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arXiv:2606.16017

Quantum Algorithm for Open-System Battery Cathodes by Modeling Multiple Strongly Coupled Holstein Polarons with Chain-Mapped Caldeira-Leggett Dynamics

Joshua M. Courtney

quant-phcond-mat.mtrl-sciphysics.chem-phSubmitted Jun 14, 2026• Announced Jun 16, 2026• 0 citations
arXivHTMLPDF

Abstract

Cathode lithiation occupies a chemical regime of tightly localized orbitals, narrow bandwidths, and strong electron-lattice coupling. The defining electrochemical observables (open-circuit voltage and differential capacity) are open-system, reservoir-equilibration quantities that closed-Hamiltonian quantum simulation cannot produce, set by exchange with electron, Li$^+$, and phonon baths. We present a fault-tolerant quantum algorithm that recovers them through a unitary chain-mapped Caldeira-Leggett embedding, rendering the baths Trotterizable. The resulting fourth-order Trotter step has a T-gate count polynomial in system size, validating its open-system dynamics against hierarchical equations of motion (HEOM) at strong coupling and the Lindblad limit at weak coupling. For single-carrier olivine LiFePO$_4$, a single voltage anchor on an otherwise DFT-fixed Hamiltonian places the differential-capacity peak within the $\pm5$ mV reproducibility of the experimental plateau. For multi-carrier spinel LiMn$_2$O$_4$, whose $1{:}1$ Mn$^{3+}$/Mn$^{4+}$ filling makes the inter-site Coulomb repulsion dynamically active, the same kernel yields a two-plateau voltage curve with a $125$ mV split, within $17\%$ of the observed $150$ mV. We deliver an end-to-end fault-tolerant resource estimate for such a multi-carrier, three-reservoir observable: $368$ logical qubits and $\sim3\times10^5$ T-gates per step, or $\sim1.7\times10^{12}$ T-gates for a full voltage curve (parallelizable over $\sim10^3$ trajectories), leaving the production-scale dynamical run as a milestone for future hardware. The same kernel reproduces macroscopic quantum coherence, two-band superconductivity, and the Mikheyev-Smirnov-Wolfenstein resonance without modification, placing dynamical battery chemistry and similar Hamiltonians within scope for fault-tolerant quantum simulation.

Comments
27 pages pre-appendix, 8 figures, 25 tables

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Cathode lithiation occupies a chemical regime of tightly localized orbitals, narrow bandwidths, and strong electron-lattice coupling. The defining electrochemical observables (open-circuit voltage and differential capacity) are open-system, reservoir-equilibration quantities that closed-Hamiltonian quantum simulation cannot produce, set by exchange with electron, Li+^++, and phonon baths. We present a fault-tolerant quantum algorithm that recovers them through a unitary chain-mapped Caldeira-Leggett embedding, rendering the baths Trotterizable. The resulting fourth-order Trotter step has a T-gate count polynomial in system size, validating its open-system dynamics against hierarchical equations of motion (HEOM) at strong coupling and the Lindblad limit at weak coupling. For single-carrier olivine LiFePO4_44​, a single voltage anchor on an otherwise DFT-fixed Hamiltonian places the differential-capacity peak within the ±5\pm5±5 mV reproducibility of the experimental plateau. For multi-carrier spinel LiMn2_22​O4_44​, whose 1:11{:}11:1 Mn3+^{3+}3+/Mn4+^{4+}4+ filling makes the inter-site Coulomb repulsion dynamically active, the same kernel yields a two-plateau voltage curve with a 125125125 mV split, within 17%17\%17% of the observed 150150150 mV. We deliver an end-to-end fault-tolerant resource estimate for such a multi-carrier, three-reservoir observable: 368368368 logical qubits and ∼3×105\sim3\times10^5∼3×105 T-gates per step, or ∼1.7×1012\sim1.7\times10^{12}∼1.7×1012 T-gates for a full voltage curve (parallelizable over ∼103\sim10^3∼103 trajectories), leaving the production-scale dynamical run as a milestone for future hardware. The same kernel reproduces macroscopic quantum coherence, two-band superconductivity, and the Mikheyev-Smirnov-Wolfenstein resonance without modification, placing dynamical battery chemistry and similar Hamiltonians within scope for fault-tolerant quantum simulation.
Variational quantum eigensolver (VQE) is a hybrid quantum-classical technique that leverages noisy intermediate scale quantum (NISQ) hardware to obtain the minimum eigenvalue of a model Hamiltonian. VQE has so far been used to simulate condensed matter systems as well as quantum chemistry of small molecules. In this work, we employ VQE methods to obtain the ground-state energy of LiCoO2_22​, a candidate transition metal oxide used for battery cathodes. We simulate Li2_22​Co2_22​O4_44​ and Co2_22​O4_44​ gas-phase models, which represent the lithiated and delithiated states during the discharge and the charge of the Li-ion battery, respectively. Computations are performed using a statevector simulator with a single reference state for three different trial wavefunctions: unitary coupled-cluster singles and doubles (UCCSD), unitary coupled-cluster generalized singles and doubles (UCCGSD) and k-unitary pair coupled-cluster generalized singles and doubles (k-UpCCGSD). The resources in terms of circuit depth, two-qubit entangling gates and wavefunction parameters are analyzed. We find that the k-UpCCGSD with k=5 produces results similar to UCCSD but at a lower cost. Finally, the performance of VQE methods is benchmarked against the classical wavefunction-based methods, such as coupled-cluster singles and doubles (CCSD) and complete active space configuration interaction (CASCI). Our results show that VQE methods quantitatively agree with the results obtained from CCSD. However, the comparison against the CASCI results clearly suggests that advanced trial wavefunctions are likely necessary to capture the multi-reference characteristics as well as the correlations emerging from high-level electronic excitations.
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.
X-ray absorption spectroscopy (XAS) is a leading technique for understanding structural changes in advanced battery materials such as lithium-excess cathodes. However, extracting critical information like oxidation states from the experimental spectra requires expensive and time-consuming simulations. Building upon a recent proposal to simulate XAS using quantum computers, this work proposes a highly-optimized implementation of the time-domain algorithm for X-ray absorption. Among a host of improvements to Hamiltonian representation, circuit implementation, and measurement strategies, three optimizations are key to the efficiency of the algorithm. The first is the use of product formulas with the compressed double factorized form of the Hamiltonian. The second is recognizing that for spectroscopy applications, it is sufficient to control the error in the eigenvalues of the (approximate) Hamiltonian being implemented by the product formula, rather than the generic error on the full time evolution operator. Using perturbation theory to estimate this eigenvalue error, we find that significantly fewer Trotter steps are needed than expected from the time evolution error bound. The third is the choice of an optimized distribution of samples that takes advantage of the exponentially decaying Lorentzian kernel. Through constant factor resource estimates, we show that a challenging model Li4_44​Mn2_22​O cluster system with 18 spatial orbitals and 22 electrons in the active space can be simulated with 100 logical qubits and less than 4×1084 \times 10^84×108 T gates per circuit. Finally, the algorithm is implemented on a simulator, and the reconstructed spectrum is verified against a classical computational reference. The low cost of our algorithm makes it attractive to use on fault-tolerant quantum devices to accelerate the development and commercialization of high-capacity battery cathodes.
This study investigates circuit depth, two-qubit gate count, and error mitigation overhead requirements for VQE simulation of hydrogen chains H4 through H16 on near-term fault-tolerant quantum hardware with surface code logical qubits within the context of quantum computing and quantum chemistry, an area of growing scientific importance given its implications for quantum hardware roadmap planning, fault-tolerant quantum chemistry milestone setting, and near-term VQE experiment design for molecular simulation. Using OpenFermion-Cirq VQE circuit construction, T-gate count estimation via Solovay-Kitaev decomposition, surface code logical qubit overhead computation with code distance d=7,11,15, and zero-noise extrapolation error mitigation cost modeling, we examine VQE trade-off between ansatz expressibility and circuit depth, with T-gate overhead scaling as O(N^4) in UCCSD and surface code cycle count scaling as O(T-count * d^2 / p_phys) determining total runtime on fault-tolerant hardware in 5 hydrogen chain systems (H4, H6, H8, H12, H16), 3 surface code distances, 4 ansatz variants (UCCSD, HEA, k-UpCCGSD, ADAPT-VQE), resource estimates for circuit depths up to 10^6 gates drawn from resource estimation performed on classical simulation using PySCF for classical reference energies and Qiskit/OpenFermion for quantum circuit construction and T-gate counting. Results indicate that H8 UCCSD requires 28 logical qubits and 2.84 million T-gates (8.4 million surface code cycles at d=11), yielding estimated 4.2 hours on a 1,000-logical-qubit machine; ADAPT-VQE reduces T-gate count by 58% at identical accuracy (p < 0.001), with ADAPT-VQE 58% T-gate reduction; H8 = 4.2 hours on 1k-logical-qubit machine as the primary quantitative benchmark. Concordance between primary and confirmatory measurement approaches exceeded 93%, validating the analytical framework. These findings contribute empirically to quantum computing and quantum chemistry and carry actionable implications for the design of programs and policies targeting quantum hardware roadmap planning, fault-tolerant quantum chemistry milestone setting, and near-term VQE experiment design for molecular simulation.
The first-principles quantum simulation of the full electronic Hamiltonian of a material model of realistic size appears daunting even in a future era of fully fault-tolerant quantum computers. In contrast, classical simulations of material excited states rely on heuristic quasiparticle frameworks, such as the GW approximation and Bethe-Salpeter equation, where only a few quasiparticles are explicitly considered. Here we give a detailed description of quasiparticle quantum simulation for multi-excitonic states within the BSE. We develop an explicit first-quantized, orbital-based block encoding of the multi-exciton BSE Hamiltonian for qubitization, and reduce its cost using integral factorization and crystal symmetries. For a materials model of size NNN, and a fixed number of quasiparticles, the resulting algorithms require O~(N)\widetilde{O}(N)O(N) Toffoli gates and O(log⁡N)O(\log N)O(logN) logical qubits, or, using space-time tradeoffs, O~(N)\widetilde{O}(\sqrt{N})O(N​) Toffoli gates and O~(N)\widetilde{O}(\sqrt{N})O(N​) logical qubits, achieving up to a sixth power reduction in Toffoli count, or up to exponential reduction in qubit count, versus standard full Hamiltonian simulation techniques that have been explored in materials simulation. Resource estimates for the classically challenging problem of triexcitonic eigenvalue estimation across a range of semiconductor models of a realistic physical size of O(103)O(10^3)O(103) atoms, show that our Toffoli-optimized implementation reduces the Toffoli count by 106−10710^6-10^7106−107 relative to full Hamiltonian simulation estimates, while a qubit-optimized implementation requires only a few hundred logical qubits. More broadly, these results show that classical heuristic frameworks will be essential in the design of future fault-tolerant quantum algorithms for materials.
We present a comprehensive end-to-end framework for simulating the real-time dynamics of chemical systems on a fault-tolerant quantum computer, incorporating both electronic and nuclear quantum degrees of freedom. An all-particle simulation is nominally efficient on a quantum computer, but practically infeasible. Hence, central to our approach is the construction of a first-quantized plane-wave algorithm making use of pseudoions. The latter consolidate chemically inactive electrons and the nucleus into a single effective dynamical ionic entity, extending the well-established concept of pseudopotentials in quantum chemistry to a two-body interaction. We explicitly describe efficient quantum circuits for initial state preparation across all degrees of freedom, as well as for block-encoding the Hamiltonian describing interacting pseudoions and chemically active electrons, by leveraging recent advances in quantum rejection sampling to optimize the implementations. To extract useful chemical information, we first design molecular fingerprints by combining density-functional calculations with machine learning techniques, and subsequently validate them through surrogate classical molecular dynamics simulations. These fingerprints are then coherently encoded on a quantum computer for efficient molecular identification via amplitude estimation. We provide an extensive analysis of the cost of running the algorithm on a fault-tolerant quantum computer for several chemically interesting systems. As an illustration, simulating the interaction between NH3\mathrm{NH_3}NH3​ and BF3\mathrm{BF_3}BF3​ (a 40-particle system) requires 808 logical qubits to encode the problem, and approximately 101110^{11}1011 Toffoli gates per femtosecond of time evolution. Our results establish a foundation for further quantum algorithm development targeting chemical and material dynamics.