Enhancing Initial State Overlap through Orbital Optimization for Faster Molecular Electronic Ground-State Energy Estimation
Pauline J. Ollitrault, Cristian L. Cortes, Jérôme F. Gonthier, Robert M. Parrish, Dario Rocca, Gian-Luca Anselmetti, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, Michael Streif
The phase estimation algorithm is crucial for computing the ground-state energy of a molecular electronic Hamiltonian on a quantum computer. Its efficiency depends on the overlap between the Hamiltonian's ground state and an initial state, which tends to decay exponentially with system size. We showcase a practical orbital optimization scheme to alleviate this issue. Applying our method to four iron-sulfur molecules, we achieve a notable enhancement, up to 2 orders of magnitude, compared to localized orbitals. Furthermore, our approach yields improved overlaps in cytochrome P450 enzyme models.
The phase estimation algorithm is crucial for computing the ground-state energy of a molecular electronic Hamiltonian on a quantum computer. Its efficiency depends on the overlap between the Hamiltonian's ground state and an initial state, which tends to decay exponentially with system size. We showcase a practical orbital optimization scheme to alleviate this issue. Applying our method to four iron-sulfur molecules, we achieve a notable enhancement, up to 2 orders of magnitude, compared to localized orbitals. Furthermore, our approach yields improved overlaps in cytochrome P450 enzyme models.
Quantum computation offers significant potential for accelerating the simulation of molecules and materials through algorithms such as quantum phase estimation (QPE). However, the expected speedup in ground-state energy estimation depends critically on the ability to efficiently prepare an initial state with high overlap with the true ground state. For strongly correlated molecules such as iron-sulfur clusters, this overlap is demonstrated to decay exponentially with system size. To alleviate this problem, we introduce an efficient classical algorithm to find entanglement-minimized orbitals (EMOs) and their orderings using spin-adapted low-bond-dimension matrix product states (MPS). The EMO basis yields a more compact ground-state representation, significantly easing initial state preparation for challenging systems. Our algorithm improves initial state overlap by nearly an order of magnitude over prior orbital optimization approaches for an iron-sulfur cluster with four irons, and is scalable to larger systems with many unpaired electrons, including the P-cluster and FeMo-cofactor in nitrogenase with eight transition metal centers. For these systems, we achieve substantial enhancements on initial state overlap by factors of O(10^{2}) and O(10^{5}), respectively, compared to results obtained using localized orbitals. Our results show that initial state preparation for these challenging systems requires far fewer resources than prior estimates suggested.
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-utilization catalysts, we perform end-to-end resource estimation using the latest version of the space-time efficient analog rotation (STAR) architecture. Our results indicate that ground-state energy estimation for active spaces of approximately 20 to 50 spatial orbitals, well beyond the reach of classical full configuration interaction, is achievable using ∼105 physical qubits, with runtimes on the order of days to weeks. These findings demonstrate that while full-fledged fault-tolerant quantum computers will be required for even larger molecular simulations, chemically meaningful quantum chemistry problems are already within reach in an experimentally relevant, early-FTQC regime.
Quantum computers can accurately compute ground state energies using phase estimation, but this requires a guiding state that has significant overlap with the true ground state. For large molecules and extended materials, it becomes difficult to find guiding states with good ground state overlap for growing molecule sizes. Additionally, the required number of qubits and quantum gates may become prohibitively large. One approach for dealing with these challenges is to use a quantum embedding method, which allows a reduction to one or multiple smaller quantum cores embedded in a larger quantum region. In such situations, it is unclear how the embedding method affects the hardness of constructing good guiding states. In this work, we therefore investigate the preparation of guiding states in the context of quantum embedding methods. We extend previous work on quantum impurity problems, a framework in which we can rigorously analyze the embedding of a subset of orbitals. While there exist results for optimal active orbital space selection in terms of energy minimization, we rigorously demonstrate how the same principles can be used to define selected orbital spaces for state preparation in terms of the overlap with the ground state. Moreover, we perform numerical studies of molecular systems relevant to biochemistry, one field in which quantum embedding methods are required due to the large size of biomacromolecules such as proteins and nucleic acids. We investigate two different embedding strategies which can exhibit qualitatively different orbital entanglement. In all cases, we demonstrate that the easy-to-obtain mean-field state will have a sufficiently high overlap with the target state to perform quantum phase estimation.
Quantum computation for chemical problems will require the construction of guiding states with sufficient overlap with a target state. Since easily available and initializable mean-field states are characterized by an overlap that is reduced for multi-configurational electronic structures and even vanishes with growing system size, we here investigate the severity of state preparation for reaction chemistry. We emphasize weaknesses in current traditional approaches (even for weakly correlated molecules) and highlight the advantage of quantum phase estimation algorithms. An important result is the introduction of a new classification scheme for electronic structures based on orbital entanglement information. We identify two categories of multi-configurational molecules. Whereas class-1 molecules are dominated by very few determinants and often found in reaction chemistry, class-2 molecules do not allow one to single out a reasonably sized number of important determinants. The latter are particularly hard for traditional approaches and an ultimate target for quantum computation. Some open-shell iron-sulfur clusters belong to class 2. We discuss the role of the molecular orbital basis set and show that true class-2 molecules remain in this class independent of the choice of the orbital basis, with the iron-molybdenum cofactor of nitrogenase being a prototypical example. We stress that class-2 molecules can be build in a systematic fashion from open-shell centers or unsaturated carbon atoms. Our key result is that it will always be possible to initialize a guiding state for chemical reaction chemistry in the ground state based on initial low-cost approximate electronic structure information, which is facilitated by the finite size of the atomistic structures to be considered.
We develop a phase-estimation method with a distinct feature: its maximal run time (which determines the circuit depth) is δ/ , where is the target precision, and the preconstant δ can be arbitrarily close to 0 as the initial state approaches the target eigenstate.The total cost of the algorithm satisfies the Heisenberg-limited scaling O( -1 ).As a result, our algorithm may significantly reduce the circuit depth for performing phase-estimation tasks on early fault-tolerant quantum computers.The key technique is a simple subroutine called quantum complex exponential least squares (QCELS).Our algorithm can be readily applied to reduce the circuit depth for estimating the ground-state energy of a quantum Hamiltonian, when the overlap between the initial state and the ground state is large.If this initial overlap is small, we can combine our method with the Fourier-filtering method developed in [Lin and Tong, PRX Quantum 3, 010318, 2022], and the resulting algorithm provably reduces the circuit depth in the presence of a large relative overlap compared to .The relative-overlap condition is similar to a spectral-gap assumption but it is aware of the information in the initial state and is therefore applicable to certain Hamiltonians with small spectral gaps.We observe that the circuit depth can be reduced by around 2 orders of magnitude in numerical experiments under various settings.
High Resolution Image Download MS PowerPoint Slide Quantum phase estimation (QPE) is a promising quantum algorithm for obtaining molecular ground-state energies with chemical accuracy. However, its computational cost, dominated by the Hamiltonian 1-norm λ and the cost of the block encoding, scales at least quadratically with the number of molecular orbitals, making it challenging to incorporate dynamic correlation by enlarging the active space. In this work, we investigate two strategies to mitigate this cost through the optimization of the basis set. First, we investigate whether adjusting the coefficients of Gaussian basis functions can minimize the 1-norm while preserving the accuracy of the ground state energy. Although this method leads to a reduction in the 1-norm up to 10%, this reduction is system-dependent and diminishes with increasing molecular size. Second, we demonstrate that employing a large-basis-set frozen natural orbital (FNO) strategy results in a substantial reduction in QPE resources without compromising accuracy. We study a data set of 58 small organic molecules and the dissociation curve of N 2, and demonstrate that an active space constructed from orbitals derived from larger basis sets captures correlation effects more effectively. This approach yields up to an 80% reduction in the 1-norm λ and also leads to a 55% reduction in the number of orbitals. Our results highlight that improving the quality, not just the size, of the orbital basis is a viable strategy for extending QPE to include dynamical correlation, making progress toward scalable and chemically accurate quantum simulations with tractable resource requirements.