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

Abstract

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.

Figures (12)

• Figure 1: Circuit for the Hadamard test to measure the expectation value $\bra{\psi}e^{-it\hat{H}}\ket{\psi}$. The real part is obtained by setting $W=I$ (identity) and measuring the ancilla in the $X$ basis. The imaginary part is obtained by setting $W=S^{\dag}$ and measuring in the $Y$ basis.
• Figure 2: Demonstration of Pauli coefficient concentration induced by UWC for the P450-Cpd I (E) model. The main panel shows the magnitudes of Pauli-LCU coefficients $\{ |c_\ell |\}$, sorted in descending order, for the original (gray solid line) and UWC-optimized (blue dashed line) Hamiltonian representations. The inset presents a magnified view of the large-weight region.
• Figure 3: Maximum per-shot Pauli-rotation gate count of partially randomized RPE with $\xi=0.01$ as a function of the number of deterministic terms, $L_D$ (up to a maximum of $L$), for the P450-Cpd I (E) model. Results are shown for the original (gray solid line) and UWC-optimized (blue dashed line) Hamiltonian representations. UWC lowers the overall cost and shifts the optimal deterministic partitioning to a smaller $L_D$, as shown in the inset, reflecting the more concentrated coefficient distribution in Fig. \ref{['fig:coeff_distribution']}.
• Figure 4: Normalized maximum per-shot Pauli-rotation gate counts for representative molecular active-space models. Results are shown for partially randomized time evolution (gray bars) and for partially randomized time evolution combined with UWC (blue hatched bars), for the initial-state parameter $\xi=0.01$. The values are normalized to the gate count for deterministic second-order Trotterization.
• Figure 5: Time-to-solution for representative molecular active-space models, utilizing a single QPU. The figure compares three schemes: second-order Trotterization (gray bars), partially randomized time evolution (light-blue cross-hatched bars), and partially randomized time evolution combined with UWC (blue hatched bars), all calculated for an initial-state parameter of $\xi=0.01$. All values are presented in days. The target accuracy $\epsilon$ and physical error rate $p_{\rm ph}$ are assumed to be $\epsilon=1.6$ mHa and $p_{\rm ph}=10^{-3}$, respectively.