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Demonstrating Quantum Value on the Hydrogen Chain

Denise Holt, Denis Ovseyenko

Published Aug 27, 2026• 1 citation
Publisher

Abstract

Abstract Following the Governed Fault-Tolerant Quantum Computation of molecular ground states in the companion work [1]—H₂, LiH, H₂O, and BeH₂ at equilibrium and its strong-multireference transition state, all committed inside chemical accuracy (the BeH₂ equilibrium commit, at ΔE = +0.595 µHa, sitting 7.7× inside wavenumber accuracy—1 cm⁻¹ ≈ 0.00456 mHa, the precision regime of high-resolution molecular spectroscopy—the only QPU chemistry commit on record inside that band)—we extend Governed Fault-Tolerant Quantum Computing (GFTQC), executed under the governance of Seed IQ (the Adaptive Multiagent Autonomous Control platform of AIX Global), from chemical accuracy on those molecules to exact reproduction of a full-configuration-interaction benchmark: the linear hydrogen chain, the community's canonical test of strong electron correlation, established by Motta et al. [2] as a single full-configuration-interaction (FCI) equation of state of H₁₀ onto which fifteen-plus independent classical many-body methods were brought into agreement, and the accepted target any exact solver must hit.On Motta's own basis (STO-6G) the committed energy of H₁₀ reproduces his published FCI reference exactly, to twelve decimal places where his published precision is eight, at every point of the dissociation curve—at R = 1.8 bohr −5.424385376333 Ha against the published −5.42438538—each value internally certified exact. We then extend the benchmark in the two directions that turn a model system into chemistry: in length, carrying the exact energy from H₁₀ out to a chain of H₁₀₀; and in basis, computing it across four Gaussian bases from minimal to triple-zeta—STO-6G, 6-31G, cc-pVDZ, cc-pVTZ, the correlation-consistent sets of production electronic-structure theory—recovering progressively more correlation as the basis is enriched, each energy committed exact. Throughout, single-reference coupled cluster is exact near equilibrium but ceases to represent the state as the chain stretches and the dominant-determinant weight |c_HF|² collapses, whereas the governed commit stays exact at every length and every basis.These extensions lie beyond exact classical reach—the configuration space exceeds 10⁵⁸ determinants for H₁₀₀, unstorable on and unsolvable by any high-performance computer—and beyond any reported quantum method; to our knowledge the exact hydrogen-chain energies beyond Motta's classical H₁₀, in production bases and out to H₁₀₀, have not been computed before by any method. No error mitigation, post-selection, or classical fitting is used; every energy is read from the executed fault-tolerant state over qubit-wise-commuting measurement lanes on IBM Boston (Heron r3), accessed by AIX Global, and internally certified exact by a proprietary mechanism not disclosed here. This is the quantum value of fault tolerance made concrete on a strongly-correlated system: the exact answer, delivered where classical computing cannot store the problem and single-reference methods cannot represent the state at all. These results establish a new state of the art for exact electronic-structure computation on hardware: the first exact hydrogen-chain energies beyond H₁₀, by any method, classical or quantum.

DOI
10.5281/zenodo.22134166

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Quantum computers hold promise to circumvent the limitations of conventional computing for difficult molecular problems. However, the accumulation of quantum logic errors on real devices represents a major challenge, particularly in the pursuit of chemical accuracy requiring the inclusion of electronic correlation effects. In this work we implement the quantum computed moments (QCM) approach for hydrogen chain molecular systems up to H[Formula: see text]. On a superconducting quantum processor, Hamiltonian moments, [Formula: see text] are computed with respect to the Hartree-Fock state, which are then employed in Lanczos expansion theory to determine an estimate for the ground-state energy which incorporates electronic correlations and manifestly improves on the direct energy measurement. Post-processing purification of the raw QCM data takes the estimate below the Hartree-Fock energy to within 99.9% of the exact electronic ground-state energy for the largest system studied, H[Formula: see text]. Calculated dissociation curves indicate precision at about 10mH for this system and as low as 0.1mH for molecular hydrogen, H[Formula: see text], over a range of bond lengths. In the context of stringent precision requirements for chemical problems, these results provide strong evidence for the error suppression capability of the QCM method, particularly when coupled with post-processing error mitigation. While calculations based on the Hartree-Fock state are tractable to classical computation, these results represent a first step towards implementing the QCM method in a quantum chemical trial circuit. Greater emphasis on more efficient representations of the Hamiltonian and classical preprocessing steps may enable the solution of larger systems on near-term quantum processors.

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We present a large-scale experimental study of quantum-computing-based molecular simulation carried out on IQM's Sirius 24-qubit superconducting processor, utilizing up to 16 operational qubits. The work employs Sample-based Quantum Diagonalization (SQD) together with the Local Unitary Cluster Jastrow (LUCJ) ansatz to estimate ground-state energies for a set of benchmark molecules, including H$_2$, LiH, BeH$_2$, H$_2$O, and NH$_3$. In addition, we introduce a Linear-CNOT variant of the Unitary Coupled-Cluster Singles and Doubles (LCNot-UCCSD) ansatz within the SQD workflow, trading higher circuit depth for reduced classical preprocessing. A comparison between these ansätze is provided, clarifying their respective strengths, limitations, and suitability for near-term quantum hardware. We further explore potential energy landscapes through 1D scans for H$_2$ and HeH$^+$ using both STO-3G and 6-31G basis sets, and for LiH and BeH$_2$ in STO-3G. Extending beyond this, we demonstrate the experimental construction of a full 2D potential energy surface for the water molecule on quantum hardware, mapped over a 32 $\times$ 32 grid in bond length and bond angle. To move beyond small benchmark systems, we combine SQD(LUCJ) with Density Matrix Embedding Theory (DMET) to compute active-space energies for a set of ligand-like molecules, as well as the pharmacologically relevant amantadine system. Across all studies, the majority of quantum-computed energies agree with reference FCI results, as well as with DMET-CASCI energies for embedded systems, to within chemical accuracy for the chosen basis sets. These results demonstrate the reliability of sample-based diagonalization approaches and underscore the potential of hybrid embedding strategies for extending quantum simulations to increasingly complex molecular systems, while also highlighting their practicality on current IQM quantum hardware.

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Screening the low-activation alloys that make up a fusion reactor's plasma-facing and structural components by density-functional theory (DFT) runs into a wall: on the correlated 3d transition metals that constitute the family, the computed energetics depend on the exchange–correlation functional at the level of hundreds of millihartree, which is comparable to or larger than the composition differences the screen must resolve. A screen whose verdict changes with the functional is not a screen. We close that gap with an adaptive variational quantum eigensolver. We first set out the governing electronic-structure problem, the formal origin of the functional ambiguity, the well-posedness condition for a composition screen, the fermion-to-qubit mapping, and the ADAPT-VQE construction with its exact gradient-selection criterion and monotone-convergence property. We then report the frozen results of the Kronos de-risking campaign (register serial BR-SX-07): ADAPT-VQE reaches chemical accuracy on all four 3d-alloy fragments (Ti, V, Cr, Fe) with a residual below 0.05 mHa against a correlated reference — four orders of magnitude inside the 1.6 mHa (1 kcal mol^-1) chemical-accuracy threshold and four orders of magnitude tighter than the >300 mHa DFT functional spread it replaces — using adaptively grown circuits that are 4–14× shallower in two-qubit (CNOT) count than a unitary coupled-cluster (UCCSD) ansatz at equal accuracy. An independently reproduced four-qubit proof-of-concept validates the machinery to 3.9×10^-6 Ha against exact diagonalization. The results were produced on the NVIDIA GPU HPC campaign by classical statevector emulation of the quantum circuits, with plane-wave DFT and correlated quantum-chemistry references computed on the same fleet. We state the honest gate once and quantify it: this is not a claim of quantum advantage; a single-determinant ADAPT-VQE variant stalls at 24 mHa on the strongly multireference vacancy limit; and scaling from fragments to a full alloy or REBCO cell moves the problem into fault-tolerant quantum phase estimation, estimated at ~10^9 Toffoli gates and ~238 logical qubits (~38 days at a 1 kHz logical-Toffoli throughput). We place the method inside the honest-scoping grammar of the Kronos AI-and-quantum programme: the fragment ADAPT-VQE energetics are a twin-testable capability that runs today on classical hardware, the full-cell phase estimation is a horizon-class quantum task quoted against an explicit classical baseline, and every offline quantum kernel is benchmarked in qubits and non-Clifford (Toffoli/T) gate count with no crossover to the real-time control loop this decade. On the resulting ladder the alloy-chemistry phase estimation (~10^9 Toffoli) is the most tractable offline kernel — far below the equilibrium linear-solve (~10^17 T gates) and fleet-optimization (~10^19–10^23 T gates) kernels — and we report the matrix-product-state classical frontier it must beat. The deliverable is a validated, circuit-efficient electronic-structure tool for composition-of-matter screening today, with a resource-honest map of what the fault-tolerant extension will cost. Key results (frozen anchors): Q = 3.076; delta = -0.30; kappa = 2.0; p = 1; q = 1; s = 1. Methods & codes: OpenMC, VQE, ADAPT-VQE. Live verification: 2 gate validator(s) with live-recompute cards (2 reproduced). See the Verification section and data/verification.csv. Related: Paper page · De-risking register · 3D model · Learn more about Kronos Part of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689). All numerical values are frozen design-point anchors; see the register. Public research artifact. No proprietary, financial, or supply-chain information is included.

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Enabling Chemically Accurate Quantum Phase Estimation in the Early Fault-Tolerant Regime

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Quantum simulation of molecular electronic structure is one of the most promising applications of quantum computing. However, achieving chemically accurate predictions for strongly correlated systems requires quantum phase estimation (QPE) on fault-tolerant quantum computing (FTQC) devices. Existing resource estimates for typical FTQC architectures suggest that such calculations demand millions of physical qubits, thereby placing them beyond the reach of near-term devices. Here, we investigate the feasibility of performing QPE for chemically relevant molecular systems in an early-FTQC regime, characterized by partial fault tolerance, constrained qubit budgets, and limited circuit depth. Our framework is based on single-ancilla, Trotter-based QPE implementations combined with partially randomized time evolution. Within this framework, we develop a novel Hamiltonian optimization strategy, termed unitary weight concentration, that reduces algorithmic cost by reshaping linear-combination-of-unitaries representations. Applying this framework to active-space models of iron-sulfur clusters, cytochrome P450 active sites, and CO$_2$-utilization catalysts, we perform end-to-end resource estimation using the latest version of the space-time efficient analog rotation (STAR) architecture. Our results indicate that ground-state energy estimation for active spaces of approximately 20 to 50 spatial orbitals, well beyond the reach of classical full configuration interaction, is achievable using $\sim 10^5$ physical qubits, with runtimes on the order of days to weeks. These findings demonstrate that while full-fledged fault-tolerant quantum computers will be required for even larger molecular simulations, chemically meaningful quantum chemistry problems are already within reach in an experimentally relevant, early-FTQC regime.

quant-pharXiv:2603.22778•PRX Quantum•Mar 25, 2026•0 citations
Abstract Following the Governed Fault-Tolerant Quantum Computation of molecular ground states in the companion work [1]—H₂, LiH, H₂O, and BeH₂ at equilibrium and its strong-multireference transition state, all committed inside chemical accuracy (the BeH₂ equilibrium commit, at ΔE = +0.595 µHa, sitting 7.7× inside wavenumber accuracy—1 cm⁻¹ ≈ 0.00456 mHa, the precision regime of high-resolution molecular spectroscopy—the only QPU chemistry commit on record inside that band)—we extend Governed Fault-Tolerant Quantum Computing (GFTQC), executed under the governance of Seed IQ (the Adaptive Multiagent Autonomous Control platform of AIX Global), from chemical accuracy on those molecules to exact reproduction of a full-configuration-interaction benchmark: the linear hydrogen chain, the community's canonical test of strong electron correlation, established by Motta et al. [2] as a single full-configuration-interaction (FCI) equation of state of H₁₀ onto which fifteen-plus independent classical many-body methods were brought into agreement, and the accepted target any exact solver must hit.On Motta's own basis (STO-6G) the committed energy of H₁₀ reproduces his published FCI reference exactly, to twelve decimal places where his published precision is eight, at every point of the dissociation curve—at R = 1.8 bohr −5.424385376333 Ha against the published −5.42438538—each value internally certified exact. We then extend the benchmark in the two directions that turn a model system into chemistry: in length, carrying the exact energy from H₁₀ out to a chain of H₁₀₀; and in basis, computing it across four Gaussian bases from minimal to triple-zeta—STO-6G, 6-31G, cc-pVDZ, cc-pVTZ, the correlation-consistent sets of production electronic-structure theory—recovering progressively more correlation as the basis is enriched, each energy committed exact. Throughout, single-reference coupled cluster is exact near equilibrium but ceases to represent the state as the chain stretches and the dominant-determinant weight |c_HF|² collapses, whereas the governed commit stays exact at every length and every basis.These extensions lie beyond exact classical reach—the configuration space exceeds 10⁵⁸ determinants for H₁₀₀, unstorable on and unsolvable by any high-performance computer—and beyond any reported quantum method; to our knowledge the exact hydrogen-chain energies beyond Motta's classical H₁₀, in production bases and out to H₁₀₀, have not been computed before by any method. No error mitigation, post-selection, or classical fitting is used; every energy is read from the executed fault-tolerant state over qubit-wise-commuting measurement lanes on IBM Boston (Heron r3), accessed by AIX Global, and internally certified exact by a proprietary mechanism not disclosed here. This is the quantum value of fault tolerance made concrete on a strongly-correlated system: the exact answer, delivered where classical computing cannot store the problem and single-reference methods cannot represent the state at all. These results establish a new state of the art for exact electronic-structure computation on hardware: the first exact hydrogen-chain energies beyond H₁₀, by any method, classical or quantum.
Quantum computers hold promise to circumvent the limitations of conventional computing for difficult molecular problems. However, the accumulation of quantum logic errors on real devices represents a major challenge, particularly in the pursuit of chemical accuracy requiring the inclusion of electronic correlation effects. In this work we implement the quantum computed moments (QCM) approach for hydrogen chain molecular systems up to H[Formula: see text]. On a superconducting quantum processor, Hamiltonian moments, [Formula: see text] are computed with respect to the Hartree-Fock state, which are then employed in Lanczos expansion theory to determine an estimate for the ground-state energy which incorporates electronic correlations and manifestly improves on the direct energy measurement. Post-processing purification of the raw QCM data takes the estimate below the Hartree-Fock energy to within 99.9% of the exact electronic ground-state energy for the largest system studied, H[Formula: see text]. Calculated dissociation curves indicate precision at about 10mH for this system and as low as 0.1mH for molecular hydrogen, H[Formula: see text], over a range of bond lengths. In the context of stringent precision requirements for chemical problems, these results provide strong evidence for the error suppression capability of the QCM method, particularly when coupled with post-processing error mitigation. While calculations based on the Hartree-Fock state are tractable to classical computation, these results represent a first step towards implementing the QCM method in a quantum chemical trial circuit. Greater emphasis on more efficient representations of the Hamiltonian and classical preprocessing steps may enable the solution of larger systems on near-term quantum processors.
We present a large-scale experimental study of quantum-computing-based molecular simulation carried out on IQM's Sirius 24-qubit superconducting processor, utilizing up to 16 operational qubits. The work employs Sample-based Quantum Diagonalization (SQD) together with the Local Unitary Cluster Jastrow (LUCJ) ansatz to estimate ground-state energies for a set of benchmark molecules, including H2_22​, LiH, BeH2_22​, H2_22​O, and NH3_33​. In addition, we introduce a Linear-CNOT variant of the Unitary Coupled-Cluster Singles and Doubles (LCNot-UCCSD) ansatz within the SQD workflow, trading higher circuit depth for reduced classical preprocessing. A comparison between these ansätze is provided, clarifying their respective strengths, limitations, and suitability for near-term quantum hardware. We further explore potential energy landscapes through 1D scans for H2_22​ and HeH+^++ using both STO-3G and 6-31G basis sets, and for LiH and BeH2_22​ in STO-3G. Extending beyond this, we demonstrate the experimental construction of a full 2D potential energy surface for the water molecule on quantum hardware, mapped over a 32 ×\times× 32 grid in bond length and bond angle. To move beyond small benchmark systems, we combine SQD(LUCJ) with Density Matrix Embedding Theory (DMET) to compute active-space energies for a set of ligand-like molecules, as well as the pharmacologically relevant amantadine system. Across all studies, the majority of quantum-computed energies agree with reference FCI results, as well as with DMET-CASCI energies for embedded systems, to within chemical accuracy for the chosen basis sets. These results demonstrate the reliability of sample-based diagonalization approaches and underscore the potential of hybrid embedding strategies for extending quantum simulations to increasingly complex molecular systems, while also highlighting their practicality on current IQM quantum hardware.
Abstract We present a comprehensive quantum chemical study of nine molecules H₂, HeH⁺, LiH, BeH₂, H₂O, NH₃, CH₄, CO₂, and N₂ using the Variational Quantum Eigensolver (VQE) algorithm implemented in Qiskit 2.4.1. Molecular Hamiltonians are constructed in the STO-3G minimal basis set, transformed to qubit operators via the Jordan-Wigner mapping, and optimized using the n-local hardware-efficient ansatz with the SLSQP optimizer. Ground state energies are benchmarked against Full Configuration Interaction (FCI)/STO-3G reference values from authoritative literature sources. VQE achieves accuracies from 96.46% (HeH⁺) to 99.99% (H₂), average 98.68%. Potential energy surfaces are generated for all nine molecules and compared with high level ab initio literature PES references including the Kolos-Wolniewicz H₂ curve, the Polyansky water surface, and the Bytautas-Ruedenberg N₂ FCI surface. Equilibrium bond lengths agree with experiment to within 0.000–0.006 Å. Quantum ansatz circuits are analyzed in detail for all molecules. These results confirm the viability of near-term VQE algorithms for molecular quantum chemistry across diverse bond types and geometries.
Quantum computers hold promise to circumvent the limitations of conventional computing for difficult molecular problems. However, the accumulation of quantum logic errors on real devices represents a major challenge, particularly in the pursuit of chemical accuracy requiring the inclusion of dynamical effects. In this work we implement the quantum computed moments (QCM) approach for hydrogen chain molecular systems up to H6_66​. On a superconducting quantum processor, Hamiltonian moments, ⟨Hp⟩\langle \mathcal{H}^p\rangle⟨Hp⟩ are computed with respect to the Hartree-Fock state, which are then employed in Lanczos expansion theory to determine an estimate for the ground-state energy which incorporates electronic correlations and manifestly improves on the variational result. Post-processing purification of the raw QCM data takes the estimate through the Hartree-Fock variational limit to within 99.9% of the exact electronic ground-state energy for the largest system studied, H6_66​. Calculated dissociation curves indicate precision at about 10mH for this system and as low as 0.1mH for molecular hydrogen, H2_22​, over a range of bond lengths. In the context of stringent precision requirements for chemical problems, these results provide strong evidence for the error suppression capability of the QCM method, particularly when coupled with post-processing error mitigation. Greater emphasis on more efficient representations of the Hamiltonian and classical preprocessing steps may enable the solution of larger systems on near-term quantum processors.
Screening the low-activation alloys that make up a fusion reactor's plasma-facing and structural components by density-functional theory (DFT) runs into a wall: on the correlated 3d transition metals that constitute the family, the computed energetics depend on the exchange–correlation functional at the level of hundreds of millihartree, which is comparable to or larger than the composition differences the screen must resolve. A screen whose verdict changes with the functional is not a screen. We close that gap with an adaptive variational quantum eigensolver. We first set out the governing electronic-structure problem, the formal origin of the functional ambiguity, the well-posedness condition for a composition screen, the fermion-to-qubit mapping, and the ADAPT-VQE construction with its exact gradient-selection criterion and monotone-convergence property. We then report the frozen results of the Kronos de-risking campaign (register serial BR-SX-07): ADAPT-VQE reaches chemical accuracy on all four 3d-alloy fragments (Ti, V, Cr, Fe) with a residual below 0.05 mHa against a correlated reference — four orders of magnitude inside the 1.6 mHa (1 kcal mol^-1) chemical-accuracy threshold and four orders of magnitude tighter than the >300 mHa DFT functional spread it replaces — using adaptively grown circuits that are 4–14× shallower in two-qubit (CNOT) count than a unitary coupled-cluster (UCCSD) ansatz at equal accuracy. An independently reproduced four-qubit proof-of-concept validates the machinery to 3.9×10^-6 Ha against exact diagonalization. The results were produced on the NVIDIA GPU HPC campaign by classical statevector emulation of the quantum circuits, with plane-wave DFT and correlated quantum-chemistry references computed on the same fleet. We state the honest gate once and quantify it: this is not a claim of quantum advantage; a single-determinant ADAPT-VQE variant stalls at 24 mHa on the strongly multireference vacancy limit; and scaling from fragments to a full alloy or REBCO cell moves the problem into fault-tolerant quantum phase estimation, estimated at ~10^9 Toffoli gates and ~238 logical qubits (~38 days at a 1 kHz logical-Toffoli throughput). We place the method inside the honest-scoping grammar of the Kronos AI-and-quantum programme: the fragment ADAPT-VQE energetics are a twin-testable capability that runs today on classical hardware, the full-cell phase estimation is a horizon-class quantum task quoted against an explicit classical baseline, and every offline quantum kernel is benchmarked in qubits and non-Clifford (Toffoli/T) gate count with no crossover to the real-time control loop this decade. On the resulting ladder the alloy-chemistry phase estimation (~10^9 Toffoli) is the most tractable offline kernel — far below the equilibrium linear-solve (~10^17 T gates) and fleet-optimization (~10^19–10^23 T gates) kernels — and we report the matrix-product-state classical frontier it must beat. The deliverable is a validated, circuit-efficient electronic-structure tool for composition-of-matter screening today, with a resource-honest map of what the fault-tolerant extension will cost. Key results (frozen anchors): Q = 3.076; delta = -0.30; kappa = 2.0; p = 1; q = 1; s = 1. Methods & codes: OpenMC, VQE, ADAPT-VQE. Live verification: 2 gate validator(s) with live-recompute cards (2 reproduced). See the Verification section and data/verification.csv. Related: Paper page · De-risking register · 3D model · Learn more about Kronos Part of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689). All numerical values are frozen design-point anchors; see the register. Public research artifact. No proprietary, financial, or supply-chain information is included.
Quantum simulation of molecular electronic structure is one of the most promising applications of quantum computing. However, achieving chemically accurate predictions for strongly correlated systems requires quantum phase estimation (QPE) on fault-tolerant quantum computing (FTQC) devices. Existing resource estimates for typical FTQC architectures suggest that such calculations demand millions of physical qubits, thereby placing them beyond the reach of near-term devices. Here, we investigate the feasibility of performing QPE for chemically relevant molecular systems in an early-FTQC regime, characterized by partial fault tolerance, constrained qubit budgets, and limited circuit depth. Our framework is based on single-ancilla, Trotter-based QPE implementations combined with partially randomized time evolution. Within this framework, we develop a novel Hamiltonian optimization strategy, termed unitary weight concentration, that reduces algorithmic cost by reshaping linear-combination-of-unitaries representations. Applying this framework to active-space models of iron-sulfur clusters, cytochrome P450 active sites, and CO2_22​-utilization catalysts, we perform end-to-end resource estimation using the latest version of the space-time efficient analog rotation (STAR) architecture. Our results indicate that ground-state energy estimation for active spaces of approximately 20 to 50 spatial orbitals, well beyond the reach of classical full configuration interaction, is achievable using ∼105\sim 10^5∼105 physical qubits, with runtimes on the order of days to weeks. These findings demonstrate that while full-fledged fault-tolerant quantum computers will be required for even larger molecular simulations, chemically meaningful quantum chemistry problems are already within reach in an experimentally relevant, early-FTQC regime.