Features disclosed in connection with any one aspect may be combined with features of any other aspect in any technically feasible manner. Brief description of the drawings

[0028] The foregoing and other objects, features and advantages of the present invention will become more apparent from the following detailed description when taken in conjunction with the accompanying drawings. [Fig.1] is a block diagram illustrating a hybrid quantum–classical workflow in which a quantum-processing unit (QPU) or GPU emulator evaluates expectation values of an electronic Hamiltonian, while a classical processing unit computes (i) a correlation-based density functional correction and (ii) a Hartree–Fock basis-set correction, the two contributions being combined to produce a corrected energy output. [Fig 2] represents a computer implemented method according to the invention. [Fig.3] represents outputs of the present invention according to (i) the correlation-based density functional correction and (ii) the Hartree–Fock basis-set correction. [Fig.4A], [Fig.4B], [Fig.4C] and [Fig.4D] are a set of logarithmic plots showing the absolute deviation of ground-state energies from a complete-basis-set reference for, respectively, H₂, LiH, H₂O and N₂ as a function of the chosen orbital basis; the trajectories compare uncorrected quantum-algorithm results with values obtained after application of the density-based basis-set correction [Fig.5A], [Fig.5B], [Fig.5C] presents, respectively, dissociation curves for H₂, LiH and N₂ computed with adaptive-VQE in small orbital bases, overlaid with curves obtained after the correction and with full-configuration-interaction curves in large Dunning bases, thereby illustrating the convergence of the corrected method along bond-stretching coordinates. [Fig 6] represents a step of applying 160 a density-based basis-set correction and computing quantum-enhanced chemical properties of the chemical. [Fig 7] is a graph representing the errors DAPT-VQE energy with respect to the FCIenergy with the number of VQE iterations for the H2O molecule with the STO-3G basis set. The starting point is the HF determinant. The number of required qubits is 12. [Fig 8] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for the water molecule using pcseg-0, SABS / V5Z-10, and 6-31G basis. The quantity ndetcorresponds to the number of determinants used for the initial state. For ndet= 1, the initial state is the Hartree-Fock determinants. For the other case, we use the CIPSI method to select the determinants. Each case required 24 qubits. [Fig 9] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for the N2molecule using N2 STO-3G basis. The starting point is the Hartree-Fock Slater determinant. Number of qubits: 16. [Fig 10] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for the N2molecule using SABS / V5Z-6 basis. The starting point is the Hartree-Fock Slater determinant. Number of qubits: 16. [Fig 11] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for LiH using STO-3G; SABS / VQZ-4, and SABS / V5Z-4 basis. The starting points are the Hartree-Fock Slater determinants. Number of qubits: 10. [Fig 12] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for LiH using pcseg-0 basis. The starting points are the Hartree- Fock Slater determinants. Number of qubits: 14. [Fig 13] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for LiH using 6-31G basis. The starting points are the Hartree- Fock Slater determinants. Number of qubits: 20. [Fig 14] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for LiH using SABS / V5Z-7 basis. The starting points are the Hartree-Fock Slater determinants. Number of qubits: 16. [Fig 15] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for LiH using SABS / V5Z-10 basis. The starting points are the Hartree-Fock Slater determinants. Number of qubits: 28. [Fig 16] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for H2using 6-31G basis. The starting points are the Hartree-Fock Slater determinants. Number of qubits: 8. [Fig 17] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for H2using cc-pVDZ. The starting points are the Hartree-Fock Slater determinants. Number of qubits: 20. [Fig 18] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for H2using SABS / V5Z-8. The starting points are the Hartree- Fock Slater determinants. Number of qubits: 24. [Fig 19] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for H8. The starting points are the Hartree-Fock Slater determinants. Number of qubits: 16. STO-3G. [Fig 20] is a graph representing the errors to the FCI energies in each basis with the number of VQE iterations for H8using 6-31G basis-set. The starting points are the Hartree-Fock Slater determinants. Number of qubits: 32.6-31G. [Fig 21] represents a system according to the invention.

[0029] Several aspects of the present invention are disclosed with reference to flow diagrams and / or block diagrams of methods, devices and computer program products according to embodiments of the invention. On the figures, the flow diagrams and / or block diagrams show the architecture, the functionality and possible implementation of devices or systems or methods and computer program products, according to several embodiments of the invention. For this purpose, each box in the flow diagrams or block diagrams may represent a system, a device, a module or code which comprises several executable instructions for implementing the specified logical function(s). In some implementations, the functions associated with the box may appear in a different order than indicated in the drawings. For example, two boxes successively shown, may be executed substantially simultaneously, or boxes may sometimes be executed in the reverse order, depending on the functionality involved. Each box of flow diagrams or block diagrams and combinations of boxes in flow diagrams or block diagrams may be implemented by special systems that perform the specified functions or actions or perform combinations of special equipment and computer instructions. Detailed Description

[0030] A description of example embodiments of the invention follows.

[0031] Hereinafter, we describe the vocabulary associated with the invention, before presenting the drawbacks of the prior art, and then finally showing in greater detail how the invention remedies them.

[0032] A “quantum circuit” denotes a sequence of gates that can be implemented on a quantum computation means to formally realize a unitary operator that acts on a given initial quantum state and produces a final quantum state. Circuits can be parametrized via gate parameters such as one or more angles of a single qubit rotation gate. A “gate” denotes an operation on a quantum system that transforms a quantum state. A quantum computation means can be either on a hardware exploiting quantum physics phenomena for the generation of physical Qubit or on a classical hardware simulating Qubit.

[0033] By “Hamiltonian” (H) or “molecular Hamiltonian” can mean, within the meaning of the invention, an operator that fully defines a quantum system. The lowest eigenstate of such operators can be called a ground state that can be the target of quantum chemistry calculations. A Hamiltonian, written in a computational basis state, can be different for each molecule. “Energy” denotes an expectation value of a Hamiltonian on a given normalized quantum state. Energy can be minimized when the state is the ground state.

[0034] A "NISQ (Noisy Intermediate Scale Quantum) device" can refer to a category of quantum computers characterized by possessing a number of qubits ranging from approximately 50 to a few hundred, where the quantum operations are inherently accompanied by notable noise levels. These devices, operative without full quantum error correction, are marked by their error susceptibility and a limited qubit coherence (Displaying reduced times of qubit stability).

[0035] As used in the context of the present invention, the expression “Variational Quantum Eigensolvers” or “VQE” can be defined as an algorithm designed for solving the eigenvalue problems of quantum systems, particularly effective on NISQ devices. The VQE algorithm can be characterized by parameterized quantum circuits where quantum circuits parameters are optimized through a feedback loop to find a system's ground state.

[0036] As used in the context of the present invention, the expression “ADAPT-VQE” can refer to an adaptive variant of the variational-quantum-eigensolver in which the ansatz is built iteratively, adding at each step the operator whose energy gradient is largest, thereby yielding a compact circuit that converges to the ground state with reduced gate count.

[0037] As used in the context of the present invention, the expression “Quantum Phase Estimation” or “QPE” can refer to a quantum algorithm that encodes the phase of a unitary time-evolution operator onto ancilla qubits, thereby yielding an estimate of the corresponding eigenvalue (energy) of the simulated Hamiltonian.

[0038] As used in the context of the present invention, the expression “density-based basis-set correction” or “DBBSC” can refer to an additive energy functional, evaluated from the electronic density, that compensates for the truncation error introduced by a finite orbital basis without increasing the number of qubits. It can comprise the non-self- consistent correlation basis-set correction and the HF (Hartree-Fock) basis-set correction. The basis-set corrections are calculated on classical computers, such as CPU (central processing unit).

[0039] As used in the context of the present invention, the expression “system-adapted basis set” or “SABS” can refer to reduced set of one-electron functions selected on-the- fly from a larger parent basis to minimize density-reconstruction error while respecting a predetermined qubit budget.

[0040] As used in the context of the present invention, the expression “complete-basis- set” or “CBS” can be defined as the ideal limit in which enlargement of the orbital basis no longer alters the calculated electronic energy, providing a basis-independent reference value.

[0041] As used in the context of the present invention, “GPU emulator” can refer to a classical graphics-processing-unit-accelerated simulator that reproduces, gate-for-gate, the execution of a quantum circuit to deliver noise-free reference results.

[0042] “Wavefunction” (^) can mean, within the meaning of the invention, a function which represents the probability density of finding a particle at a given location, preferably on Hilbert space. A target wave function can be an approximation of the ground state of a Hamiltonian.

[0043] “Ansatz" or "ansatz wavefunction" can mean, within the meaning of the invention, a subroutine consisting of a sequence of gates applied to specific wires. An ansatz wavefunction can capture the most important contributions to the electronic correlation energy and, at the same time, is capable of being represented on rather shallow quantum circuits. The initial ansatz can also be called the current ansatz which is the ansatz which will be grown by appending the selected operators. As the process ispreferably iterative, the process begin with an initial ansatz while after at least an iteration a current ansatz (the current ansatz can generally be the new ansatz of the previous iteration) is used.

[0044] A "hybrid quantum-classical system", according to the invention can be a computing architecture that combines quantum and classical processing capabilities, where for example quantum processor is tasked with state preparation and measurement while classical processor manages optimization and data processing, creating a feedback loop with the quantum processor.

[0045] "Quantum circuit measurements", according to the invention, can involve the process of obtaining specific quantum state properties through the observation of qubits, where each measurement typically results in a probabilistic collapse of the quantum state into one of several possible states, influencing subsequent quantum operations.

[0046] "Adaptive algorithms" in the context of the invention can be considered as algorithms that adjust their operational parameters or procedures based on feedback concerning their performance.

[0047] "Gradient-free optimization" in the context of the invention can be considered as an optimization approach that bypasses the computational overhead associated with gradient calculations by employing alternative strategies such as heuristic or probabilistic techniques to determine the direction of optimization, significantly reducing computational demands especially in environments with high noise levels.

[0048] An "operator pool" can, in the context of the invention can comprise a predefined set of quantum gates or operations from which selections are made during the execution of a quantum algorithm to optimize performance, particularly pivotal in adaptive and variational quantum algorithms.

[0049] “Parameterized unitary operators" are defined as quantum operators that influence the state of qubits within a quantum circuit through adjustable parameters, typically involving angles in rotation gates that modify the state transformation properties of the operators.

[0050] "Greedy algorithms" in the context of the invention can be defined as algorithms that make successive choices that appear to be optimal at each step, aimed at achieving a local optimum with an assumption that these local optima will lead to a globally optimal solution.

[0051] "Ground state approximation" in the context of the invention can refer to a method of estimating the lowest energy state of a quantum system, which is found several technical applications, for example in quantum chemistry for predicting molecular configurations and interactions.

[0052] As used in the context of the present invention, the expression “chemical system” can refer any finite collection of nuclei and electrons treatable by the quantum-chemical basis-set framework disclosed. Thus it include, atom, molecule, or combination of molecules.

[0053] By “process”, “compute”, “determine”, “display”, “extract”, “compare” or more broadly “executable operation” can mean, within the meaning of the invention, an action performed by a computing device or a processor unless the context indicates otherwise. In this regard, the operations relate to actions and / or processes of a data processing system, for example a computing system or an electronic computing device, which manipulates and transforms the data represented as physical (electronic) quantities in the memories of the computing system or other devices for storing, transmitting or displaying information. In particular, calculation operations are carried out by the processor of the device, the produced data are entered in a corresponding field in a data memory and this field or these fields can be returned to a user for example through a Human Machine Interface formatting such data. These operations may be based on applications or software.

[0054] The terms or expressions “application”, “software”, “program code”, and “executable code” mean any expression, code or notation, of a set of instructions intended to cause a data processing to perform a particular function directly or indirectly (for example after a conversion operation into another code). Exemplary program codes may include, but are not limited to, a subprogram, a function, an executable application, a source code, an object code, a library and / or any other sequence of instructions designed for being performed on a computing system.

[0055] By “processor” is meant, within the meaning of the invention, at least one hardware circuit configured to perform operations according to instructions contained in a code. The hardware circuit may be an integrated circuit. Examples of a processor include, but are not limited to, a central processing unit, a graphics processor, an application- specific integrated circuit (“ASIC” according to Anglo-Saxon terminology), and a programmable logic circuit. A single processor or several other units may be used toimplement the invention.

[0056] By “coupled” is meant, within the meaning of the invention, connected, directly or indirectly, with one or more intermediate elements. Two elements may be coupled mechanically, electrically or linked by a communication channel.

[0057] The expression “human-machine interface”, within the meaning of the invention, corresponds to any element allowing a human being to communicate with a computer, in particular and without that list being exhaustive, a keyboard and means allowing in response to the commands entered on the keyboard to perform displays and optionally to select with the mouse or a touchpad item displayed on the screen. Another embodiment is a touch screen for selecting directly on the screen the elements touched by the finger or an object and optionally with the possibility of displaying a virtual keyboard.

[0058] Electronic-structure algorithms executed on quantum processors typically map one spin-orbital of a molecular Hamiltonian to one logical qubit. Although this linear mapping permits exploration of exponentially large Hilbert spaces, present and near-term fault- tolerant devices provide only tens to hundreds of logical qubits. In such hardware regimes, the slow convergence of total energies and first-order properties with respect to one- electron basis-set size remains the principal accuracy bottleneck.

[0059] The present disclosure remedies this limitation by coupling any wave-function- based quantum solver with a density-based basis-set correction (DBBSC) computed on classical computation means. The correction can for example comprise a correlation basis-set correction and a Hartree–Fock (HF) energy term equal to the difference between HF energies obtained in an extended reference basis and in the finite basis used on the quantum processor. Several operating modes can be implemented to use this invention. In the following detailed description two strategies will be described in detail, the “strategy 1” applies the correction a posteriori to the energy returned by the quantum solver, whereas “strategy 2” embeds a potential derived from the functional derivative of the correlation term (e.g. a one-electron correction potential or a two-electron correction potential) into the Hamiltonian and iterates until self-consistency of the corrected property is reached. Both modes maintain the original qubit count because the correction is evaluated on classical computer (e.g. CPU, GPU).

[0060] The present invention, as it is described is after, allows one to determine a physical quantity (energy) of a real-world system via simulation. In particular, the invention can beused to determine or approximate the energy of a molecular system more efficiently / accurately without direct physical experimentation on the molecular system. Advantageously, the invention addresses that problem by combining quantum computational capabilities with classical computational capabilities. Thus the present invention reduce hardware limitations. In particular, it implementation allows one to work on a molecular system with a much lower circuit depth of quantum computation means.

[0061] In particular, the invention can address the current bottleneck through a partition of the workload between quantum and classical resources, as summarized in the block diagram of FIG.1. In an embodiment, the quantum processor (or a GPU-accelerated emulator) evaluates the ground-state energy. The classical processor then applies a density-based basis-set correction (DBBSC) that compensates the missing short-range correlation and the Hartree–Fock (HF) contribution which are absent from the finite basis. The two contributions are added algebraically, leaving the qubit count unchanged.

[0062] The workflow comprises the following stages: i) providing a Hamiltonian (describing a chemical system geometry) and a basis set for the chemical system; ii) Applying, on the quantum computation means (QPU), a quantum solver on a initial quantum state prepared to represent the chemical system according to the Hamiltonian and using the provided basis set; and iii) Compute, on classical computation means, a density-based basis-set correction to modify the density-dependent terms in the Hamiltonian. COMPUTER IMPLEMENTED METHOD

[0063] According to a first aspect, the invention relates to a computer implemented method 100 for computing quantum-enhanced chemical properties of a chemical system. The invention allows for example the determination of a physical energy of a chemical system using a quantum computing device. The chemical system can be a molecule or a material composed of interacting particles.

[0064] Preferably, said computer comprises quantum computational means. The computer can be a Hybrid Quantum / classical computation means. Also, the computer implementing the method of the invention can comprise hardware equipment that are physically distant but connected by a communication network. As it is described hereafter the quantum computational means can be quantum hardware or classical hardware configured to simulate quantum computing. The quantum hardware can for example be selected among quantum computers based on trapped ions, superconducting quantumcomputers, neutral atoms in optical lattices, quantum dot computer spin-based or spatial- based, Bose-Einstein condensate-based quantum computer, quantum wells computers, nuclear magnetic resonance quantum computer, cavity quantum electrodynamics, optical quantum computer, or diamond-based quantum computer.

[0065] As shown in figure 2, the method 100 according to the invention comprises the following steps: - Providing a Hamiltonian 120 of the chemical system subject of the quantum- enhanced chemical computing; - Providing a basis set 130 for the chemical system; - Preparing an initial quantum state 140 to represent the chemical system according to the Hamiltonian and using the provided basis set; and - Applying a quantum solver 150 on the initial quantum state.

[0066] As shown in figure 2, the method 100 according to the invention can also comprise computing, on classical computation means, a density-based basis-set correction 160 to modify the density-dependent terms in the Hamiltonian. The step 160 can correspond to applying, on classical computation means, a density-based basis-set correction to modify the density-dependent terms in the Hamiltonian, preferably comprising a Hartree-Fock basis-set correction term.

[0067] As shown in figure 2, the method 100 according to the invention can also comprise computing quantum-enhanced chemical properties 170 of the chemical system.

[0068] In particular, the method can comprise the steps of providing a Hamiltonian of the chemical system subject of the quantum-enhanced chemical computing, providing a basis set for the chemical system, preparing an initial quantum state on a quantum computation means to represent the chemical system according to the Hamiltonian and using the provided basis set, applying, on the quantum computation means, a quantum solver on the initial quantum state; and applying, on classical computation means, a density-based basis-set correction to modify the density-dependent terms in the Hamiltonian combined with a Hartree-Fock basis-set correction and computing quantum-enhanced chemical properties of the chemical system

[0069] The method 100 can further comprise a step of defining a given number of qubits 110 that will be involved in the computer implemented method 100, a step of providing a basis set having a number of sets of functions for each valence atomic orbital which is defined according to the given number of qubits, a step of defining compact adaptativebasis-sets.

[0070] In particular, the method 100 can be performed on a hybrid quantum-classical computer, the computer comprising a quantum computing component having a plurality of qubits (at least the given number of qubits), and a classical computing component comprising at least one processor and a non-transitory computer-readable memory, the non-transitory computer-readable memory storing computer instructions.

[0071] A method 100 according to the invention can comprise a step of defining a given number of qubits 110 that will be involved in said method 100. This step 110 allows for the definition a qubits budget, thereby setting an explicit upper bound on the size of the system-adapted basis set and, consequently, on the overall quantum-circuit resources required. The given qubits or the qubit budget can be denoted nqubitsor nq.

[0072] For example, the given number of qubits is at least 10, preferably at least 20, more preferably at least 40, and even more preferably at least 100 given qubits. The given number of qubits is at most 5000, preferably at most 4000, more preferably at most 2000, and even more preferably at most 1000 given qubits. Preferably, the number of given qubits is comprised between 10 and 5000 given qubits, preferably 20 to 4000, more preferably 40 to 2000, and even more preferably 100 to 1000 given qubits.

[0073] A method 100 according to the invention comprises a step of providing a Hamiltonian 120 of the chemical system subject of the quantum-enhanced chemical computing. Providing a Hamiltonian

[0074] For example, the method according to the invention can comprise receiving, by a classical computing mean, data defining a second-quantised electronic Hamiltonian for the chemical system.

[0075] A method 100 according to the invention comprises a step of providing a basis set 130 for the chemical system. Preferably, the basis set is a finite orbital basis set for the Hamiltonian. For example, this step can comprise the selection selecting, by a classical computing mean, of a finite basis set for representing electronic wavefunctions of the chemical system. Preferably, as detailed in examples, this step can comprise the generation of atomic-orbital (AO) basis sets under a basis size budget. More preferably,this step comprises extracting subsets of a given “large” basis-set (typically comprising more than three sets of functions for each valence atomic orbital) for any index subset achieving a target size and minimal accuracy loss using a step of solving an optimisation problem under constraints.

[0076] Preferably, the basis set is selected from a large reference basis set which is a correlation-consistent basis (e.g. aug-cc-pV5Z) and the basis-set correction functional is computed for use with that reference basis. In a particular embodiment, the basis set is selected among: sto-3G, 6-31g, pc-seg0, cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z, or cc- pV6Z.

[0077] Advantageously, the basis set is adapted to the given number of qubits. Hence, in a preferred embodiment of the present invention, the step of providing a basis set 130 for the chemical system, can comprise a step of generating a finite truncated orbital basis limited by a predetermined qubit budget.

[0078] The number of qubits used in the quantum computation is preferably limited to a predetermined budget, and the method adjusts the density-based basis-set correction and / or selected basis set to achieve a target accuracy under this constraint.

[0079] During the step of providing a basis set 130 for the chemical system, a basis set is can be selected or build according a qubit budget. In particular, the step can further comprises selecting a subset of M basis functions from a larger reference basis by minimizing an error measure between a reference electron density and the electron density representable in the subset, where the subset size M is determined by the predetermined qubit budget.

[0080] In particular, as described in examples, the definition of truncated versions of the Dunning basis sets can be done applying a procedure, . such as a greedy procedure, for discarding elements of the full AO-product set that spans the space containing the reference density , which admit the expansion:with D = CC⊤the density matrix and C the orbital coefficients in the AO basis.

[0081] A method 100 according to the invention comprises a step of preparing an initial quantum state 140 on a quantum computation means. This step allows to represent the chemical system according to the Hamiltonian and using the provided basis set. The step 140 of preparing the initial quantum state can comprise configuring at least one qubit register and at least one ancilla qubit. In a preferred embodiment, the initial quantum state can refer to an initial wavefunction estimate consistent with the Hamiltonian. Moreover, this step can comprise preparing, on a quantum processor that includes a plurality of physical qubits, an initial quantum state encoding the Hamiltonian in the selected basis set.

[0082] The step of preparing the initial quantum state 140 on the quantum computation means comprises encoding the Hamiltonian in the basis set. Preparing the initial quantum state can comprise configuring at least one qubit register and at least one ancilla qubit.

[0083] This step can comprise initializing or preparing, by a quantum processor, an initial quantum state encoding a Hamiltonian or model of the chemical system (e.g. encoding information about the molecule’s electrons or orbitals in qubits).

[0084] In particular, this step can comprise preparing, on the quantum computation means, a parameterized quantum state (e.i. wavefunction ansatz) representing the chemical system’s electrons, preferably the state being defined in a finite orbital basis limited by a predetermined qubit budget.

[0085] When preparing the initial quantum state 140 on the quantum computation means, the Hamiltonian can be encoded on the quantum computational means by mapping each spin-orbital of a one-electron basis set to a corresponding qubit.

[0086] The computer-implemented method according to the invention, wherein the initial quantum state refers to a wavefunction estimate consistent with the provided Hamiltonian.

[0087] A method 100 according to the invention comprises a step of applying 150, on the quantum computation means, a quantum solver on the initial quantum state. In particular, the method 100 can comprise applying a set of quantum gates to the involved qubits and measuring an outcome state. For example, this step can comprise applying a sequence of quantum logic gates on a quantum computation means to evolve the quantum state according to the Hamiltonian (simulating the molecular interactions).

[0088] Advantageously, the step of applying 150 is done on quantum computation means coupled to classical computation means.

[0089] In a particular embodiment, the quantum solver is configured to solve the Schrödinger equation to determine the fundamental properties of the quantum system of the molecular target.

[0090] The quantum solver can refer to a quantum computing algorithm and apparatus that outputs the energy of a given system. The quantum solver can be selected among: quantum diagonalization solvers such as Quantum Phase Estimation solver (QPE Solver) or Variational Quantum Eigensolver (VQE Solver); quantum stochastic solvers such as quantum Monte Carlo solver or stochastic variational solver; quantum coupled-cluster solvers such as coupled-cluster singles and doubles solver or UCCSDT solver, or non- iterative dUCCD or tensor networks aided solvers. In particular, it can be described as a quantum computing device executing a ground-state energy finding algorithm. In a more preferred embodiment, the quantum solver 150 comprises executing an adaptive variational quantum eigensolver (ADAPT-VQE) that incrementally expands the wavefunction ansatz.

[0091] This step can also comprise measuring one or more observables of the evolved quantum state to obtain measurement results (e.g. measuring qubit states to gather information about the chemical system’s energy). Furthermore, it can comprise processing the measurement data to calculate or estimate a quantum-enhanced chemical property of the chemical system such as the energy value of the chemical system (for instance, computing an expectation value or optimizing parameters based on the measurement outcomes).

[0092] In a preferred embodiment, applying a quantum solver 150 comprises executing, on the quantum computation means, the quantum solver configured to generate an approximate electronic wavefunction and at least one expectation value derived therefrom.

[0093] The quantum solver can preferably be configured to solve the Schrödinger equation to determine the fundamental properties of the quantum system of the molecular target. For example, this step can comprise executing, on quantum computation means, a quantum eigensolver routine that includes: applying a parameterized quantum circuit to the initial state; measuring expectation values of Hamiltonian terms; and transmitting corresponding measurement data to classical computation means.

[0094] A method 100 according to the invention comprises a step of computing a density-based basis-set correction 160. This density-based basis-set correction can further be used to modify the Hamiltonian. In particular, this density-based basis-set correction can further be used to modify the density-dependent terms in the Hamiltonian.

[0095] Advantageously, the step of computing a density-based basis-set correction 160 is implemented on classical computation means. Advantageously, the basis-set correction is performed entirely on a classical computation means (e.g. CPU or GPU) and does not require additional qubits beyond those used for the quantum wavefunction calculation.

[0096] Furthermore, preferably, the density-based basis-set correction comprises computing a self-consistent correction based on energy and density.

[0097] This step can also be described as applying 160, on classical computation means, a density-based basis-set correction to modify the density-dependent terms in the Hamiltonian. In particular, while numerous standard functions exist (LDA, GGA, etc.), this can comprise applying a density-based correction using a density functional approximation to adjust the computed energy. For example, the method comprises generating an electron density distribution for at least part of the chemical system and applying a correction to the energy values using a predefined density functional that takes said electron density as input.

[0098] Preferably, the density-based basis-set correction comprises estimate a correlation energy term (for example using a density functional theory calculation on the electron density of the molecule), which is added to the energy computed by the computation means.

[0099] Preferably, the step of computing 160 a density-based basis-set correction and computing quantum-enhanced chemical properties of the chemical system comprises computing, on the classical computation means, the density-based basis-set correction for the integrals in the molecular orbitals basis set, the correction computation including a functional of the electronic density. Moreover, the step of computing 160 a density-based basis-set correction and computing quantum-enhanced chemical properties of the chemical system can comprise computing, on the classical computation means, the density-based basis-set correction for the integrals in the molecular orbitals basis set, the correction computation including a functional of the electronic density and a basis-set adjustment term, preferably said basis-set adjustment term being for energy contributions that are not taken into account in the previous correction potential (for example, theHartree-Fock error is not corrected in the correlation functional).

[0100] In a preferred embodiment of the invention, the density-based basis-set correction comprises a Hartree–Fock correction term, more preferably the Hartree–Fock correction term compensates for basis-set in the Hartree–Fock energy.

[0101] In particular, the step of computing 160 a density-based basis-set correction and computing quantum-enhanced chemical properties of the chemical system comprises computing, on the classical computation means, the density-based basis-set correction for the integrals in the molecular orbitals basis set, the correction computation including a functional of the electronic density and a Hartree-Fock correction estimated as the difference between the Hartree-Fock property computed on a large basis-set and the Hartree-Fock property of the basis set of interest.

[0102] In some embodiments, the step of computing or applying 160 the density-based basis-set correction and computing quantum-enhanced chemical properties of the chemical system comprises: - Computing 161, on the classical computation means, a density‑based basis‑set correction; and - Combining 162 an approximate expectation value with the density-based basis- set correction to output a quantum-enhanced chemical properties of the chemical system that approaches a complete‑basis‑set value.

[0103] In some embodiments, the step of computing or applying 160 a density-based basis-set correction and computing quantum-enhanced chemical properties of the chemical system comprises updating the quantum computation means using the computed basis-set correction and re-evaluating the energy and density on the quantum computation means.

[0104] In some embodiments, the density-based basis-set correction comprises: a correlation energy functional that tends to zero as the basis set approaches the complete-basis-set limit; and a basis-set correction term, that preferably compensates for basis-set in the Hartree– Fock energy.

[0105] In preferred embodiments, the density-based basis-set correction comprises two contributions: (i) a correlation energy functional of the quantum-computed electron density,preferably derived from a parameterized density functional that approaches zero in the complete-basis-set limit; and (ii) a basis-set Hartree–Fock correction which compensates for the lack of completeness in the orbital basis, preferably by using a larger reference basis for the Hartree–Fock energy.

[0106] As mentioned, several implementation of the invention can be deduced from the present disclosure. Hereafter, we will describe in details two strategies to implement the invention.

[0107] STRATEGY 1

[0108] In the context of the invention, computing 160 the density-based basis-set correction can comprise computing a-posteriori correction based only on energy.

[0109] Preferably, in strategy 1, the density-based basis-set corrections are applied a posteriori after the step of applying the quantum solver; preferably, the method comprises adding a Hartree–Fock basis-set correction and a correlation basis-set correction (computed on the classical computation means) to the energy obtained from the quantum solver.

[0110] In particular, in the computer-implemented method according to the invention, the computing at least one quantum-enhanced chemical properties 170 comprises the addition of the density-based basis-set correction to an approximation of the full- configuration interaction energy calculated when applying the quantum solver (150) on the prepared quantum state in the basis-set.

[0111] In particular, according to the strategy 1, the step of applying the quantum solver 150 comprises executing, on the quantum computation means, a wave-function algorithm that returns a ground-state electronic energy for the Hamiltonian expressed in the provided basis set; and the method comprises computing a corrected ground-state electronic energy using the ground-state electronic energy for the Hamiltonian expressed in the provided basis set and the density-based basis-set correction. This is illustrated in figure 1 and figure 3.

[0112] In a preferred embodiment of the invention, the density-based basis-set correction comprises (i) a correlation energy functional of the electron density and (ii) an energy correction term (for example a Hartree–Fock energy correction term that compensates for basis-set in the Hartree–Fock energy).

[0113] Preferably, the non- self-consistent correlation basis-set correction is functional of the one-electron density and preferably, it is built to correct the basis-set due to short- range electron correlation in such a way that the correction correctly vanishes in the CBSlimit.

[0114] As it is detailed in the examples, the step of computing at least one quantum- enhanced chemical property 170 preferably comprises forming a corrected ground-state electronic energy. In a more preferred embodiment, the corrected energy being obtained by adding to the ground-state electronic energy:the density-based basis-set correctiondetermined in step (160) and, optionally, a Hartree–Fock basis-set shiftto the Hartree–Fock ground-state electronic energy.

[0115] As it is detailed in the examples, the at least one quantum-enhanced chemical property can be an expectation value of a one-body operator and the method comprises: a) measuring, on the quantum computation means, an expectation value for the one- body operator in the provided basis set (130); b) computing, on the classical computation means, a density-based basis-set correction for that operator, preferably by evaluating the density-based basis-set- correction functional with the matrix elements of the one-body operator substituted for the Hamiltonian matrix elements; and c) forming a corrected expectation value by adding the density-based basis-set correction, and, optionally, a Hartree–Fock basis-set shift for the same operator obtained in a larger reference basis set, to the numerical value produced in step (a). STRATEGY 2

[0116] In the context of the invention, in a preferred embodiment, applying the quantum solver 150 and computing 160 the density-based basis-set correction are repeated iteratively. This implementation relates mainly to the strategy 2 and is explained in great details in the examples.

[0117] Preferably, each iteration comprises repeating the quantum solver with an updated Hamiltonian until convergence of the computed energy is reached. In particular, density- based basis-set correction is re-inserted into the Hamiltonian before the quantum solver isrerun, the iteration loop being repeated until convergence, thereby yielding corrected energies and density-derived one-body observables.

[0118] For example, the steps of applying the quantum solver 150 and computing 160 the density-based basis-set correction are repeated iteratively, with an updated (density- based basis-set corrected) Hamiltonian communicated from the classical computation means to the quantum computation means after each iteration. Hence, the method can comprise iteratively updating, on the classical computation means, the circuit parameters until a convergence criterion on the measured energy is met.

[0119] In particular, each iteration can comprise: - computing a correction potential (e.g. a one-electron correction potential or a two- electron correction potential) as the functional derivative of the density-based basis-set correction with respect to the electron density, - adding said electron correction potential to the Hamiltonian, and - repeating the quantum solver with the updated Hamiltonian until convergence of the computed energy is reached.

[0120] For example, the quantum computation and classical computation can be executed concurrently and iteratively; Preferably at each iteration, the classical computation means calculates an updated correction potential based on the latest quantum density and sends it to the quantum computation means, while the quantum computation means provides updated energy / density information to the classical computation means.

[0121] In some embodiments, as details in examples, expectation values can be computed when applying the quantum solver 150 on the prepared quantum state in the basis-set; and the computed density-based basis-set correction comprises a functional of the electron density obtained from the expectation values. Furthermore, the quantum solver being re-executed with an upgraded Hamiltonian with the computed density-based basis-set correction.

[0122] Advantageously, the step 160 is done on an optimized Hamiltonian generated by the quantum solver; in particular, the Hartree-Fock basis-set correction and non-self- consistent correlation basis-set correction (or the first order correlation), are computed on classical computing units and added to the expectation value out of the quantum solver.

[0123] According to a particular embodiment of the method 100 of present invention, the steps of applying 150 the quantum solver and applying 160 the density-based basis-set correction are done iteratively and a density-corrected Hamiltonian is transferred from the classical computation means to the quantum computation means for furthermeasurements

[0124] As illustrated in Figure 6, Advantageously, the step 160 can comprise the steps of calculating 161, on classical computation means, a density-based basis-set correction by evaluating a functional of an electron density obtained from the quantum solver, and / or a step of combining 162 the approximate expectation value with the correction to output a quantum-enhanced chemical properties of a chemical system that approaches a complete-basis-set value.

[0125] The step of calculating 161, on classical computation means, a density-based basis-set correction by evaluating a functional of an electron density obtained from the quantum solver allows for a compensation of basis-set incompleteness without allocating additional qubits or extending circuit depth. Indeed, as the functional is evaluated entirely on CPUs / GPUs, all quantum-hardware demands remain fixed; yet the residual energy error observed for minimal or user-budgeted bases is reduced by up to an order of magnitude, in a 24-qubit V5Z-10 basis.

[0126] The step of combining 162 the approximate expectation value with the correction to output a quantum-enhanced chemical properties of a chemical system that approaches a complete-basis-set value allows for a CBS-quality electronic energies and first-order properties while preserving the original qubit count. Empirical data show that ground-state energies for small molecules (H₂, He, LiH) converge within the chemical-accuracy threshold (1.6 mHa) using only 20 – 28 logical qubits, whereas an uncorrected FCI computation would require more than 100 qubits.

[0127] As illustrated in Figure 2, a method according to the invention can also comprise a step of computing at least one quantum-enhanced chemical property 170 of the chemical system. This step allows the hybrid workflow to deliver physical observables (such as ground-state electronic energies, dissociation constant or dipole moments) that are numerically close to the complete-basis-set limit while preserving the qubit count and circuit depth set during the quantum-solver execution.

[0128] In particular, according to an embodiment of the present invention, the computing quantum-enhanced chemical property of the chemical system comprises the ground-state and excited-state energies, dissociation curves, wavefunctions, and / or dipole moment.

[0129] In a preferred embodiment, computing at least one quantum-enhanced chemical property 170 comprises adding the density-based basis-set correction to the approximatefull-configuration-interaction energy obtained when the quantum solver 150 is applied to the prepared quantum state in the selected basis set.

[0130] Hence, the invention provides a computer-implemented method that allows determination of a molecular energy with improved accuracy or feasibility, thereby aiding in analysis of chemical systems. The invention can be particularly useful on a the technical context of chemical engineering or reaction design.

[0131] In particular, the invention relates to computer-implemented method 100, wherein the at least one quantum-enhanced chemical property comprises a corrected ground-state electronic energy. For computing a corrected ground-state electronic energy, the method preferably comprises (a) receiving, from the quantum computation means 10, a numerical ground-state energy obtained for the Hamiltonian in the provided basis set; (b) adding, on the classical computation means 40, a density-based basis-set correction to the a numerical ground-state energy to obtain a corrected ground-state energy; and (c) storing the resulting ground-state energy and wave-function ansatz in computer-readable memory in computer-readable memory.

[0132] In particular, the invention relates to computer-implemented method 100, wherein the at least one quantum-enhanced chemical property comprises a corrected dipole moment. For computing a corrected dipole moment, the method preferably comprises : a) obtaining, from quantum computing and classical computing means, a corrected wave- function and a one-electron density (this can for example be obtained following iterative strategy such as the ones described in strategy 2); b) computing, on classical computing means, an expectation value of the dipole moment operator; and c) computing, on classical computing means, an Hartree-Fock correction to the dipole moment, for example by computing the difference between the Hartree-Fock dipole moment computing on a large basis set and the Hartree-Fock dipole moment computing on the basis-set of interest. The method can further include adding to the correlation corrected dipole moment, the Hartree-Fock correction to the dipole moment. This method particularly benefits from the correction procedure following strategy 2).

[0133] According to another aspect, the present invention relates to a computer- implemented method to compute a dissociation energy in a molecular system. The computer-implemented method to compute a dissociation energy in a molecular system comprises the following steps:i) executing the steps of a method 100 according to invention for a first nucleargeometry of the molecular system corresponding to an equilibrium bond length; ii) re-executing the steps of the method 100 according to the invention for at leastone additional nuclear geometry of the molecular system in which the bonded atoms are separated by a distance sufficient to represent dissociation; and iii) computing, on the classical computation means 40, a numerical differencebetween the corrected ground-state energies generated in steps (i) and (ii).

[0134] A computer-implemented method to compute a dissociation energy in a molecular system comprises a step of i) executing the steps of a method 100 according to invention for a first nuclear geometry of the molecular system corresponding to an equilibrium bond length. The step i) allows for establishing a reference ground-state electronic energy of the bound molecular configuration with complete-basis-set-level accuracy while retaining the reduced qubit and gate-depth budget defined for the hybrid quantum–classical workflow.

[0135] A computer-implemented method to compute a dissociation energy in a molecular system comprises a step of ii) re-executing the steps of the method 100 according to the invention for at least one additional nuclear geometry of the molecular system in which the bonded atoms are separated by a distance sufficient to represent dissociation.

[0136] A computer-implemented method to compute a dissociation energy in a molecular system comprises a step of iii) computing, on the classical computation means 40, a numerical difference between the corrected ground-state energies generated in steps (i) and (ii). COMPUTER SYSTEM

[0137] According to another aspect, the present invention relates to a data-processing system. The data-processing system completes the energy estimation within a total quantum-gate execution time that is shorter than a coherence time specified for the quantum computation means 10.

[0138] A data-processing system can comprise (i) a quantum computation means 10 with physical qubits arranged in a two-dimensional grid having defined connectivity.

[0139] A data-processing system can comprise (ii) a classical computation means configured to transmit control pulses to, and receive measurement data from, quantumcomputation means 10.

[0140] A data-processing system can comprise (iii) a non-transitory storage medium storing program code which, when executed by the classical computation means, causes performance of the method 100 according to the invention.

[0141] In another aspect, the invention relates to a computer system 1 configured to implement the method 100 according to the invention and / or a computer-implemented method to compute a dissociation energy in a molecular system according to the invention.

[0142] The figure 21 is a schematic block diagram illustrating various hardware components that may be utilized a computer system 1 according to the invention. In particular, the computer system 1 comprises: one or more quantum computation means 10; and one or more classical computation means 40.

[0143] In a preferred embodiment, the system 1 can implement the following steps of the computer implemented method 100 for computing quantum-enhanced chemical property of a chemical system: - Providing a Hamiltonian 120 of the chemical system subject of the quantum- enhanced chemical computing, preferably said Hamiltonian being expressed in second-quantised form; - Providing a basis set 130 for the chemical system, preferably a finite basis set; - Preparing an initial quantum state 140, on the quantum computation means 10, to represent the chemical system according to the Hamiltonian and using the provided basis set; - Applying, on the quantum computation means 10, a quantum solver 150 on the prepared quantum state in the basis set; and - Computing, on classical computation means 40, a density-based basis-set correction 160 to modify the density-dependent terms in the Hamiltonian ; and preferably computing at least one quantum-enhanced chemical property 170 of the chemical system.

[0144] Further, the computer system 1 can comprise: one or more memory components 20, one or more communication interfaces 30; and / or one or more user interfaces 50.

[0145] The computer system 1 can include one or more classical binary computers coupled to one or more quantum computers. The one or more conventional binarycomputers can be configured to receive one or more computing tasks via an input port and to output corresponding computational results via an output port.

[0146] The one or more quantum computers can be configured to execute one or more quantum circuits that are generated from the one or more tasks to generate corresponding output results for the one or more classical binary computers to use to generate the corresponding computational results.

[0147] The computer system 1 comprise quantum computation means 10, for example said quantum computation means 10 being configured for quantum chemical simulations. In particular, the quantum computation means comprise a quantum circuit 11 for quantum chemical simulation obtainable, preferably obtained, by a method according to the invention. The invention quantum computation means 10 can be integrated in a computing system 1 as described hereafter. Also, the computer implemented methods according to the invention can be implemented on a computing system 1.

[0148] The memory component 20 may comprise any computer readable medium known in the art including, for example, a volatile memory, such as a static random access memory (SRAM) and a dynamic random-access memory (DRAM), and / or a non-volatile memory, such as read-only memory, flash memories, hard disks, optical disks and magnetic tapes. The memory component 20 may include a plurality of instructions or modules or applications for performing various functions. Thus, the memory component 20 can implement routines, programs, or matrix-type data structures. Preferably, the memory component 20 may comprise a medium readable by a computing system in the form of a volatile memory, such as a random-access memory (RAM) and / or a cache memory. The memory component 20, like the other modules, can for example be connected with the other components of the computing system 1 via a communication bus and one or more data carrier interfaces.

[0149] Furthermore, the computing system 1 can also comprise a communication interface 30. The communication interface 30 is preferably configured to transmit data on at least one communication network and may implement a wired or wireless communication. The computer system 1 can communicate with other devices or computing systems and in particular with clients thanks to the communication interface 30. A communication interface 30 according to the invention is in particular configured to exchange data with third-party devices or systems.

[0150] A computing system 1 comprise one or more classical computation means 40. Aclassical computation means 40 may be operably coupled to the memory component 20 to execute instructions, encoded in programs, for carrying out the presently disclosed techniques, more particularly to perform the method according to the invention. Advantageously, the classical computation means are selected among GPUs and / or CPUs. In a particular embodiment, the classical computational means comprise CPU, GPU or ASIC, preferably configured to support the computational demands and parallel processing requirements of hybrid quantum-classical algorithms.

[0151] The encoded instructions may be stored in any suitable article of manufacture (such as the memory component 20) that includes at least one tangible non-transitory, computer-readable medium that at least collectively stores these instructions or routines. In this manner, the memory component 20 may contain a set of instructions that, when executed by the classical computation means 40, performs the method of the invention.

[0152] The memory component 20 may include any number of databases or similar storage media that can be queried from the classical computation means 40 as needed to perform the method of the invention.

[0153] These different modules or components are separated in Figure 21, but the invention may provide various types of arrangement, for example a single module cumulating all the functions described here. Similarly, these modules or components may be divided into several electronic boards or gathered on a single electronic board. A computing system 1 according to the invention can be incorporated into a computing system and able to communicate with one or several external devices such as a keyboard, a pointer device, a display, or any device allowing a user to interact with the system 1.

[0154] The computing system 1 may also be configured to communicate with or via a human-machine-interface. Thus, in one embodiment of the present invention, the computing system 1 can be coupled to a human interface machine (HMI). The HMI may be used to allow the transmission of parameters to the devices or conversely make available to the user the values of the data measured or calculated by the device.

[0155] In general, the HMI is communicatively coupled to a processor and includes a user output interface and a user input interface. The user output interface may include an audio and display output interface and various indicators such as visual indicators, audible indicators and haptic indicators. The user input interface 50 may include a keyboard, a mouse, or another navigation module such as a touch screen, a touchpad, a stylus inputinterface, and a microphone for inputting audible signals such as a user speech, data and commands that can be recognized by the classical computation means 40.

[0156] Preferably, implementing the invention involves manipulating (e.g. initializing, flipping…) quantum states on a specific quantum computation means, like an ion-trap, a superconducting or a photonic qubit system. As mentioned, the invention can be used on the one hand in improving the proper functioning of any quantum computer and on the other hand in several field of applications using convention, hybrid or quantum computers.

[0157] For example, the invention can be used to initialize a quantum state to represent a molecular system. In particular, it can set up a quantum state, either a ground state or a superposition, that corresponds to a particular molecular configuration.

[0158] Also, advantageously in a context of molecular analysis, the present invention can be used to apply molecular Hamiltonian's interactions to a prepared quantum state. It's can be used when for analysing chemical properties and / or simulating molecular dynamics. The application of this invention to physical qubits can be done using several quantum systems such as superconducting qubits systems, trapped ions qubits systems or photonic qubits systems.

[0159] The implementation can for example comprise initializing the qubits in a quantum register. This can involve cooling the superconducting qubits to their ground state, typically using dilution refrigerators that bring the system close to absolute zero temperature. Superconducting quantum computers often use transmon qubits, which are weakly anharmonic oscillators. Coupling multiple transmon qubits using resonant or dispersive interactions can facilitate the control and target operations of the quantum gate or circuit of the present invention. In particular, applying microwave pulses to the control qubits can bring them into the desired state. These pulses should be calibrated in duration, amplitude, and phase to achieve the correct quantum state. The invention can be further implemented by sequentially activating interactions between the control qubits and the target qubit. This can be achieved using a series of controlled-phase gates, which are native to superconducting systems, followed by single-qubit rotations to transform the CZ into CNOT operations. As superconducting qubits are prone to errors due to decoherence and operational inaccuracies, the invention can benefit from implement quantum error correction protocols, such as a surface code, to detect and correct errors during the gate operation. Finally, the implementation should comprise a measure of the state of the qubits. Superconducting qubit measurements typically involve state-dependent frequencyshifts, which are detected using resonant circuits.

[0160] The implementation can for example comprise cooling trapped ions to their motional ground state using laser cooling techniques, such as Doppler cooling followed by sideband cooling. Preferably, implementation of the present invention in a trapped ions can use tightly focused laser beams to individually address ions in the trap. This allows for selective manipulation of control and target ions for the quantum gates or quantum circuit according to the invention. The invention can comprise using a combination of single-ion operations and multi-ion entangling operations. Single-ion operations can be achieved through Rabi oscillations induced by laser pulses while multi-ion entangling operations can be implemented using Mølmer-Sørensen gates, which entangle the internal states of the ions via their collective motional modes. Thus, implementing the invention gates and quantum circuit should comprise by sequentially applying controlled operations across the ion chain, with predetermined timing and synchronization of laser pulses. The eventual issues of decoherence and operational error can be addressed using techniques like dynamical decoupling and sympathetic cooling, where auxiliary ions are used for cooling without disturbing the computational qubits. Finally, the invention can comprise a measure of the state of the ions where ions are illuminated with a laser, and the emitted photons are detected, like in a state-dependent fluorescence.

[0161] The implementation can for example comprise a generation of entangled photon pairs using spontaneous parametric down-conversion or other quantum dot-based sources. In such system, the control and target qubits can be encoded into different degrees of freedom of the photons, such as polarization, path, or orbital angular momentum modes. The invention can implement the gates and quantum circuit of the invention using linear optical elements like beam splitters, phase shifters, and wave plates. These elements manipulate the photonic qubits to perform the necessary quantum gates. For example, photons can be entangled using quantum interference effects at beam splitters and other optical elements. Due to the probabilistic nature of optical quantum computing, the invention can preferably use ancilla photons and post-selection techniques to achieve a higher success rate. Finally, the photons can be detected using single-photon detectors, such as avalanche photodiodes. Preferably, the measurement results are then post-processed to account for any non-deterministic operations.

[0162] Advantageously, the system 1 according to the invention, can implement the method according to the invention. In particular, the system 1 may implement all features of the method according to the invention, including those which are optional,advantageous, or described as preferred, either individually or in any technically compatible combination.

[0163] In another aspect, the invention relates to one or more computer-readable medium with instructions that, when executed by a control processor in operative communication with a quantum processor, perform the methods according to the invention.

[0164] Preferably, the computer-readable medium is a tangible non-transitory computer- readable medium.

[0165] For the purposes of this disclosure, computer-readable medium may include any instrumentality or aggregation of instrumentalities that may retain data and / or instructions for a period of time. Computer-readable medium may include, for example, without limitation, storage medium such as a direct access storage device (e.g. a hard disk drive or floppy disk drive), a sequential access storage device (e.g. a tape disk drive), compact disk, CD-ROM, DVD, RAM, ROM, electrically erasable programmable read-only memory (EEPROM), and / or flash memory; as well as communications medium such as wires, optical fibers, microwaves, radio waves, and other electromagnetic and / or optical carriers; and / or any combination of the foregoing.

[0166] In particular, any combination of one or more computer-readable medium may be used. In the context of this document, a computer-readable medium may be any tangible medium that may contain, or store, a program for use by or in connection with an instruction execution system, apparatus, or device. A computer-readable medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared or semiconductor system, apparatus or device, or any suitable combination of the foregoing. More specific examples (a non-exhaustive list) of the computer-readable storage medium would include: a hard disk, a random-access memory (RAM).

[0167] Computer program code for performing operations for aspects of the present invention may be written in any combination of one or more programming languages, including an object-oriented programming language such as Java, C ++, or similar, the programming language "C" or similar programming languages, a scripting language such as Perl, or similar languages, and / or functional languages such as Meta Language. Program code can run entirely on a user's computer, partly on a user's computer, and partly on a remote computer or entirely on the computer or remote server. In the latter scenario, the remote computer can be connected to a user's computer by any type ofnetwork, including a local area network (LAN) or a wide area network (WAN).

[0168] These computer program instructions may be stored on a computer readable medium that can direct a computing device (i.e. computer, server ...), so that the instructions stored in the computer-readable medium produce a computing device configured to implement the invention.

[0169] For example, a computing device may be a personal computer, a network storage device, or any other suitable device and may vary in size, shape, performance, functionality, and price. The computing device may include random access memory (RAM), one or more processing resources such as a central processing unit (CPU) or hardware or software control logic, ROM, and / or other types of nonvolatile memory. Additional components of the computing device may include one or more disk drives, one or more network ports for communication with external devices as well as various input and output (I / O) devices, such as a keyboard, a mouse, and a video display. The computing device may also include one or more buses operable to transmit communications between the various hardware components.

[0170] The invention can be the subject of numerous variants and applications other than those described above. In particular, unless otherwise indicated, the different structural and functional characteristics of each of the implementations described above should not be considered as combined and / or closely and / or inextricably linked to each other, but on the contrary as simple juxtapositions. In addition, the structural and / or functional characteristics of the various embodiments described above may be the subject in whole or in part of any different juxtaposition or any different combination.I. EXAMPLES In these examples, we will describe the integration of the DBBSC method with quantum algorithms to expedite reaching the CBS limit and achieve chemical accuracy on complex molecular systems. This strategy, which nativelylimits the required qubit counts, can be applied to quantum algorithms that tackle the ground-state quantum chemistryproblem, such as the QPE or VQE algorithms. While QPE can guarantee ground-state energy with arbitrary high precision given a carefully chosen initial state, it demands large quantum circuits and will only be viable in the FTQC era. Conversely, VQE lacks convergence guarantees but employs smaller circuits, aligning better with this study’s aim of advancing short-term quantum computers toward practical quantum-chemistry applications. The paper is organized as follows. In Section IV, we present two variants of the application of the DBBSC method to wave-function QC calculations, denoted as Strategy 1 and Strategy 2: (1) a basis-set correction a posteriori added to the solution of the quantum algorithm, integrating two contributions, namely a basis-set correlation density- functional correction and a basis-set Hartree-Fock (HF) correction, and (2) a self-consistent scheme integrating the DBBSC method to the quantum algorithm that dynamically modifies the one-electron density used in the basis-set correlation correction. Strategy 1 offers the possibility to correct any wave-function QC energy calculation through a simple additive correction. Strategy 2 enables one to self-consistently access to an improved electronic density, offering both improved energies and first-order molecular properties. Also, we introduce a new type of system-adapted basis sets for Gaussian-type orbitals (GTOs), with size comparable to minimal basis sets. These new methodologies enable us to perform the first investigation of DBBSC corrections in the minimal basis-set regime. In Section II, we provide relevant tests of our approach for different atomic and molecular systems using several families of basis sets. We carry out numerical simulations using graphics-processing-unit (GPU) accelerated QC sparse emulation on up to 32 qubits, exploring the applicability of this method to converge ground-state energies, dissociation curves, and dipole moments. We consistently observe significant improvements over typical quantum- algorithm approaches, reaching an accuracy level that would have otherwise required hundreds of qubits. We thus expect the present approach to become a standard part of quantum-enhanced wave-function calculations, particularly the approach of Strategy 1 which can immediately provide large improvements to existing results with relatively simple efforts. Finally, Section III contains our conclusions. Additional details and results are provided in the Supplementary Information (SI). II. RESULTS AND DISCUSSION We computed the ground-state energies, dissociation curves, and dipole moments for the N2, H2O, LiH, and H2molecules using both basis-set correction strategies. For all systems except H2, the 1s molecular orbitals were frozen, i.e. we use the frozen-core approximation. Correspondingly, we use the frozen-core version of the DBBSC method, in which in particular the contribution to the density coming from the core electrons is neglected in the basis-set correlation density functional. The error in each calculation is quantified as the deviation from the CBS limit, which itself is determined using a two-point extrapolation scheme based on the cc-pVQZ and cc-pV5Z basis sets. For thedipole-moment calculations we employed only Strategy 2, using an expectation value over the dipole-moment operator,avoiding a finite-difference approach prone to numerical errors . We remind the reader that these calculations are to be interpreted in the framework of perfect (noiseless) logical qubits, and therefore the discussed errors can be only attributed to the method used. The Appendix presents the quantum circuits that were used for all computations and provides the number of CNOT gates required to perform for the algorithm as well as the number of ADAPT-VQE iterations. The classical computations, including basis-set corrections and the calculation of reference energies and dipole moments, were carried out using Quantum Package 2.0. In this software, the FCI energies are approximated by the energy from a CIPSI wave function to which a second-order perturbation-theory (PT2) correction is added. Given the demonstrated nearly-FCI quality of this approximation for the systems we studied, we simplify our terminology by referring to this approach as simply FCI rather than CIPSI+PT2. The SABS are labeled VXZ-Y where X is respectively D, T, Q, 5, or 6 for cc-pVDZ, cc-pVTZ, up to cc-pV6Z basis sets, and Y is the target size. We report in the SI the classically computed energies for the N2, H2O, LiH, and H2molecules.A. Ground-state EnergiesGround-state energies of the H2, LiH, H2O, and N2 molecules close to their respective equilibrium geometries arepresented in Figure 4 and Table I. Details regarding the ADAPT-VQE iterations and the associated “ADAPT” valuesare provided in the SI, as well as additional tests on atoms and hydrogen chains.In Figure 4, a general trend is observed: the basis-set corrected ground-state energies with the small basis sets,i.e. STO-3G, 6-31G, or pcseg-0, align with values between the cc-pVDZ and cc-pVTZ basis-set levels, while requiringmuch less qubits than these latter basis sets. As we can see from Table I, both strategies provide the same quantitativeimprovements. For the best cases, the basis-set corrected ground-state energies have errors in the order of tens of mHa: 40 mHafor H2O with 24 qubits, 60 mHa for N2 with 16 qubits, and less than 10 mHa for LiH and H2 with less than 10 qubits.The results stay consistent for the other systems available in the SI (H4, H6, H8). For H2 using the cc-pVDZ basisset, the basis-set corrected ground-state energy reaches chemical accuracy relative to the CBS limit, while requiring20 logical qubits. Similar convergence to the CBS limit is observed in the SI for He and Be. Without the basis-setcorrection, reaching a similar accuracy would have required typically more than a hundred of qubits.In the initial work on the DBBSC method, it was found that basis-set corrected ground-state energies of atoms reachchemical accuracy with the cc-pVTZ basis set. Similarly, for the molecules studied here, if one had access to a fewhundred logical qubits required for the cc-pVTZ basis set, achieving chemical accuracy would be theoretically feasiblefor all cases. To support this claim, we report in the SI classically computed basis-set corrected FCI ground-stateenergies using Dunning basis sets. The error relative to the CBS limit for the LiH molecule is 0.2 mHa with thecc-pVTZ basis set. For H2O and N2, we achieve errors of 3 mHa and 1 mHa, respectively, with the cc-pVTZ basisset. Let us now discuss the results obtained with our SABS for LiH, N2, and H2O. They were chosen to match the sizesof the small basis sets previously discussed. Specifically for H2O, employing the V5Z-10 basis set (24 qubits) achievesresults slightly superior to those obtained with the 6-31G basis set. For H2O we observed a reduction in the HF basis-set correction by 40 mHa when moving from the 6-31G to the V5Z-10 basis set, by over 100 mHa when comparingthe STO-3G to the V5Z-4 basis set for the LiH molecule, and more than an Hartree for the N2 molecule moving fromthe STO-3G to the V5Z-6 and V5Z-11 basis sets. Based on these findings, we anticipate that further exploration ofthis strategy could lead to a basis-set correction scheme requiring only the correlation basis-set correction term. Ofcourse, the interest of SABS is to be systematically improvable toward the parent Dunning basis set. It is importantto note that this SABS systematic improvement is not limited to quantum algorithms such as ADAPT-VQE and isalso present for the HF and FCI calculations. Since the SABS approach strongly reduces the computational cost whilemaintaining accuracy, it should offer further applications of the CIPSI approach in classical quantum chemistry. Inpractice, a cc-pVTZ-like quality is reached with a V5Z-11 basis set (32 qubits) for N2, and with a V5Z-11 basis set(30 qubits) for H2O. A V5Z-10 basis set (28 qubits) achieves a cc-pV5Z-like ground-state energy accuracy for LiH.Overall, the SABS always provide the best ADAPT energies with the self-consistent correlation basis-set correction(before addition of the HF basis-set correction).Overall, concerning the basis sets, it is important to point out a key anomaly, i.e. the remarkable performance ofthe minimal STO-3G basis set. This was expected as Pople already discussed the outperformance of this basis set inthe seventies . Indeed, Davidson and Feller detailed in their 1986 review the existence of error compensations andstated that “the smaller the basis the more ab initio calculations assume an empirical flavor”. In practice, amongthe available minimal basis-set possibilities, STO-3G proved to be the most robust and cost-efficient choice for thebasis-set corrections. Also, in the DBBSC schemes, the STO-3G basis set highly benefits from the HF basis-setcorrection which reduces as the basis set and number of qubits increases. When compared to SABS, for example,STO-3G always display smaller self-consistent basis-set corrected ADAPT values and larger HF basis-set corrections.In practice, Pople also noted that STO-3G is especially good for energies around the equilibrium geometry, and largerbasis sets are clearly required to describe accurately the notoriously more difficult dissociation curves and dipolemoments. Finally, one last aspect of the analysis of such large basis-set simulations is related to the convergence of ADAPT-VQE. Indeed, in VQE-type computations, there is no formal guarantee of convergence and ADAPT-VQE belongsto such heuristic family of methods. If, for systems like H2, LiH or N2 (), convergence is obtained in a few dozenof iterations, for more complex systems such as H2O, the number of required iterations strongly increases when adouble-zeta basis set is used. Clearly, Figure 9 shows that more than a thousand iterations is required to achieve fullconvergence starting from the HF initial state. This heuristic aspect makes that no anticipation of the exact requirednumber of iterations can be made. This may be manageable on a real quantum processor, but the use of classicalemulation makes each iteration more costly in term of time-to-solution than the previous one, limiting the overallconvergence capabilities. It is possible to force convergence by replacing the HF starting point by a CIPSI one. Figure8 shows that chemical accuracy can be easily reached with a well-converged CIPSI solution. However, ADAPT-VQEstruggles to improve the solution which is expected due to the enormous size of the parameter space. Furthermore, the CIPSI-based initial state already contains significant amount of correlation. This represents a hard challenge for the ADAPT-VQE procedure, which now needs to more carefully pick the next ansatz operator to improve the existing quantum state. In any case, starting from an unconverged CIPSI wave function is a robust solution to strongly reduce the overall computational time. One example is given for H2O and the V5Z-10 basis set which initially led to a not fully converged result. With a better CIPSI starting point, it is possible to recover a few milli-Hartrees (see footnotes in Table I). It is also possible to change the operator pool but the point here is that convergence becomes challenging when tackling complex electronic structures and such computations would not be possible without GPU-accelerated emulation. B. Dissociation Curves We pursue with the dissociation energies reported in Figure 4. Clearly, the simple ADAPT-VQE / STO-3G level of theory appears quite far from an accurate description of the dissociation (compared to the calculations in the largest basis sets), which is achieved with FCI / triple-zeta (and beyond) levels. The non-corrected ADAPT-VQE values are represented with bold lines whereas the corrected values are reported with dashed curves. These dissociation curves are extremely challenging as they involve several regimes of correlation going from weak correlation happening typically at short distances to strong correlation effects at large distances. First, we notice that for all the cases, the basis-set corrected values around the equilibrium are always substantially closer to the large cc-pV5Z reference values than the uncorrected ones and improve with the basis-set size. For larger distances, the basis-set requirements appear even more stringent. In practice, the three dissociation curves manage to be converged to high accuracy: we obtain nearly a triple-zeta quality for N2using the basis-set correction with the V5Z-6 basis set(16 qubits). A triple-zeta-like quality can be achieved using the larger V5Z-11 basis-set at the price of more qubits (32 qubits, see Table II). In the same line, nearly a cc-pV5Z quality for H2using the basis-set correction with the V5Z-8 basis set, and nearly a cc-pV5Z quality for LiH using the basis-set correction with a V5Z-7 basis set. For H2, the basis-set corrected cc-pVDZ curve matches the cc-pV5Z reference up to a distance of 2.5̊A and a slight discrepancy appears at long distances. This is consistent with classical computations leading to a convergence to the CBS limit only when the basis-set correction is applied to the cc-pVTZ basis set (114 qubits). It is possible to fix this issue more affordably by using a V5Z-8 basis set requiring only 24 qubits. For LiH, it is also possible to fix the small discrepancies observed on the V5Z-7 basis-set dissociation curve using a larger V5Z-10 basis set (28 qubits) which reaches the CBS limit. Finally, it is important to highlight the good performance of the DBBSC method for the prediction of the dissociationcurve of the triple-bonded N2molecule. Indeed, such computation is well-documented in the literature and known asextremely difficult as it requires both a multi-reference treatment and various weak correlation effects going from short-range to charge polarization effects. At the cost of a minimal basis set (i.e. 16 qubits), our N2DBBSC computations achieved nearly a cc-pVTZ quality, providing an accuracy which would have required around 100 logical qubits in the context of a brute-force simulation. To the best of our knowledge, these results are the most accurate using a quantum algorithm when compared to the recent results from the literature. Indeed, a group of researchers managed to predict this dissociation curve using a Local Unitary Cluster Jastrow (LUCJ) ansatz coupled to double-zeta basis sets (6-31G and cc-pVDZ). The computation required the use of massive computational resources, namely hundreds of compute nodes of the Fugaku classical supercomputer coupled to a QPU. Alternatively, in the context of the fermionic quantum emulator, N2simulations used ADAPT-VQE coupled to the 6-31G and def2-SVP (56 qubits) basis sets to perform computation via approximate Matrix Product States (MPS). In the present work, we achieve a better accuracy with far less qubits and, again, these results can be systematically improved using larger SABS. Indeed, access to a cc-pVTZ-like basis set accuracy is possible using the next (larger) SABS in term of size, i.e. V5Z-11 (32 qubits). C. Dipole Moments We also explore the idea to extend the basis-set correction scheme in QC calculations of molecular properties such as the dipole moment. As pointed out by Halkier et al., the dipole moment also suffers from a slow basis-set convergence, which can be thought of as an indirect impact of the missing short-range correlation effects in a finite basis set. Therefore, it is relevant to apply the basis-set correction to the dipole moment. We report in Table II the dipole moments of H2O and LiH. We calculate the basis-set corrected dipole moment as expectation value over the dipole-moment operator over the last ADAPT-VQE wave function coming out of the self-consistent basis-set correction method. We also add a posteriori the HF basis-set correction to the dipole moment, calculated as thedifference between the HF dipole moment in the aug-cc-pV5Z basis set and the HF dipole moment in the considered basis set. From Table II, we see that the correlation basis-set correction is not sufficient to converge the dipole moments. However, the addition of the HF basis-set correction significantly improves the basis-set convergence. For both molecules, strong improvements are observed with respect to the ADAPT-VQE / STO-3G level. Furtherimprovements are observed as the size of the basis set increases. Finally, we remark the ADAPT-VQE slow convergencein term of iterations is also the source of small errors in the dipole moments. III. CONCLUSIONS We demonstrated the applicability of the DBBSC method to QC algorithms for quantum chemistry. Using ADAPT- VQE, these approaches were shown to be able to systematically improve minimal basis-set results by predicting ground-state energies that are intermediate between double-zeta and triple-zeta FCI qualities. Overall, the presented self-consistent basis-set corrected ground-state energies are very close to their non-self-consistent counterparts. As the latter a posteriori basis-set correction approach can be very easily applied to any wave-function QC calculation performed on real quantum hardware, it provides an affordable improvement strategy for current QC chemistrycomputations. The self-consistent basis-set correction scheme is still useful since it permits the calculation of propertiessuch as dipole moments thanks to the availability of improved QC densities. An additional reduction of the required basis-set size is provided by our approach. Beside being fast, as basis-set can be generated within seconds, such a “black-box” pivoted-Cholesky strategy for the on-the fly generation of basis sets has been shown to be competitive and often superior to available standard choices. It offers systematic computational savings for large basis sets, reducing significantly the qubit requirements. Thus usefulness is not limited to QC algorithms as they also offer systematically improvable solutions for performing reference classical computations. Indeed, user-defined truncated versions of the Dunning basis sets can match a qubit budget, they also offer a reduced cost access to improved accuracy for any quantum-chemistry method. Our methodology allows us to compute chemically meaningful energies and properties on systems that would have required far more than 100 logical qubits. For example, the computation of the H2total energy at the FCI / cc-pV5Z level that would have required more than 220 logical qubits, can be achieved here with only 24 qubits using our basis-set correction scheme and our technique. Overall, we were able to converge four systems to the FCI / CBS limit including He, Be, H2, and LiH. We were also able to provide accurate dissociation curves for H2, LiH, and N2. Such computations required the use of only a single GPU. Since most quantum-chemistry studies are presently out of reach of quantum computers, this research opens the path to more affordable quantitative quantum-chemistry simulations of small molecules using QC algorithms. In particular, the present a posteriori basis-set corrections can be easily added to any type of STO-3G VQE fermionic computations on real hardware allowing one to improve significantly their accuracy at very little computational cost. Since adaptive simulations on real hardware are making some progress while hardware itself improves, basis-set corrected simulations should be progressively possible on future quantum computers providing a route to FCI / CBS quality computations. This strategy is particularly suited for resources demanding computations that converge slowly and require large basis sets associated to large qubits counts. Finally, the DBBSC method is not limited to ground-state computations and can be extended to excited states via a linear-response formalism. This will be the subject of future QC research. In this context, beside the state-of-the-art ADAPT-VQE hybrid quantum-classical algorithm, it would be interestingto revisit accurate fixed wave-function ansätze, such as UCCSDT and others, to analyze their convergence when coupled to the DBBSC method. In practice, the presented basis-set correction framework is not restricted to a given QC wave-function ansatz and can leverage any future QC algorithmic improvements. Our present DBBSC methodology still requires to use qubits either through quantum hardware or through the use of a classical quantum emulator. The qubit count is therefore our main limitation. The computations of this paper used up to 32 logical qubits and represent a proof-of-concept study of what one can presently do with quantum emulation to prepare the advent of fault-tolerant quantum computing, see Appendix for detailed resource estimations. However, state-vector simulations have limitations due to memory. Indeed, if they are theoretically possible up to 40-50 qubits on very large exascale supercomputers, they start to become relatively unpractical for quantum-chemistry simulations when reaching 36 qubits due to the high computational resources and time-to-solution requirements. To explore further the chemical electronic space with quantum algorithms, we are currently upgrading our Hyperion-1 framework to increase our emulated qubit counts by going beyond the state-vector formalism thanks to various elements from the CUDA-Q SDK developed by NVIDIA. We are also presently testing various implementations of the DBBSC algorithms on available quantum hardware. To conclude, by reducing the number of qubits required to reach the CBS limit, we expect to tackle predictive real-world quantum-chemistry applications with strategies applicable to both NISQ and FTQC algorithms.IV. METHODSThe overall methodological procedure is illustrated in Figures 1 and 2. The procedure starts with the definitionof the system, and a standard basis set or another method for basis set selecting based on a qubit budget. Afterdefining the second-quantized Hamiltonian, the quantum state is prepared on a quantum processing unit (QPU) or aGPU-accelerated quantum emulator. The basis-set corrections are calculated on the classical CPU using one of thetwo DBBSC variants labeled as Strategy 1 and Strategy 2.A. Density-based basis-set correction methodIn the infinite-dimensional (antisymmetric) N -electron Hilbert space, H =∧N L2(R3 × {↑, ↓},C), we consider anatomic or molecular system with HamiltonianĤ = T̂ + Ŵee + V̂ne, (1)where T̂ is the kinetic-energy operator, Ŵee is the Coulomb electron-electron operator, and V̂ne is the nuclei-electronpotential operator. The exact ground-state energy is defined asE0 =Ψm∈iWn ^Ψ, HΨ̂^, (2)∧where W = {Ψ ∈ N H1(R3 × {↑, ↓},C) | ^Ψ,Ψ^ = 1} is the space of admissible wave functions, H1 is the first-orderSobolev space, and ^·, ·^ designates the standard inner product of H. In quantum-chemistry calculations, we normallyintroduce a one-electron basis set B ⊂ H1(R3 × {↑, ↓},C) and we work in the finite-dimensional N -electron Hilbertspace generated by this basis set, i.e. HB =∧N span(B). The FCI ground-state energy is then defined asEBFCI = min ^Ψ, HΨ̂^, (3)Ψ∈Wwhere WB = {Ψ ∈ HB | ^Ψ,Ψ^ = 1} isFCIground-state energy tends to the exact FCIslow due to short-range electron correlation.In the DBBSC method, for the given basis set B, we introduce the following approximation to the ground-stateenergy B ( B ) E0 = min ^Ψ, HΨ̂^ + Ē[nΨ] , (4)Ψ∈Wwhere EB̄[nΨ] is a basis-set correctionsetcorrection density functional must vanish in the CBS limit so that 0 properly converges to the exact ground-stateenergy, i.e. EB→CBS0 = E0, but it must be such that it accelerates the convergence to the CBS limit. Based on thePerdew-Burke-Ernzerhof (PBE) correlation density functional, such a basis-set correction density functional can beconstructed in a semilocal form∫ EB̄[n] = ēB(n(r),∇n(r))dr, (5)where n(r) and ∇n(r) are the densityr, and the function ēB(n,∇n), including howthe dependence on the basis set B is included. We will now discuss how the DBBSC method can be adapted towave-function QC calculations.1. Strategy 1: a posteriori basis-set correctionIn Strategy 1, we use a non-self-consistent approximation to Eq. (4). The idea is to calculate the FCI energyEBFCI, or a good approximation to it, on a quantum computer, and then add a posteriori the basis-set correlationcorrection of Eq. (5) and a HF basis-set correction, both calculated on a classical computer. This leads to the followingapproximation to the ground-state energyEB = EB+ EB BB 1FCI ̄[nHF] + ∆EHF. (6)As in previous works, the basis-set correlation correction is evaluated at the HF density in the basis set B, leadingto a calculation of EB̄[nB ] with a marginal computational cost with respect to the cosB HF t of calculating EFCI. ThePBE-based basis-set correlation correction only corrects for basis-set incompleteness errors due to short-range electroncorrelation. However, when using small basis sets, a significant part of the basis-set error also comes from the factthe HF part of the energy is not converged to the CBS limit. We add a HF basis-set correction. In the present work,we choose it simply as the difference between the HF energy in the CBS limit, that we estimate as the value E5ZB HFobtained with the cc-pV5Z basis set, and the HF energy EHF in the basis set B∆EB5ZBHF = EHF − EHF. (7)We note that it is also possible to avoid performing a HF calculation with the cc-pV5Z basis set for estimating theCBS limit of the HF energy by using a complementary auxiliary basis set.Let us emphasize again that the PBE-based basis-set correction mentioned above only takes care of the correlationpart and therefore does not correct the basis-set error of the HF energy. Hence, there is no double counting betweenthe two basis-set corrections.2. Strategy 2: Self-consistent basis-set correctionIn Strategy 2, we use the self-consistent version of the DBBSC method in Eq. (4), using the basis-set correlationcorrection of Eq. (5) in the self-consistent part of the calculation, and we only add a posteriori the fixed HF basis-setcorrection. This leads to the following approximation to the ground-state energy that can be applied to any variationalquantum ansatzEB( =min ^Ψ, HΨ̂ B) B2 ^ + Ē[nΨ] + ∆E . (8)Ψ∈WHFThe minimization in Eq. (8) leads=(9)where P̂B is the projector on the N - theeffective HamiltonianHˆ̃B[n] = Ĥ + V ˆ̄ B[n]. (10)Here, V ˆ̄ B[n] is the basis-set correction one-electron potential operator∫ Vˆ̄ B[n] = v̄B[n](r) n(̂r)dr, (11)where v̄B[n](r) = δEB̄[n] / δn(r) is the correction with respect to the density, andn(̂r) is the density operator.The idea is now to solve iteratively Eq. (9) on a quantum computer. The potential v̄B[n](r), and therefore theHamiltonian H ˆ̃B[n], is iteratively updated with the density nB oB th if the wave-function ansatz solution Ψi of the iiteration. The convergence criterion is reached when the difference between the last two iterated energy eigenvaluesEBi is less than 0.1 mHa. In practice, this is fulfilled after two iterations.Finally, the self-consistent basis-set corrected energy EB2 can be directly computed from the last energy eigenvalueEB and density nB = nΨ coming out from the quantum solver as follows∫EB = EB + EB̄[nB] − v̄B[nB] B B2 (r)n (r)dr + ∆EHF, (12)Rwhere the HF basis-set correction ∆EBHF is identical to the one used in Strategy 1.In Strategy 2, the density of the QC wave-function ansatz Ψ needs to be calculated. For this, we calculate theone-particle density matrixn = ^Ψ, ↠â Ψ^, (13)where ↠and â are the creation andspin σ ∈ {↑, ↓}. This is achieved with a Jordan-Wigner mapping of the operator â†p,σâq,σ and the state Ψ preparedusing the parameterized ansatz. The density matrix is then used to classically update the Hamiltonian in Eq.(10)and to calculate dipole moments.3. GPU-accelerated QPU emulation and shift of the basis-set correction to classical resourcesThe DBBSC method does not increase the qubit count as the basis-set correction is performed classically using adensity functional and a HF correction. In practice, such computations present no advantage to be performed at theQC level since they would consume a large number of qubits without accuracy benefit over its classical counterpart.Therefore a hybrid QC wave-function / classical DBBSC approach is highly preferable. In this work, VQE / DBBSCcomputations are performed classically on GPU-accelerated computing nodes, the GPUs replacing the quantumprocessing unit (QPU) through the use of a classical quantum emulator. The basis-set correction contributions canbe then shifted to the unused CPUs to maximize the computational efficiency while the computationally challengingVQE algorithm is fully offloaded on GPUs for optimal time-to-solution performances. Of course, the proposed hybridschemes are also fully tractable on a real quantum computer, as the basis-set correction contributions can be entirelyshifted to classical resources harnessing GPUs / CPUs to preserve the qubit count while the core wave-function ansatzis maintained on the QPU.B. System-adapted basis-set generationWithin the DBBSC method, the natural choice for basis sets is the Dunning correlation-consistent basis-set familythat offers a path to a reliable convergence to the CBS limit, as demonstrated in initial DBBSC classical quantum-chemistry studies. For QC calculations, such basis sets are unfortunately usually out of reach of quantum hard-ware / emulators as they are associated with very large qubit counts. In the present section, we address the taskof generating atomic-orbital (AO) basis sets under a basis-size budget, with controllable accuracy. We propose amathematical formulation of this goal in terms of a constraint optimization problem, as follows.Given a molecule composed of Natm atoms, with fixed nuclear coordinates {ra}1≤a≤N , letN⋃ B= {χµ}1≤µ≤N = {χaµ(r − ra)}1≤µ≤Ndenote a spatial basis set of atom- density, denoted by nB0 , is availableafter a converged ground-state. Our main purpose is to a posterioriextract subsets of the given “large” basis set B, denoted by BI = {χµ}µ∈I ⊆ B for any index subset I ⊆ {1, ... , Nbas},achieving (i) a target size, and (ii) minimal accuracy loss on the given density nB0 used as a reference. In the presentcomputing scheme, since the DBBSC method uses the cc-pV5Z basis set for the HF basis-set correction, we chooseB = cc-pV5Z. In practice, this choice is motivated by the fact that such a quintuple-zeta basis set fairly approacheschemical accuracy compared to quasi-exact all-electron numeric atom-centered orbital computations. Hence, givena target basis size M smaller than Nbas, we seek the optimal index subset, denoted by IM , that minimizes the bestapproximation error of the density nB0 over the set of BI -representable one-electron densities, for any I ⊆ {1, ... , Nbas}with |I| = M , where | · | denotes the cardinal of a set, or, in other words, we solve an optimization problem underconstraints: I:= arg mi B BM nmin∥n0 − n ∥, (14)nwhere ∥ · ∥ is a give∫n no∫rm for functions overproduct ^u, v^c := ′RRdrdru(r)v(r′)|r − r′|−1. Let us emphasize that Eq. (14) requires knowledge of the densitynB0 , which is assumed to be pre-computed on a classical computer. Such quantity is available in the framework of theDBBSC scheme since HF computations are systematically performed at the cc-pV5Z level to estimate the CBS limitof the HF energy.The problem in Eq. (14) can be solved as follows. Our approach is to identify a greedy procedure for discardingelements of the full AO-product set that spans the space containing the reference density nB0 , which admits theexpansion N ,where D = CC⊤ is the density matrix and C of the coefficients in the AO basis of the Noccoccupied molecular orbitals. We plan to achieve this by eliminating linear dependencies present in the AO-productset. The pivoted Cholesky decomposition (PCD) of a matrix is an algebraic tool for eliminating linear dependencies occurring between matrix rows (resp. columns), which may be interpreted as an iterative greedy procedure for discarding elements that do not contribute to the full row (resp. column) space, up to an orthogonal projection error tolerance. PCD has been previously applied to the auxiliary basis-set generation for density fitting. In the present work, we employ PCD within a new scheme, named system-adapted basis-set generation, for solving the problem in Eq. (14). Let us formulate our scheme in detail. Prior to the AO-product selection and in order to ensure orbital symmetry of the resulting SABS, we pre-process the initial basis B and first contract all angular components (e.g. all three p-type components p N x,py,pz) of GTOs. To this end, we consider the partition {Bi}i=1of {1, ... , Nbas}, eachBibeing an index block containing all angular components of a single GTO in B, and the Norb×Nbascontraction matrix P , defined for any 1 ≤ i ≤ Norb as Pij = 1 if j ∈ Bi and zero otherwise. Next, we definethe four-index tensor T with entries N∑ Tpqrs=PpµPqνDµν ^χµχν , χκχλ^c DκλPrκPsλ,µ,ν,κ,λ=1and fold pairwise its first two Gram matrix of weightedAO products, denoted by G,we discard rows and columns of Gcorresponding to products on same denote the resulting submatrixA. Now, PCD is applied to A, using the machine epsilon as a tolerance threshold, yielding an index set sorted inpivot-descending order. Given a target M , we define the selected AO-product index set, denoted by SCholM , as theM -first pivot indices. Lastly, we recover the underlying AO index set⋃ JM={p1, p2},(p ,p )∈Sand post-process it in order to ensure that set is assigned to all atoms of the same chemicaltype. For this purpose, the new basis setis the union of parameters of those GTOs inBJ that are centered on any atom of that type. This yields a solution IM ⊇ JM to our problem in Eq. (14) and theresulting SABS is BI .Note that our generation scheme directly fixes the size M of the selected products, i.e. |SCholM | = M . The actual sizeof the AO basis set, equal to |IM|, is only implicitly controlled during our procedure. In practice, as numerical results show, |IM| is very close to M for s- and p-type basis sets, while it remains the same order as M for higher angular- momentum orbital types. Overall, the BS generation approach is extremely fast and offers access to compact basis sets, specifically adapted to a given system and user-defined qubit budget. We note that the adaptation and generation of basis sets to the molecular geometry has been the subject of several publications exploring other strategies. Among them, research in the context of quantum computing focusing on the need to limit the qubit requirements through basis-set reoptimization. Alternatively, a basis-set-free approach has been proposed through an adaptive representation using pair-natural orbitals which was tested up to 22 qubits. C. Computational Details In the present study, we perform ADAPT-VQE computations using the Qubit-Excitation-Based pool of operators, which is considered a standard. Additional details about the ADAPT-VQE methodology can be found in the SI. The communication of the Hamiltonian from the CPU to the QPU / GPU is done by using standard FCIDUMP files to communicate the one- and two-electron integrals to the QPU / GPU software in order to construct the fermion operators. TREXIO files are also useful to communicate a wider range of relevant information. ADAPT-VQE computations were performed using the Hyperion-1 GPU-accelerated state-vector sparse emulator up to 32 qubits. Hyperion-1 uses classical computing systems and is grounded on an efficient multi- GPU ensemble of fast custom sparse linear-algebra libraries accelerating Hyperion-1’s exact / noiseless simulations. In this paper, computations were performed on NVIDIA DGX A100 nodes (8x 80GB A100 GPUs per node) and NVIDIA DGX H100 nodes (4x & 8x 80GB H100 GPUs per node). QC calculations being strongly memory-dependent, a single GPU can carry out a 20-qubit ADAPT-VQE simulation depending on the nature (i.e. Hamiltonian sparsity) of the system whereas a single node (8 GPUs) can handle up to 28 qubits. Multi-node computations are required beyond such a qubit count. Further details about Hyperion-1 and its full capabilities will be given in a forthcoming publication. Convergence for all ADAPT-VQE computations were set to 10−6Ha. Most computations started from a HF initialstate. For selected ones (indicated in the text and Tables), we started the ADAPT-VQE procedure from a roughconfiguration-interaction perturbatively-selected-iteratively (CIPSI) initial state (converged to only 10−2 Ha) to savesome computational time within Hyperion-1. This reflects a commonly adopted strategy where a multi-determinantinitial state is employed instead of a single HF determinant to increase the ground-state support in the initial state.The Quantum State Preparation of such classically-derived CIPSI wave functions has been studied in the context ofVQE and QPE. Also, we report in Table III, the walltime required for several ground-state energy calculations. Asone can see, the walltime is not only a function of the number of qubits, sparsity is also important. For example, theHamiltonian of the H12 molecule is sparser than the one of the water molecule resulting in a faster convergence. Besidethe increased computing power of H100 GPUs leading to improved time-to-solution, our results also highlight theimportance of fast node-to-node interconnects when performing large-scale quantum emulation. Indeed, the benefitof H100 over A100 is striking for the largest 32 qubits simulations on 16 nodes where an improvement of a factor3 was observed on the DGX H100 systems. Such speedup is therefore also partially related to higher node-to-nodebandwidth observed on DGX H100 versus A100 systems.V. APPENDIX: CNOT COUNTSCNOT-gate counts for qubit and qubit-excitation-based (QEB) operator pools are presented in Table IV.TABLE I: Ground-state energies (in Ha) for H2O, N2, LiH, H2, and H8 calculated by FCI, ADAPT-VQE (denotedas ADAPT), and basis-set corrected ADAPT-VQE according to Strategy 1, denoted as A+PBE+∆HF, and toStrategy 2, denoted as SC(A+PBE) and SC(A+PBE)+∆HF (i.e., without and with the HF basis-set correction,respectively). Here, PBE refers to the PBE-based correlation basis-set correction and ∆HF refers to the HFbasis-set correction, and SC stands for “self-consistent”. The frozen-core approximation has been used for H2O, N2,and LiH. The CBS limits are estimated by two-point extrapolations from cc-pVQZ and cc-pV5Z calculations.H2O Nqubits FCI ADAPT A+PBE+∆HF SC(A+PBE) SC(A+PBE)+∆HFSTO-3G 12 -75.01250 -75.01250 -76.30232 -75.197880 -76.30191pcseg-0 24 -75.90855 -75.90843a -76.33681 -76.03999 -76.332796-31G 24 -76.11995 -76.11989b -76.28035 -76.23717 -76.32025V5Z-10 24 -76.12626 -76.12409 -76.32705 -76.27418c -76.32505V5Z-11 30 -76.15902 -76.15165 -76.33704 -76.28622 -76.33570cc-pVDZ 46 -76.24165 - - - -cc-pVTZ 114 -76.33250 - - - -cc-pVQZ 228 -76.35985 - - - -cc-pV5Z 400 -76.36877 - - - -CBS - -76.37812 - - - -N2 Nqubits FCI ADAPT A+PBE+∆HF SC(A+PBE) SC(A+PBE)+∆HFSTO-3G 16 -107.65251 -107.65251 -109.36630 -107.86974 -109.36661V5Z-6 16 -108.88869 -108.88869 -109.34552 -109.09850 -109.34608V5Z-11 32 -108.89413 -109.11566 -109.37278 -109.27385 -109.37281cc-pVDZ 52 -109.27698 - - - -cc-pVTZ 116 -109.37527 - - - -cc-pVQZ 216 -109.40558 - - - -cc-pV5Z 360 -109.41505 - - - -CBS - -109.42498 - - - -LiH Nqubits FCI ADAPT A+PBE+∆HF SC(A+PBE) SC(A+PBE)+∆HFSTO-3G 10 -7.88218 -7.88218 -8.02160 -7.89590 -8.02119pcseg-0 14 -7.98139 -7.98139 -8.0160 -7.99166 -8.015616-31G 20 -7.99800 -7.99800 -8.01668 -8.00806 -8.01611V5Z-4 10 -7.99287 -7.99287 -8.01758 -8.00562 -8.01690V5Z-7 16 -7.99793 -7.99793 -8.01710 -8.01109 -8.01643V5Z-10 28 -8.01302 -8.01302 -8.02575 -8.02134 -8.02540cc-pVDZ 26 -8.01438 - - - -cc-pVTZ 86 -8.02234 - - - -cc-pVQZ 190 -8.02386 - - - -cc-pV5Z 290 -8.02433 - - - -CBS - -8.02482 - - - -H2 Nqubits FCI ADAPT A+PBE+∆HF SC(A+PBE) SC(A+PBE)+∆HFSTO-3G 4 -1.13415 -1.13415 -1.17606 -1.15590 -1.175946-31G 8 -1.15003 -1.15003 -1.16911 -1.16196 -1.16913cc-pVDZ 20 -1.16275 -1.16275 -1.17239 -1.16858 -1.17246V5Z-8 24 -1.16613 -1.16613 -1.17315 -1.17170 -1.17320cc-pVTZ 56 -1.17041 - - - -cc-pVQZ 120 -1.17182 - - - -cc-pV5Z 220 -1.17223 - - - -CBS - -1.17265 - - - -H8 Nqubits FCI ADAPT A+PBE+∆HF SC(A+PBE) SC(A+PBE)+∆HFSTO-3G 16 -4.24339 -4.24320 -4.45483 -4.32764 -4.453546-31G 32 -4.37032 -4.35752 -4.43488 -4.41275 -4.43502cc-pVDZ 80 -4.42756 - - - -cc-pVTZ 222 -4.47121 - - - -cc-pVQZ 474 -4.47702 - - - -cc-pV5Z 864 - - - - -CBS - - - - - -a 500 iterations with a 14668-determinant CIPSI initial state.bc 1000 iterations with a 10879-determinant CIPSI initial state.The value is -76.27636 when using a 11016-determinant CIPSI initial state.TABLE II: Dipole moments (in atomic units) of LiH and H2O calculated by FCI and self-consistently basis-setcorrected ADAPT-VQE, denoted as SC(A+PBE) and SC(A+PBE)+∆HF (without and with and the HF basis-setcorrection, respectively). Here, ∆HF corresponds to the HF basis-set correction to the dipole moment, calculated asthe difference between the HF dipole moment in the aug-cc-pV5Z basis set and the HF dipole moment in theconsidered basis set. More data are available in the SI.LiH N FCI SC(A+PBE) SC(A+PBE)+∆HFSTO-3G 10 -1.81835 -1.86299 -2.31321pcseg-0 14 -2.33313 -2.37289 -2.256506-31G 20 -2.16646 -2.20768 -2.23674V5Z-4 10 -2.37818 -2.44145 -2.22886V5Z-7 16 -2.23095 -2.27789 -2.25618V5Z-10 28 -2.24997 -2.27458 -2.31438cc-pVDZ 26 -2.25566 - -cc-pVTZ 86 -2.29998 - -cc-pVQZ 190 -2.30361 - -cc-pV5Z 290 -2.30647 - -H O N FCI SC(A+PBE) SC(A+PBE)+∆HFSTO-3G 12 -0.63584 -0.67084 -0.77162pcseg-0 24 -0.95822 -0.99450 -0.770656-31G 24 -0.99020 -1.01898 -0.76342V5Z-10 24 -0.99305 -1.01887 -0.77634V5Z-11 30 -0.99185 -1.02170 -0.77737cc-pVDZ 46 -0.76073 - -cc-pVTZ 114 -0.75013 - -cc-pVQZ 228 -0.74994 - -cc-pV5Z 400 -0.74241 - -TABLE III: Walltime (in minutes) required to grow the wave-function ansatz of size Nadapt for a given molecularsystem using Hyperion-1 state-vector emulator on Ngpus NVIDIA GPUs. The results have been obtained on A100(80 GB) and H100 (80 GB) GPUs using CUDA Toolkit 12.0 and NVIDIA HPC SDK 23.3.Molecule / Basis set Nadapt Nqubits Ngpus A100 walltime [min] H100 walltime [min]H2O / 6-31G 500 24 1 644 503H12 / STO-3G 500 24 1 174 134H14 / STO-3G 300 28 8 283 184H16 / STO-3G 100 32 128 450 147TABLE IV: CNOT-gate counts for qubit and qubit-excitation-based (QEB) operator pools. The numbers of CNOTgates, NCNOT, are evaluated as N3 single + 13Ndouble for the QEB pool, and as N2 single + 6Ndouble for the qubit pool,where Nsimple is the number of single-qubit operators and Ndouble is the number of two-qubit operators. Nop is thenumber of operators in the final wave-function ansatz, and Nadapt iter is the number of ADAPT-VQE iterationsachieved and at which we collect the values in the Table.H2 Nqubits NCNOT (if QEB) NCNOT (if qubit) Nop Nadapt iterSTO-3G 4 13 6 1 16-31G 8 71 34 7 7cc-pVDZ 20 233 110 21 21V5Z-8 24 395 186 35 35H4 Nqubits NCNOT (if QEB) NCNOT (if qubit) Nop Nadapt iterSTO-3G 8 207 98 19 19H6 Nqubits NCNOT (if QEB) NCNOT (if qubit) Nop Nadapt iterSTO-3G 12 2437 1134 199 199H8 Nqubits NCNOT (if QEB) NCNOT (if qubit) Nop Nadapt iterSTO-3G 16 12677 5870 999 9996-31G 32 8855 4090 685 685He Nqubits NCNOT (if QEB) NCNOT (if qubit) Nop Nadapt iterpc-seg0 4 19 10 3 36-31G 4 19 10 3 3cc-pVDZ 10 58 28 6 6Be Nqubits NCNOT (if QEB) NCNOT (if qubit) Nop Nadapt iterSTO-3G 8 39 18 3 3pc-seg0 10 58 28 6 66-31G 16 207 98 19 19cc-pVDZ 26 292 136 26 26LiH Nqubits NCNOT (if QEB) NCNOT (if qubit) Nop Nadapt iterSTO-3G 10 90 44 10 10pc-seg0 14 258 124 26 266-31G 20 511 242 47 47VQZ-4 10 90 44 10 10V5Z-4 10 90 44 10 10V5Z-7 16 291 138 27 27V5Z-10 28 1083 506 91 91H2O Nqubits NCNOT (if QEB) NCNOT (if qubit) Nop Nadapt iterSTO-3G 12 729 342 63 636-31G 24 12657 5862 999 999V5Z-11 30 12647 5858 999 999N2 Nqubits NCNOT (if QEB) NCNOT (if qubit) Nop Nadapt iterSTO-3G 16 4868 2256 386 386V5Z-6 16 9714 4492 758 758V5Z-11 32 11718 5420 916 916I. MAPPING QUANTUM CHEMISTRY TO QUANTUM COMPUTERS AND QUANTUM SOLVERIn a given spin-orbital basis set B, the molecular electronic Hamiltonian can be expressed in second-quantizationas ∑ ∑ Ĥ =hpq↠† † pâq+ wpqrsâpâqâsâr, (1)p,q p,q,r,swhere the indices p, q, r, s run over theand hpq and wpqrs are the one-are usually employed to mapthe representation of the second-quantized Hamiltonian Ĥ on a quantum computer, with each qubit encoding a spin-orbital of the system. The molecular Hamiltonian Ĥ now expressed as linear combination of Pauli products, variousquantum algorithms such as the Variational Quantum Eigensolver (VQE) and Quantum Phase Estimation (QPE) cancompute the ground state of this molecular Hamiltonian. In this paper, a VQE-inspired algorithm is used. The VQEminimizes the Hamiltonian’s expectation value with respect to a parameterized ansatz wave function, in a classical-quantum hybrid approach. The challenge is constructing an ansatz wave function balancing accuracy with a shallowquantum-circuit representation for NISQ devices. To address such a challenge, the adaptive derivative-assembledpseudo-Trotter variational quantum eigensolver (ADAPT-VQE) has emerged as a standard, by proposing an ansatzthat is dynamically grown through an iterative process, resulting in an increased accuracy with shallower circuits thantraditional VQE ansatze, for which the structure is predetermined before simulations.In the present study, we use the Qubit-Excitation-Based pool of operators, which is considered a standard. Notethat one issue of ADAPT-VQE is linked to its classical optimization procedure that can encounter barren plateausgenerating very large numbers of parameters and therefore limiting practical convergence. In such cases, a variantof ADAPT-VQE can be used: the Overlap-ADAPT-VQE that allows one to grow wave functions by maximizingtheir overlap with an intermediate target wave function. That way, when barren plateaus are encountered, Overlap-ADAPT-VQE can reduce the number of optimization parameters and produce an ultra-compact ansatz suitable forhigh-accuracy initialization of a new ADAPT procedure able to converge faster to full configuration interaction (FCI).II. DATAA. GeometriesTABLE I: Geometries (in Å) of the molecular systems studied present in the paper.System x y zNN 0. 0. 0.2 N0. 0. 1.0977H OO 0. 0. 0.11732 H0. 0.7572 -0.4692H 0. -0.7572 -0.4692LiHLi 0. 0. 0.H 0. 0. 1.5949Hn H 0. 0. 0.0H 0. 0. 0.8... H0. 0 0.8(n − 1)B. Total ground-state energies with standard basis setsTABLE II: Ground-state energies (in Ha) for He, Be, FH, LiH, and H2O at the HF, near-FCI (CIPSI+PT2),ADAPT-VQE (denoted as ADAPT), non-self-consistent basis-set corrected near-FCI (denoted as FCI+PBE+∆HF),and self-consistent basis-set corrected ADAPT-VQE without the HF basis-set correction (denoted asSC(ADAPT+PBE)). The values of the PBE-based correlation basis-set correction and of the HF basis-set correctionare also given. The frozen-core approximation is used for Be, FH, LiH, and H2O. The CBS limits are estimated bytwo-point extrapolations from cc-pVQZ and cc-pV5Z calculations. The number of iterations for the ADAPT-VQEcalculations are given in bracket.He N HF ∆HF PBE FCI FCI+PBE+∆HF ADAPT SC(ADAPT+PBE)STO-3G 2 -2.80778 -0.05384 -0.03182 -2.80778 -2.89344 - -pc-seg0 4 -2.83405 -0.02757 -0.01812 -2.84979 -2.89548 -2.84979 [3] -2.867916-31G 4 -2.85516 -0.00646 -0.01950 -2.87016 -2.89612 -2.87016 [3] -2.88960cc-pVDZ 10 -2.85516 -0.00646 -0.01168 -2.88759 -2.90573 -2.88759 [6] -2.89927cc-pVTZ 28 -2.86115 -0.00047 -0.00420 -2.90023 -2.9049 - -cc-pVQZ 60 -2.86151 -0.00011 -0.00167 -2.90241 -2.90419 - -cc-pV5Z 110 -2.86162 0 -0.00084 -2.90315 -2.90399 - -CBS - - - - -2.90392 - - -Be N HF ∆HF PBE FCI FCI+PBE+∆HF ADAPT SC(ADAPT+PBE)STO-3G 8 -14.35188 -0.22113 -0.00789 -14.40332 -14.63234 -14.40333 [3] -14.41120pc-seg0 10 -14.53608 -0.03693 -0.00439 -14.57712 -14.61844 -14.57712 [6] -14.581606-31G 16 -14,56676 1,456,661 -0.00360 -14.61274 1,456,647 -14.61274

[0019] -14.61641cc-pVDZ 26 -14.57234 -0.00067 -0.00187 -14.61684 -14.61938 -14.61684

[0024] -14.61884cc-pVTZ 58 -14.57287 -0.00014 -0.00082 -14.61842 -14.61938 - -cc-pVQZ 108 -14.57297 -0.00004 -0.00038 -14.61895 -14.61937 - -cc-pV5Z 180 -14.57301 0 -0.00021 -14.61908 -14.61929 - -CBS - - - - -14.61921 - - -LiH N HF ∆HF PBE FCI FCI+PBE+∆HF ADAPT SC(ADAPT+PBE)STO-3G 10 -7.86203 -0.12529 -0.01413 -7.88218 -8.0216 -7.88218

[0010] -7.89590pc-seg0 14 -7.96337 -0.02395 -0.01066 -7.98139 -8.016 -7.98139

[0026] -7.991666-31G 20 -7.97927 -0.00805 -0.01063 -7.99800 -8.01668 -7.99800

[0047] -8.00806VQZ-4 10 -7.97603 -0.01129 -0.01344 -7.99281 -8.01754 -7.99281

[0010] -8.00557V5Z-4 10 -7.97604 -0.01128 -0.01343 -7.99287 -8.01758 -7.99287

[0010] -8.00562V5Z-7 16 -7.98198 -0.00534 -0.01383 -7.99793 -8.0171 -7.99793

[0027] -8.01658V5Z-10 28 -7.98326 -0.00406 -0.00867 -8.01302 -8.02575 -8.01302

[0091] -8.02134cc-pVDZ 36 -7.98373 -0.00359 -0.00418 -8.01438 -8.02215 - -cc-pVTZ 86 -7.98665 -0.00067 -0.00145 -8.02234 -8.02446 - -cc-pVQZ 190 -7.98718 -0.00014 -0.00063 -8.02386 -8.02463 - -cc-pV5Z 290 -7.98732 0 -0.00035 -8.02433 -8.02468 - -CBS - - - - -8.02482 - - -H O N HF ∆HF PBE FCI FCI+PBE+∆HF ADAPT SC(ADAPT+PBE)STO-3G 12 -74.96302 -1.10403 -0.185798 -75.01250 -76.302328 -75.01250

[0063] -75.19788pc-seg0 24 -75.77425 -0.2928 -0.135581 -75.90855 -76.336931 -75.90842

[1000] -76.039996-31G 24 -75.98397 -0.08308 -0.07738 -76.11995 -76.28041 -76.11989

[1000] -76.23717V5Z-10 24 -76.01618 -0.05087 -0.15209 -76.12626 -76.32922 -76.12409

[1000] -76.27418V5Z-11 30 -76.01756 -0.04948 -0.13591 -76.15902 -76.34441 -76.15165

[0541] -76.28622cc-pVDZ 46 -76.02677 -0.04028 -0.07406 -76.24165 -76.35599 - -cc-pVTZ 114 -76.05713 -0.00992 -0.03063 -76.33250 -76.37305 - -cc-pVQZ 228 -76.06479 -0.00226 -0.01471 -76.35985 -76.37682 - -cc-pV5Z 400 -76.06705 0 -0.00801 -76.36877 -76.37678 - -CBS - - - - -76.37812 - - -N N HF ∆HF PBE FCI FCI+PBE+∆HF ADAPT SC(ADAPT+PBE)STO-3G 16 -107.49589 -1.49687 -0.21692 -107.65253 -109.36632 -107.65251

[0386] -107.86974V5Z-6 16 -108.74518 -0.24758 -0.20925 -108.88869 -109.34552 -108.88869

[0759] -109.09850cc-pVDZ 52 -108.95412 -0.03864 -0.08984 -109.27698 -109.40546 - -cc-pVTZ 116 -108.98347 -0.00929 -0.03695 -109.37527 -109.42151 - -cc-pVQZ 216 -108.99108 -0.00168 -0.01788 -109.40558 -109.42514 - -cc-pV5Z 360 -108.99276 0 -0.00996 -109.41505 -109.42501 - -CBS - - - - -109.42498 - - -a From the 19th iteration, the ADAPT-VQE iterations kept choosing the same operator, thus the energy stops varying from iteration 19.bc From the 47th iteration, the ADAPT-VQE iterations kept choosing the same operator, thus the energy stops varying from iteration 47.Initial state has 10879 determinants.d Initial state has 14668 determinants.TABLE III: Ground-state energies (in Ha) for hydrogen chains with atomic distances of 0.8 Å at the HF, near-FCI(CIPSI+PT2), ADAPT-VQE (denoted as ADAPT), non-self-consistent basis-set corrected near-FCI (denoted asFCI+PBE+∆HF), and self-consistent basis-set corrected ADAPT-VQE without the HF basis-set correction(denoted as SC(ADAPT+PBE)). The values of the PBE-based correlation basis-set correction and of the HFbasis-set correction are also given. The CBS limits are estimated by two-point extrapolations from cc-pVQZ andcc-pV5Z calculations. The number of iterations for the ADAPT-VQE calculations are given in bracket.H2 Nqubits HF ∆HF PBE FCI FCI+PBE+∆HF ADAPT SC(ADAPT+PBE)STO-3G 4 -1.11085 -0.02004 -0.02187 -1.13415 -1.17606 -1.13415 [1] -1.155906-31G 8 -1.12371 -0.00717 -0.01191 -1.15003 -1.16911 -1.15003 [7] -1.16196cc-pVDZ 20 -1.12700 -0.00388 -0.00576 -1.16275 -1.17239 -1.16275

[0021] -1.16858V5Z-8 24 -1.12938 -0.00150 -0.00552 -1.16613 -1.17315 -1.16613

[0035] -1.17170cc-pVTZ 56 -1.13029 -0.00059 -0.00177 -1.17041 -1.17277 - -cc-pVQZ 120 -1.13075 -0.00014 -0.00074 -1.17182 -1.17270 - -cc-pV5Z 220 -1.13089 0 -0.00037 -1.17223 -1.17260 - -CBS - - - - -1.17265 - - -H4 Nqubits HF ∆HF PBE FCI FCI+PBE+∆HF ADAPT SC(ADAPT+PBE)STO-3G 8 -2.12139 -0.05672 -0.04299 -2.16756 -2.26727 -2.16756

[0019] -2.21040cc-pVDZ 40 -2.16785 -0.01026 -0.01188 -2.24884 -2.27098 - -cc-pVTZ 112 -2.17696 -0.00115 -0.00371 -2.26848 -2.27333 - -cc-pVQZ 240 -2.17782 -0.00029 -0.00157 -2.27134 -2.27320 - -cc-pV5Z 438 -2.17811 0 -0.00078 -2.27222 -2.27299 - -CBS - - - - -2.27315 - - -H6 Nqubits HF ∆HF PBE FCI FCI+PBE+∆HF ADAPT SC(ADAPT+PBE)STO-3G 12 -3.13461 -0.09203 -0.04299 -3.20441 -3.33943 -3.20441

[0200] -3.268586-31G 24 -3.20987 -0.01677 -0.02653 -3.29582 -3.33912 - -cc-pVDZ 60 -3.20989 -0.01675 -0.01801 -3.33763 -3.37239 - -cc-pVTZ 168 -3.22483 -0.00181 -0.00566 -3.36939 -3.37686 - -cc-pVQZ 360 -3.22619 -0.00045 -0.00241 -3.37379 -3.37665 - -cc-pV5Z 650 -3.22664 0 -0.00120 - - - -CBS - - - - - - - -H8 Nqubits HF ∆HF PBE FCI FCI+PBE+∆HF ADAPT SC(ADAPT+PBE)STO-3G 16 -4.14962 -0.1259 -0.08573 -4.24339 -4.45502 -4.24320

[1000] -4.327646-31G 32 -4.25325 -0.02227 -0.05509 -4.37032 -4.44768 -4.35752

[0685] -4.41275cc-pVDZ 80 -4.25244 -0.02308 -0.02419 -4.42756 -4.47483 - -cc-pVTZ 222 -4.27304 -0.00248 -0.00762 -4.47121 -4.48131 - -cc-pVQZ 474 -4.27490 -0.00062 -0.00327 - - - -cc-pV5Z 864 -4.27552 0 - - - - -CBS - - - - - - - -H10 Nqubits HF ∆HF PBE FCI FCI+PBE+∆HF ADAPT SC(ADAPT+PBE)STO-3G 20 -5.16558 -0.15903 -0.10712 -5.28355 -5.5497 - -cc-pVDZ 100 -5.29511 -0.0295 -0.03040 -5.51771 -5.57761 - -cc-pVTZ 278 -5.32146 -0.00315 -0.00959 -5.57298 -5.58572 - -cc-pVQZ 590 -5.32381 -0.0008 -0.00412 - - - -cc-pV5Z 1074 -5.32461 0 - - - - -CBS - -5.32545 - - - - - -C. Total ground-state energies The target sizes vary from minimal basis size (STO-3G) to the full size of the original AO basis set. TABLE IV: [1 / 2] Ground-state energies (in Ha) of the H2molecule calculated by HF, near-FCI (CIPSI+PT2), and self-consistent basis-set corrected near-FCI without and with the HF basis-set correction (denoted as SC(FCI+PBE) and SC(FCI+PBE)+∆HF, respectively). VXZ-Y corresponds to the cc-pVXZ basis transformed to a basis of Y AO functions following the building procedure. The notation VXZ-[Yn-Yn+a] means that the values are the same for the basis sets VXY-Yn, VXY-Yn+1, ..., VXY-Yn+a−1, and VXY-Yn+a. The FCI / CBS limit is -1.13103 Ha. H2 Basis set # active MOs HF FCI SC(FCI+PBE) SC(FCI+PBE)+∆HFVDZ-[2-3] 2 -1.09635 -1.11372 -1.13412 -1.16880VDZ-[4-5] 4 -1.12286 -1.14810 -1.16186 -1.17002VDZ-6 10 -1.12700 -1.16275 -1.16857 -1.17260VTZ-[2-3] 2 -1.09723 -1.11461 -1.13498 -1.16878VTZ-[4-5] 4 -1.12226 -1.14639 -1.16134 -1.17010VTZ-[6-7] 6 -1.12513 -1.15191 -1.16293 -1.16883VTZ-[8-9] 12 -1.12949 -1.16278 -1.16649 -1.16802VTZ-[10-12] 18 -1.13027 -1.16890 -1.17188 -1.17263VTZ-13 18 -1.13027 -1.16890 -1.17188 -1.17263VTZ-14 28 -1.13029 -1.17040 -1.17215 -1.17289VQZ-[2-3] 2 -1.09751 -1.11491 -1.13527 -1.16879VQZ-4 4 -1.12096 -1.14421 -1.16002 -1.17008VQZ-[5-7] 6 -1.12552 -1.15237 -1.16356 -1.16906VQZ-[8-9] 12 -1.13005 -1.16638 -1.17068 -1.17166VQZ-10 18 -1.13019 -1.16856 -1.17246 -1.17330VQZ-[11-12] 20 -1.13040 -1.16907 -1.17252 -1.17314VQZ-[13-15] 26 -1.13067 -1.16987 -1.17214 -1.17250VQZ-[16-19] 36 -1.13074 -1.17092 -1.17218 -1.17246VQZ-20 46 -1.13074 -1.17158 -1.17258 -1.17286VQZ-21 46 -1.13074 -1.17158 -1.17259 -1.17287VQZ-22 60 -1.13074 -1.17182 -1.17257 -1.17285V5Z-2 2 -1.09759 -1.11499 -1.13535 -1.16879V5Z-[3-6] 4 -1.12106 -1.14572 -1.15779 -1.16776V5Z-7 10 -1.12579 -1.16014 -1.16560 -1.17083V5Z-8 12 -1.12938 -1.16613 -1.17170 -1.17335V5Z-9 14 -1.13016 -1.16720 -1.17199 -1.17286V5Z-[10-12] 20 -1.13018 -1.16816 -1.17246 -1.17331V5Z-[13-15] 26 -1.13078 -1.16986 -1.17232 -1.17256V5Z-16 28 -1.13079 -1.17006 -1.17234 -1.17257V5Z-17 28 -1.13079 -1.17006 -1.17232 -1.17255V5Z-18 38 -1.13081 -1.17101 -1.17290 -1.17312V5Z-19 48 -1.13085 -1.17167 -1.17290 -1.17308V5Z-20 54 -1.13087 -1.17175 -1.17274 -1.17290V5Z-21 54 -1.13087 -1.17175 -1.17274 -1.17289V5Z-22 54 -1.13087 -1.17175 -1.17270 -1.17286V5Z-[23-26] 68 -1.13087 -1.17194 -1.17281 -1.17296V5Z-[27-28] 78 -1.13088 -1.17205 -1.17268 -1.17283V5Z-[29-31] 92 -1.13088 -1.17216 -1.17260 -1.17275V5Z-32 110 -1.13088 -1.17222 -1.17252 -1.17267TABLE V: [2 / 2] Ground-state energies (in Ha) of the H2molecule calculated by HF, near-FCI (CIPSI+PT2), and self-consistent basis-set corrected near-FCI without and with the HF basis-set correction (denoted as SC(FCI+PBE) and SC(FCI+PBE)+∆HF, respectively). VXZ-Y corresponds to the cc-pVXZ basis transformed to a basis of Y AO functions following the building procedure. The notation VXZ-[Yn-Yn+a] means that the values are the same for the basis sets VXY-Yn, VXY-Yn+1, ..., VXY-Yn+a−1, and VXY-Yn+a. The FCI / CBS limit is -1.13103 Ha. H2 Basis set # active MOs HF FCI SC(FCI+PBE) SC(FCI+PBE)+∆HFV6Z-[2-3] 2 -1.09760 -1.11500 -1.13536 -1.16879V6Z-4 4 -1.12438 -1.15020 -1.16311 -1.16976V6Z-5 10 -1.12791 -1.16391 -1.17072 -1.17383V6Z-[6-8] 12 -1.12829 -1.16455 -1.17064 -1.17337V6Z-[9-10] 18 -1.12832 -1.16503 -1.17045 -1.17315V6Z-11 20 -1.12979 -1.16704 -1.17220 -1.17344V6Z-12 26 -1.13068 -1.16963 -1.17239 -1.17273V6Z-[13-17] 28 -1.13077 -1.16993 -1.17243 -1.17268V6Z-18 38 -1.13081 -1.17129 -1.17301 -1.17323V6Z-19 52 -1.13081 -1.17151 -1.17293 -1.17315V6Z-20 52 -1.13081 -1.17151 -1.17294 -1.17316V6Z-21 54 -1.13082 -1.17156 -1.17292 -1.17313V6Z-22 54 -1.13082 -1.17156 -1.17294 -1.17314V6Z-23 54 -1.13082 -1.17156 -1.17293 -1.17314V6Z-24 60 -1.13087 -1.17173 -1.17276 -1.17291V6Z-25 70 -1.13088 -1.17181 -1.17285 -1.17299V6Z-26 70 -1.13088 -1.17181 -1.17285 -1.17299V6Z-27 80 -1.13089 -1.17203 -1.17270 -1.17283V6Z-28 80 -1.13089 -1.17203 -1.17269 -1.17283V6Z-29 86 -1.13089 -1.17205 -1.17266 -1.17279V6Z-30 86 -1.13089 -1.17205 -1.17267 -1.17280V6Z-31 100 -1.13090 -1.17217 -1.17257 -1.17270V6Z-[32-33] 100 -1.13090 -1.17217 -1.17268 -1.17281V6Z-[34-37] 118 -1.13090 -1.17222 -1.17266 -1.17279V6Z-[38-39] 140 -1.13090 -1.17224 -1.17261 -1.17273V6Z-[40-45] 150 -1.13090 -1.17226 -1.17256 -1.17269V6Z-[46-79] 168 -1.13090 -1.17229 -1.17254 -1.17267TABLE VI: [1 / 3] Ground-state energies (in Ha) of the LiH molecule calculated by HF, near-FCI (CIPSI +PT2), and self-consistent basis-set corrected near-FCI without and with the HF basis-set correction (denoted as SC(FCI+PBE) and SC(FCI+PBE)+∆HF, respectively). VXZ-Y corresponds to the cc-pVXZ basis transformed to a basis of Y AO functions following the building procedure. The notation VXZ-[Yn-Yn+a] means that the values are the same for the basis sets VXY-Yn, VXY-Yn+1, ..., VXY-Yn+a−1, and VXY-Yn+a. The FCI / CBS limit is -8.02482 Ha. LiH Basis set # active MOs HF FCI SC(FCI+PBE) SC(FCI+PBE)+∆HFVDZ-[4-7] 5 -7.97539 -7.99185 -8.00469 -8.01677VDZ-8 9 -7.98017 -7.99894 -8.00953 -8.01683VDZ-9 12 -7.98147 -8.00858 -8.01321 -8.01922VDZ-10 13 -7.98232 -8.01002 -8.01456 -8.01972VDZ-11 13 -7.98232 -8.01002 -8.01460 -8.01975VDZ-12 18 -7.98372 -8.01437 -8.01851 -8.02226VTZ-[4,5] 5 -7.97596 -7.99267 -8.00544 -8.01696VTZ-[6-8] 8 -7.98028 -7.99634 -8.00932 -8.01652VTZ-9 11 -7.98340 -7.99981 -8.01260 -8.01668VTZ-10 12 -7.98447 -8.00140 -8.01392 -8.01693VTZ-11 17 -7.98509 -8.00438 -8.01549 -8.01788VTZ-12 20 -7.98592 -8.01781 -8.02355 -8.02510VTZ-[13-15] 21 -7.98606 -8.01837 -8.02278 -8.02420VTZ-16 26 -7.98627 -8.01898 -8.02319 -8.02440VTZ-[17,18] 33 -7.98643 -8.01973 -8.02368 -8.02473VTZ-19 38 -7.98647 -8.02061 -8.02286 -8.02386VTZ-[20-23] 41 -7.98652 -8.02151 -8.02302 -8.02397VTZ-[24-41] 42 -7.98661 -8.02212 -8.02359 -8.02446VTZ-42 43 -7.98664 -8.02234 -8.02378 -8.02461VQZ-[4,5] 5 -7.97603 -7.99281 -8.00557 -8.01702VQZ-[6-9] 8 -7.97985 -7.99605 -8.00895 -8.01657VQZ-[10,11] 12 -7.98304 -8.00059 -8.01089 -8.01532VQZ-12 15 -7.98404 -8.01500 -8.02051 -8.02394VQZ-[13,14] 16 -7.98469 -8.01585 -8.02132 -8.02411VQZ-15 21 -7.98561 -8.01772 -8.02277 -8.02463VQZ-16 22 -7.98575 -8.01817 -8.02285 -8.02458VQZ-17 25 -7.98651 -8.01900 -8.02364 -8.02461VQZ-18 30 -7.98659 -8.02043 -8.02331 -8.02419VQZ-[19-21] 35 -7.98661 -8.02065 -8.02345 -8.02432VQZ-22 36 -7.98696 -8.02107 -8.02385 -8.02437VQZ-23 39 -7.98705 -8.02276 -8.02443 -8.02485VQZ-[24,25] 44 -7.98712 -8.02288 -8.02448 -8.02483VQZ-[26-28] 51 -7.98714 -8.02316 -8.02473 -8.02507VQZ-[29,30] 58 -7.98714 -8.02326 -8.02478 -8.02511VQZ-31 61 -7.98715 -8.02341 -8.02455 -8.02488VQZ-32 62 -7.98715 -8.02350 -8.02462 -8.02494VQZ-33 63 -7.98716 -8.02357 -8.02468 -8.02500VQZ-34 72 -7.98717 -8.02366 -8.02472 -8.02503VQZ-[35-38] 79 -7.98717 -8.02381 -8.02458 -8.02489VQZ-39 84 -7.98717 -8.02391 -8.02453 -8.02484D. Dissociation curvesTABLE VII: [2 / 3] Ground-state energies (in Ha) of the LiH molecule calculated by HF, near-FCI (CIPSI +PT2), and self-consistent basis-set corrected near-FCI without and with the HF basis-set correction (denoted as SC(FCI+PBE) and SC(FCI+PBE)+∆HF, respectively). VXZ-Y corresponds to the cc-pVXZ basis transformed to a basis of Y AO functions following the building procedure. The notation VXZ-[Yn-Yn+a] means that the values are the same for the basis sets VXY-Yn, VXY-Yn+1, ..., VXY-Yn+a−1, and VXY-Yn+a. The FCI / CBS limit is -8.02482 Ha. LiH Basis set # active MOs HF FCI SC(FCI+PBE) SC(FCI+PBE)+∆HFV5Z-[4-6] 5 -7.97604 -7.99287 -8.00561 -8.01705V5Z-[7,8] 8 -7.98198 -7.99793 -8.01108 -8.01657V5Z-9 11 -7.98238 -7.99900 -8.01179 -8.01689V5Z-10 14 -7.98326 -8.01302 -8.02133 -8.02554V5Z-[11,12] 15 -7.98391 -8.01389 -8.02195 -8.02552V5Z-13 16 -7.98420 -8.01421 -8.02227 -8.02555V5Z-14 19 -7.98543 -8.01533 -8.02336 -8.02541V5Z-15 20 -7.98630 -8.01710 -8.02423 -8.02541V5Z-16 21 -7.98651 -8.01757 -8.02375 -8.02471V5Z-17 22 -7.98657 -8.01778 -8.02382 -8.02472V5Z-[18,19] 27 -7.98673 -8.01954 -8.02377 -8.02452V5Z-20 32 -7.98673 -8.01970 -8.02385 -8.02460V5Z-[21,22] 35 -7.98683 -8.02212 -8.02454 -8.02519V5Z-23 38 -7.98714 -8.02247 -8.02487 -8.02520V5Z-[24,25] 39 -7.98724 -8.02298 -8.02493 -8.02517V5Z-26 44 -7.98724 -8.02317 -8.02506 -8.02529V5Z-[27-29] 49 -7.98728 -8.02326 -8.02510 -8.02530V5Z-[30-32] 56 -7.98728 -8.02338 -8.02517 -8.02536V5Z-33 60 -7.98729 -8.02362 -8.02499 -8.02517V5Z-34 67 -7.98730 -8.02384 -8.02484 -8.02502V5Z-35 68 -7.98732 -8.02387 -8.02487 -8.02503V5Z-36 68 -7.98732 -8.02387 -8.02485 -8.02501V5Z-[37-39] 75 -7.98732 -8.02389 -8.02487 -8.02503V5Z-40 78 -7.98732 -8.02390 -8.02472 -8.02488V5Z-41 83 -7.98732 -8.02403 -8.02465 -8.02481V5Z-42 90 -7.98732 -8.02412 -8.02453 -8.02469V5Z-43 90 -7.98732 -8.02412 -8.02455 -8.02471V5Z-44 90 -7.98732 -8.02412 -8.02448 -8.02464V5Z-45 90 -7.98732 -8.02412 -8.02452 -8.02467V5Z-46 90 -7.98732 -8.02408 -8.02469 -8.02484V5Z-47 99 -7.98732 -8.02406 -8.02466 -8.02481V5Z-48 99 -7.98732 -8.02413 -8.02472 -8.02488V5Z-49 99 -7.98732 -8.02412 -8.02472 -8.02487V5Z-50 99 -7.98732 -8.02413 -8.02468 -8.02483V5Z-51 99 -7.98732 -8.02409 -8.02470 -8.02486V5Z-52 106 -7.98732 -8.02415 -8.02466 -8.02482V5Z-53 115 -7.98732 -8.02420 -8.02443 -8.02459V5Z-54 115 -7.98732 -8.02420 -8.02452 -8.02468V5Z-55 115 -7.98732 -8.02420 -8.02452 -8.02468V5Z-56 120 -7.98732 -8.02427 -8.02451 -8.02466V5Z-57 129 -7.98732 -8.02430 -8.02445 -8.02461V5Z-58 129 -7.98732 -8.02430 -8.02449 -8.02464V5Z-59 129 -7.98732 -8.02430 -8.02450 -8.02465V5Z-60 129 -7.98732 -8.02430 -8.02448 -8.02464TABLE VIII: [3 / 3] Ground-state energies (in Ha) of the LiH molecule calculated by HF, near-FCI (CIPSI+PT2), and self-consistent basis-set corrected near-FCI without and with the HF basis-set correction (denoted as SC(FCI+PBE) and SC(FCI+PBE)+∆HF, respectively). VXZ-Y corresponds to the cc-pVXZ basis transformed to a basis of Y AO functions following the building procedure. The notation VXZ-[Yn-Yn+a] means that the values are the same for the basis sets VXY-Yn, VXY-Yn+1, ..., VXY-Yn+a−1, and VXY-Yn+a. The FCI / CBS limit is -8.02482 Ha. LiH Basis set # active MOs HF FCI SC(FCI+PBE) SC(FCI+PBE)+∆HFV5Z-61 129 -7.98732 -8.02430 -8.02450 -8.02466V5Z-62 129 -7.98732 -8.02430 -8.02454 -8.02469V5Z-63 140 -7.98732 -8.02432 -8.02442 -8.02458V5Z-64 140 -7.98732 -8.02432 -8.02446 -8.02462V5Z-65 140 -7.98732 -8.02432 -8.02430 -8.02445V5Z-66 140 -7.98732 -8.02432 -8.02449 -8.02464V5Z-67 140 -7.98732 -8.02432 -8.02435 -8.02451V5Z-68 140 -7.98732 -8.02432 -8.02447 -8.02462V5Z-69 140 -7.98732 -8.02432 -8.02433 -8.02449V5Z-70 140 -7.98732 -8.02432 -8.02425 -8.02440V5Z-71 140 -7.98732 -8.02432 -8.02428 -8.02443V5Z-72 140 -7.98732 -8.02432 -8.02437 -8.02453V5Z-73 140 -7.98732 -8.02432 -8.02451 -8.02467V5Z-74 140 -7.98732 -8.02432 -8.02432 -8.02448V5Z-75 140 -7.98732 -8.02432 -8.02426 -8.02441V5Z-76 140 -7.98732 -8.02432 -8.02440 -8.02456V5Z-77 140 -7.98732 -8.02432 -8.02430 -8.02445V5Z-78 140 -7.98732 -8.02432 -8.02428 -8.02444V5Z-79 140 -7.98732 -8.02432 -8.02433 -8.02449V5Z-80 140 -7.98732 -8.02432 -8.02431 -8.02447V5Z-81 145 -7.98732 -8.02431 -8.02399 -8.02415TABLE IX: [1 / 2] Ground-state energies (in Ha) of the H2O molecule calculated by HF, near-FCI (CIPSI +PT2), and self-consistent basis-set corrected near-FCI without and with the HF basis-set correction (denoted as SC(FCI+PBE) and SC(FCI+PBE)+∆HF, respectively). VXZ-Y corresponds to the cc-pVXZ basis transformed to a basis of Y AO functions following the building procedure. The notation VXZ-[Yn-Yn+a] means that the values are the same for the basis sets VXY-Yn, VXY-Yn+1, ..., VXY-Yn+a−1, and VXY-Yn+a. The FCI / CBS limit is -76.37812 Ha. H2O Basis set # active MOs HF FCI SC(FCI+PBE) SC(FCI+PBE)+∆HFVDZ-5 6 -75.89241 -75.93669 -76.11929 -76.29628VDZ-6 7 -75.90086 -75.95934 -76.13269 -76.30124VDZ-7 9 -75.96623 -76.02968 -76.19811 -76.30128VDZ-8 9 -75.96623 -76.02968 -76.19811 -76.30128VDZ-9 12 -75.98044 -76.11620 -76.25492 -76.34389VDZ-10 12 -75.98044 -76.11620 -76.25499 -76.34396VDZ-11 18 -76.01858 -76.17283 -76.29347 -76.34430VDZ-12 18 -76.01858 -76.17283 -76.29347 -76.34430VDZ-13 18 -76.01858 -76.17283 -76.29347 -76.34430VDZ-14 18 -76.01858 -76.17283 -76.29347 -76.34430VDZ-15 23 -76.02677 -76.24164 -76.31537 -76.35801VTZ-5 6 -75.90857 -75.95344 -76.13540 -76.29623VTZ-6 6 -75.90857 -75.95344 -76.13540 -76.29623VTZ-7 8 -75.97635 -76.02689 -76.20462 -76.29768VTZ-8 8 -75.97635 -76.02689 -76.20461 -76.29767VTZ-9 11 -75.99693 -76.11476 -76.26502 -76.33749VTZ-10 12 -76.00181 -76.13304 -76.27671 -76.34430VTZ-11 12 -76.00181 -76.13304 -76.27661 -76.34420VTZ-12 18 -76.03306 -76.17994 -76.31259 -76.34894VTZ-13 20 -76.03462 -76.18539 -76.31508 -76.34987VTZ-14 20 -76.03462 -76.18539 -76.31508 -76.34987VTZ-15 20 -76.03462 -76.18539 -76.31508 -76.34987VTZ-16 26 -76.04399 -76.20157 -76.31989 -76.34530VTZ-17 31 -76.05250 -76.25599 -76.34643 -76.36333VTZ-18 31 -76.05250 -76.25599 -76.34643 -76.36333VTZ-19 36 -76.05538 -76.28894 -76.34552 -76.35954VTZ-20 39 -76.05554 -76.30272 -76.35240 -76.36626VTZ-21 39 -76.05554 -76.30298 -76.35240 -76.36626VTZ-22 49 -76.05664 -76.31067 -76.35578 -76.36854VTZ-23 56 -76.05705 -76.32636 -76.36003 -76.37239VQZ-5 6 -75.91275 -75.95778 -76.13974 -76.29640VQZ-6 6 -75.91275 -75.95778 -76.13974 -76.29640VQZ-7 6 -75.91275 -75.95778 -76.13974 -76.29640VQZ-8 8 -75.97626 -76.02635 -76.20482 -76.29796VQZ-9 11 -75.99855 -76.11074 -76.26425 -76.33510VQZ-10 12 -76.00570 -76.12947 -76.27719 -76.34089VQZ-11 14 -76.02138 -76.14741 -76.29062 -76.33865VQZ-12 14 -76.02138 -76.14741 -76.29062 -76.33865VQZ-13 20 -76.04618 -76.19039 -76.31636 -76.33958VQZ-14 20 -76.04618 -76.19039 -76.31636 -76.33958VQZ-15 23 -76.04840 -76.21652 -76.32652 -76.34753VQZ-16 28 -76.05728 -76.28404 -76.34910 -76.36123VQZ-17 34 -76.05950 -76.29193 -76.35589 -76.36579VQZ-18 39 -76.06046 -76.30063 -76.36204 -76.37098VQZ-19 39 -76.06046 -76.30063 -76.36205 -76.37099VQZ-20 39 -76.06046 -76.30065 -76.36204 -76.37098VQZ-21 45 -76.06207 -76.30361 -76.36256 -76.36990VQZ-22 45 -76.06207 -76.30361 -76.36256 -76.36990VQZ-23 50 -76.06247 -76.31192 -76.35916 -76.36609TABLE X: [2 / 2] Ground-state energies (in Ha) of the H2O molecule calculated by HF, near-FCI (CIPSI+PT2), and self-consistent basis-set corrected near-FCI without and with the HF basis-set correction (denoted as SC(FCI+PBE) and SC(FCI+PBE)+∆HF, respectively). VXZ-Y corresponds to the cc-pVXZ basis transformed to a basis of Y AO functions following the building procedure. The notation VXZ-[Yn-Yn+a] means that the values are the same for the basis sets VXY-Yn, VXY-Yn+1, ..., VXY-Yn+a−1, and VXY-Yn+a. The FCI / CBS limit is -76.37812 Ha. H2O Basis set # active MOs HF FCI SC(FCI+PBE) SC(FCI+PBE)+∆HFV5Z-5 6 -75.91383 -75.95891 -76.14081 -76.29638V5Z-6 6 -75.91383 -75.95891 -76.14081 -76.29638V5Z-7 6 -75.91383 -75.95891 -76.14081 -76.29638V5Z-8 8 -75.99384 -76.04603 -76.22026 -76.29583V5Z-9 9 -76.00422 -76.07508 -76.23959 -76.30477V5Z-10 12 -76.01618 -76.12626 -76.27636 -76.32958V5Z-11 15 -76.01756 -76.15902 -76.29319 -76.34504V5Z-12 15 -76.01756 -76.15902 -76.29321 -76.34506V5Z-13 17 -76.02137 -76.16455 -76.29813 -76.34616V5Z-14 23 -76.05007 -76.21406 -76.33178 -76.35112V5Z-15 28 -76.05818 -76.27612 -76.35656 -76.36779V5Z-16 28 -76.05818 -76.27612 -76.35656 -76.36779V5Z-17 30 -76.05939 -76.27878 -76.35751 -76.36752V5Z-18 30 -76.05939 -76.27878 -76.35751 -76.36752V5Z-19 30 -76.05939 -76.27878 -76.35751 -76.36752V5Z-20 30 -76.05939 -76.27878 -76.35751 -76.36752V6Z-5 6 -75.91390 -75.95899 -76.14088 -76.29639V6Z-6 6 -75.91390 -75.95899 -76.14088 -76.29639V6Z-7 6 -75.91390 -75.95899 -76.14088 -76.29639V6Z-8 8 -75.99599 -76.04790 -76.22379 -76.29720V6Z-9 9 -76.00502 -76.07392 -76.24098 -76.30537V6Z-10 12 -76.01665 -76.14766 -76.29046 -76.34322V6Z-11 18 -76.04304 -76.19235 -76.32124 -76.34761V6Z-12 18 -76.04304 -76.19233 -76.32127 -76.34763V6Z-13 21 -76.04482 -76.19875 -76.32480 -76.34939V6Z-14 22 -76.04584 -76.20378 -76.32573 -76.34930V6Z-15 24 -76.04619 -76.20739 -76.32728 -76.35050V6Z-16 24 -76.04619 -76.20740 -76.32728 -76.35050V6Z-17 29 -76.05504 -76.26201 -76.35272 -76.36709V6Z-18 32 -76.05517 -76.27309 -76.35519 -76.36943V6Z-19 38 -76.05960 -76.28117 -76.35808 -76.36789V6Z-20 38 -76.05960 -76.28166 -76.35808 -76.36789TABLE XI: Ground-state energies (in Ha) of the N2molecule calculated by HF, near-FCI (CIPSI+PT2), and self-consistent basis-set corrected near-FCI without and with the HF basis-set correction (denoted as SC(FCI +PBE) and SC(FCI+PBE)+∆HF, respectively). VXZ-Y corresponds to the cc-pVXZ basis transformed to a basis of Y AO functions following the building procedure. The notation VXZ-[Yn-Yn+a] means that the values are the same for the basis sets VXY-Yn, VXY-Yn+1, ..., VXY-Yn+a−1, and VXY-Yn+a. The FCI / CBS limit is -109.42498 Ha. N2 Basis set # active MOs HF FCI SC(FCI+PBE) SC(FCI+PBE)+∆HFVQZ-10 14 -108.82485 -109.01533 -109.20135 -109.37101VQZ-11 16 -108.88278 -109.10163 -109.27284 -109.38458VQZ-[12-14] 22 -108.90058 -109.14692 -109.29652 -109.39046VQZ-[15-16] 24 -108.90226 -109.15252 -109.29290 -109.38516VQZ-[17-20] 34 -108.98089 -109.31664 -109.39284 -109.40647VQZ-[21-25] 44 -108.98341 -109.32951 -109.38867 -109.39977VQZ-26 58 -108.98412 -109.35139 -109.39949 -109.40989VQZ-27 68 -108.98955 -109.36564 -109.41443 -109.41940V5Z-10 14 -108.80656 -109.01099 -109.18262 -109.37057V5Z-[11-13] 16 -108.89413 -109.12739 -109.28517 -109.38556V5Z-14 26 -108.97053 -109.28372 -109.38187 -109.40585V5Z-[15-17] 32 -108.97450 -109.29460 -109.39378 -109.41380V5Z-18 34 -108.97679 -109.30157 -109.39547 -109.41320V5Z-19 40 -108.97763 -109.31094 -109.39763 -109.41451TABLE XII: Ground-state energies (in Ha) used for the dissociation curves of H2 at the HF, near-FCI(CIPSI+PT2), self-consistent basis-set corrected ADAPT-VQE without the HF basis-set correction (denoted asSC(A+PBE)), and with the HF basis-set correction (denoted as SC(A+PBE)+∆HF). Distances are in Å. VXZstands for the standard cc-pVXZ basis set.Basis set Distance HF FCI SC(A+PBE) SC(A+PBE)+∆HFVDZ 0.5 -1.04880 -1.07937 -1.08802 -1.104140.6 -1.10689 -1.13917 -1.14667 -1.156160.7 -1.12692 -1.16090 -1.16748 -1.173330.8 -1.12700 -1.16275 -1.16858 -1.172460.9 -1.11639 -1.15408 -1.15930 -1.162181.0 -1.10015 -1.14007 -1.14481 -1.147241.5 -1.00219 -1.06153 -1.06513 -1.067872.0 -0.92191 -1.01759 -1.02089 -1.025222.5 -0.86533 -1.00313 -1.00631 -1.012383.0 -0.82645 -0.99955 -1.00268 -1.01043V5Z-8 0.5 -1.05683 -1.08907 -1.09726 -1.105350.6 -1.11193 -1.14569 -1.15284 -1.157290.7 -1.13028 -1.16548 -1.17178 -1.174280.8 -1.12938 -1.16614 -1.17170 -1.173210.9 -1.11825 -1.15679 -1.16174 -1.162761.0 -1.10177 -1.14243 -1.14689 -1.147701.5 -1.00378 -1.06363 -1.06675 -1.067912.0 -0.92434 -1.01944 -1.02219 -1.024092.5 -0.86969 -1.00467 -1.00739 -1.009113.0 -0.83308 -1.00100 -1.00373 -1.00484VTZ 0.5 - -1.10087 - -0.6 - -1.15352 - -0.7 - -1.17101 - -0.8 - -1.17041 - -0.9 - -1.16041 - -1.0 - -1.14576 - -1.5 - -1.06617 - -2.0 - -1.02046 - -2.5 - -1.00467 - -3.0 - -1.00072 - -V5Z 0.5 -1.06492 -1.10397 - -0.6 -1.11638 -1.15581 - -0.7 -1.13278 -1.17297 - -0.8 -1.13089 -1.17223 - -0.9 -1.11927 -1.16216 - -1.0 -1.10258 -1.14748 - -1.5 -1.00493 -1.06795 - -2.0 -0.92624 -1.02186 - -2.5 -0.87140 -1.00550 - -3.0 -0.83420 -1.00124 - -TABLE XIII: Ground-state energies (in Ha) used for the dissociation curves of LiH at the HF, near-FCI(CIPSI+PT2), self-consistent basis-set corrected ADAPT-VQE without the HF basis-set correction (denoted asSC(A+PBE)), and with the HF basis-set correction (denoted as SC(A+PBE)+∆HF). Distances are in Å. VXZstands for the standard cc-pVXZ basis set.Basis set Distance HF FCI SC(A+PBE) SC(A+PBE)+∆HF6-31G 0.5 -7.14721 -7.16953 -7.17775 -7.231490.75 -7.66281 -7.68441 -7.69458 -7.722921.0 -7.87136 -7.89039 -7.90091 -7.917941.25 -7.95144 -7.96925 -7.97975 -7.991821.5 -7.97686 -7.99510 -8.00533 -8.014281.75 -7.97824 -7.99809 -8.00780 -8.014732.0 -7.96887 -7.99121 -8.00025 -8.006172.25 -7.95492 -7.98053 -7.98884 -7.994612.5 -7.93936 -7.96907 -7.97663 -7.982912.75 -7.92365 -7.95844 -7.96522 -7.972533.0 -7.90850 -7.94951 -7.95550 -7.96420V5Z-7 0.5 -7.10996 -7.13045 -7.14080 -7.231790.75 -7.65310 -7.66747 -7.68139 -7.719431.0 -7.87189 -7.88480 -7.89919 -7.915701.25 -7.95443 -7.96789 -7.98184 -7.990921.5 -7.97979 -7.99488 -8.00827 -8.014291.75 -7.98055 -7.99819 -8.01091 -8.015542.0 -7.97042 -7.99152 -8.00347 -8.007842.25 -7.95565 -7.98110 -7.99226 -7.997292.5 -7.93924 -7.97003 -7.98041 -7.986812.75 -7.92265 -7.95988 -7.96953 -7.977833.0 -7.90660 -7.95145 -7.96043 -7.97104VDZ 0.5 - -7.19664 - -0.75 - -7.71273 - -1.0 - -7.91535 - -1.25 - -7.99037 - -1.5 - -8.01264 - -1.75 - -8.01278 - -2.0 - -8.00361 - -2.25 - -7.99108 - -2.5 - -7.97809 - -2.75 - -7.96613 - -3.0 - -7.95594 - -V5Z 0.5 -7.20095 -7.24405 - -0.75 -7.69115 -7.73130 - -1.0 -7.88839 -7.92707 - -1.25 -7.96350 -8.00106 - -1.5 -7.98581 -8.02285 - -1.75 -7.98517 -8.02236 - -2.0 -7.97479 -8.01280 - -2.25 -7.96068 -8.00003 - -2.5 -7.94564 -7.98672 - -2.75 -7.93096 -7.97443 - -3.0 -7.91720 -7.96344 - -TABLE XIV: Ground-state energies (in Ha) used for the dissociation curves of N2 at the HF, near-FCI(CIPSI+PT2), self-consistent basis-set corrected ADAPT-VQE without the HF basis-set correction (denoted asSC(A+PBE)), and with the HF basis-set correction (denoted as SC(A+PBE)+∆HF). Distances are in Å. VXZstands for the standard cc-pVXZ basis set.Basis set Distance HF FCI SC(A+PBE) SC(A+PBE)+∆HFSTO-3G 0.8 -106.68080 -106.76594 -106.99161 -108.828820.9 -107.18719 -107.29271 -107.51568 -109.181751.0977 -107.49589 -107.65251 -107.86975 -109.366621.2 -107.48778 -107.67707 -107.88963 -109.350161.5 -107.27245 -107.58147 -107.78293 -109.222002.0 -106.87150 -107.45512 -107.65901 -109.155672.5 -106.61696 -107.44041 -107.63628 -109.18056V5Z-6 0.8 -107.89375 -107.97171 -108.19082 -108.815080.9 -108.39476 -108.49175 -108.70783 -109.166341.0977 -108.74519 -108.88869 -109.09851 -109.346081.2 -108.76756 -108.93985 -109.14120 -109.321951.5 -108.63207 -108.90483 -109.09718 -109.176622 -108.32055 -108.81060 -109.00808 -109.055682.5 -108.10572 -108.80176 -109.00012 -109.05564VDZ 0.8 - -108.66035 - -0.9 - -109.05746 - -1.1 - -109.27657 - -1.2 - -109.26410 - -1.5 - -109.12459 - -2.0 - -108.98115 - -2.5 - -108.95868 - -VTZ 0.8 - -108.82771 - -0.9 - -109.18511 - -1.1 - -109.36884 - -1.2 - -109.34807 - -1.5 - -109.20218 - -2.0 - -109.04838 - -2.5 - -109.01953 - -V5Z 0.8 -108.51801 - - -0.9 -108.85326 - - -1.0977 -108.99276 - - -1.2 -108.94831 - - -1.5 -108.71151 - - -2.0 -108.36816 - - -2.5 -108.16125 - - -E. Dipole momentsTABLE XV: Dipole moments (in atomic units) of LiH and H2O computed as expectation values of HF andnear-FCI (CIPSI) wave functions, and the near-FCI dipole moments with the HF basis-set correction ∆HF =aug-cc-pV5ZHF − dBHF, where daug-cc-pV5Z HF is the aug-cc-pV5Z HF dipole moment and dB HF is the HF dipole moment inthe basis set B considered. aug-cc-pVXZ values are provided for informational purposes.LiH Nqubits HF FCI FCI + ∆HFSTO-3G 10 -1.91107 -1.81835 -2.26857pcseg-0 14 -2.47768 -2.33313 -2.216746-31G 20 -2.33223 -2.16646 -2.19552V5Z-4 10 -2.57388 -2.37818 -2.16559V5Z-7 16 -2.38299 -2.23095 -2.20925V5Z-10 28 -2.32149 -2.24997 -2.28977cc-pVDZ 36 -2.33994 -2.25566 -2.27701cc-pVTZ 86 -2.35488 -2.29998 -2.30639cc-pVQZ 190 -2.35762 -2.30361 -2.30728cc-pV5Z 290 -2.36017 -2.30647 -2.30759aug-cc-pVDZ 62 -2.37055 -2.32496 -2.3157aug-cc-pVTZ 136 -2.36235 -2.31028 -2.30922aug-cc-pVQZ 250 -2.36152 -2.30792 -2.30769aug-cc-pV5Z 412 -2.36129 -2.30787 -2.30787H2O Nqubits HF FCI FCI + ∆HFSTO-3G 12 -0.67878 -0.63584 -0.73662pcseg-0 24 -1.00341 -0.95822 -0.734376-31G 24 -1.03512 -0.9902 -0.73464V5Z-10 24 -1.02208 -0.99305 -0.75053V5Z-11 30 -1.02389 -0.99185 -0.74752cc-pVDZ 46 -0.80943 -0.76073 -0.73086cc-pVTZ 114 -0.79709 -0.74858 -0.73105cc-pVQZ 228 -0.79006 -0.74409 -0.73359cc-pV5Z 400 -0.78783 -0.74241 -0.73414aug-cc-pVDZ 80 -0.78671 -0.72703a -0.71988aaug-cc-pVTZ 181 -0.78038 -0.72364a -0.72282aaug-cc-pVQZ 342 -0.77955 -0.72695a -0.72696aaug-cc-pV5Z 572 -0.77956 -0.72815a -0.72815aa CCSD(T) values from Ref. [? ].

Claims

1. Claims1. A computer implemented method (100) for computing quantum-enhancedchemical property of a chemical system comprising the following steps: -Providing a Hamiltonian (120) of the chemical system subject of the quantum-enhanced chemical computing, preferably said Hamiltonian being expressed in second-quantised form; -Providing a basis set (130) for the chemical system, preferably a finite basisset; -Preparing an initial quantum state (140), on the quantum computation means(10), to represent the chemical system according to the Hamiltonian and using the provided basis set; -Applying, on the quantum computation means (10), a quantum solver (150) onthe prepared quantum state in the basis set; and -Computing, on classical computation means (40), a density-based basis-setcorrection (160) to modify the density-dependent terms in the Hamiltonian ; and preferably computing at least one quantum-enhanced chemical property (170) of the chemical system.

2. The computer-implemented method (100) according to claim 1, wherein computingthe density-based basis-set correction (160) comprises computing a-posteriori correction based only on energy.

3. The computer-implemented method (100) according to anyone of the previousclaims, wherein computing at least one quantum-enhanced chemical property (170) comprises the addition of the density-based basis-set correction to anapproximation of the full-configuration interaction energy calculated when applying the quantum solver (150) on the prepared quantum state in the basis set.

4. The computer-implemented method (100) according to anyone of the previousclaims, wherein the density-based basis-set correction comprises (i) a correlation energy functional of the electron density and (ii) an energy correction term.

5. The computer-implemented method (100) of claim^2, wherein computing thedensity-based basis-set correction (160) comprises computing a self-consistent correction based on energy and density.

6. The computer-implemented method (100) according to anyone of the previous claims, wherein applying the quantum solver (150) and computing the density- based basis-set correction (160) are repeated iteratively, with an updated Hamiltonian communicated from the classical computation means to the quantum computation means (10) after each iteration.

7. The computer-implemented method (100) according to anyone of the previous claims, wherein expectation values are computed when applying the quantum solver (150) on the prepared quantum state in the basis set; and the computed density-based basis-set correction comprises a functional of the electron density obtained from the expectation values; and the quantum solver being re-executed with an upgraded Hamiltonian with the computed density-based basis-set correction.

8. The computer-implemented method (100) of claim^2, wherein the step of computing the density-based basis-set correction (160) and computing at least one quantum-enhanced chemical property (170) of the chemical system comprises: - computing (161), on the classical computation means (40), a density‑based basis‑set correction; and - combining (162) an approximate expectation value with the density-based basis-set correction to output a quantum-enhanced chemical properties of the chemical system that approaches a complete‑basis‑set value.

9. The computer-implemented method (100) of claim^2, wherein the step of computing a density-based basis-set correction (160) and computing at least one quantum-enhanced chemical property (170) of the chemical system comprises updating the quantum computation means (10) using the computed basis-set correction and re-evaluating the energy and density on the quantum computation means (10).

10. The computer-implemented method (100) of claim^2, wherein the steps of applying (150) the quantum solver and applying the density-based basis-set correction (160) and computing quantum-enhanced chemical property (170) of the chemical system are repeated iteratively; preferably each iteration comprising:- computing a correction potential as the functional derivative of the density- based basis-set correction with respect to the electron density, - adding said electron correction potential to the Hamiltonian, and - repeating the quantum solver with the updated Hamiltonian until convergence of the computed energy is reached.

11. The computer-implemented method (100) according to anyone of the previous claims, wherein computing quantum-enhanced chemical properties of the chemical system comprises the ground-state, dissociation curves, wavefunctions, and / or dipole moment.

12. The computer-implemented method (100) of any one of the preceding claims, wherein the at least one quantum-enhanced chemical property comprises a corrected ground-state electronic energy, the method further including: (a) receiving, from the quantum computation means (10), a numerical ground- state energy obtained for the Hamiltonian in the provided basis set; (b) adding, on the classical computation means (40), a density-based basis-set correction to the a numerical ground-state energy to obtain a corrected ground- state energy; and (c) storing the resulting ground-state energy and wave-function ansatz in computer-readable memory in computer-readable memory.

13. A computer-implemented method (100) of any one of the preceding claims, wherein the at least one quantum-enhanced chemical property comprises a corrected dipole moment, wherein it comprises the following steps: - obtaining, from quantum computing and classical computing means, a corrected wave-function and a one-electron density; - computing, on classical computing means, an expectation value of the dipole moment operator; - computing, on classical computing means, an Hartree-Fock correction to the dipole moment, for example by computing the difference between the Hartree- Fock dipole moment computing on a large basis set and the Hartree-Fock dipole moment computing on the basis-set of interest; and - adding to the correlation corrected dipole moment, the Hartree-Fock correction to the dipole moment.

14. A computer-implemented method to compute a dissociation energy in a molecularsystem, said method comprising the steps of: i) executing the steps of a method (100) according to anyone of previous claimsfor a first nuclear geometry of the molecular system corresponding to an equilibrium bond length; ii) re-executing the steps of the method (100) according to anyone of previousclaims for at least one additional nuclear geometry of the molecular system in which the bonded atoms are separated by a distance sufficient to represent dissociation; and iii) computing, on the classical computation means (40), a numerical differencebetween the corrected ground-state energies generated in steps (i) and (ii).

15. A computer-readable medium with instructions that, when executed by a classicalcomputation means (40) in operative communication with a quantum computation means (10), perform the method (100) of anyone of the claims 1 to 13 and / or a computer-implemented method to compute a dissociation energy in a molecular system according to claim 14.

16. A computer system (1), comprising :- one or more quantum computation means (10); and- one or more classical computation means (40),the computer system (1) being configured to implement a method 100 according to anyone of the claims 1 to 13 and / or a method according to claim 14.

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