The cost of quantum algorithms for biochemistry: A case study in metaphosphate hydrolysis

TL;DR

The paper addresses the question of how much quantum resource is required to simulate a biologically important reaction—metaphosphate hydrolysis within ATP hydrolysis—across three leading quantum algorithms: VQE/ADAPT-VQE, quantum Krylov, and quantum phase estimation. It combines DUCC downfolding to create a tractable active-space Hamiltonian, hardware-aware compilation, and classical simulations to produce end-to-end resource estimates for NISQ-to-FASQ regimes, complemented by a complete dataset of Hamiltonians and code. The key finding is that variational approaches (especially ADAPT-VQE) demand significantly fewer quantum resources and are potentially feasible on near-term devices, whereas Krylov and QPE require enormous two-qubit gate counts and deeper circuits, challenging near-term hardware. The work thus provides actionable guidance for algorithmic and hardware development in quantum biochemistry and offers a valuable benchmark dataset for future improvements.

Abstract

We evaluate the quantum resource requirements for ATP/metaphosphate hydrolysis, one of the most important reactions in all of biology with implications for metabolism, cellular signaling, and cancer therapeutics. In particular, we consider three algorithms for solving the ground state energy estimation problem: the variational quantum eigensolver, quantum Krylov, and quantum phase estimation. By utilizing exact classical simulation, numerical estimation, and analytical bounds, we provide a current and future outlook for using quantum computers to solve impactful biochemical and biological problems. Our results show that variational methods, while being the most heuristic, still require substantially fewer overall resources on quantum hardware, and could feasibly address such problems on current or near-future devices. We include our complete dataset of biomolecular Hamiltonians and code as benchmarks to improve upon with future techniques.

Figures (7)

• Figure 1: Energy of the metaphosphate hydrolysis reaction as a function of its reaction coordinate according to several different classical electronic structure theories. These theories differ most at the transition state, which is an indication of the quantum mechanical difficulty of the problem.
• Figure 2: Number of two-qubit gates required to reach chemical accuracy for a Hydrogen chain with $n_\text{H} = n_Q / 2$ atoms. ADAPT gate counts are exact and Krylov/QPE gate counts are upper bounds.
• Figure 3: Number of CNOTs required for one Trotter step with three methods, the "naive" strategy directly exponentiating each Pauli string in the Hamiltonian, the "grouped" strategy where terms are first sorted into commuting sets and then exponentiated, and the Paulihedral strategy li2022paulihedral. All strategies consider a gateset of $\{U, \text{CNOT}\}$ where $U$ is an arbitrary single-qubit gate and CNOT can be performed between any pair of qubits. The Hamiltonian here is an $n \times n$ spinless Fermi-Hubbard model (which acts on $n^2$ qubits). While the naive and grouped strategies are upper bounds, we see the Paulihedral method produces circuits with significantly fewer two-qubit gates as the problem is scaled. For this reason we use the Paulihedral method when evaluating overall resources for quantum Krylov and quantum phase estimation (which require time evolution as a subroutine).
• Figure 4: Accuracy of quantum Krylov applied to a $n = 6$ qubit linear chain of Hydrogen atoms as a function of the subspace dimension $d$ and the number of Trotter steps. For reference, quantum Krylov with exact time evolution (infinite Trotter steps) is shown. The initial state $|\phi\rangle$ is chosen such that the overlap with the true ground state $|\psi\rangle$ is $|\langle \phi | \psi \rangle|^2 = 0.85$.
• Figure 5: Estimation of the required subspace dimension $d$ and number of Trotter steps $n_T$ for quantum Krylov. Points show values computed by exact simulation to reach chemical accuracy on hydrogen chains of $n_H = n_Q / 2$ atoms. We fit a line to these points to extrapolate to the metaphosphate system size (vertical dashed line), and take these as estimated lower bounds for the more complicated metaphosphate systems.