Rapid Dissipative Ground State Preparation at Chemical Transition States

TL;DR

This work addresses the challenge of obtaining accurate ground-state energies along reaction paths, where transition-state geometries exhibit strong multi-reference character that strains classical methods—and where standard quantum approaches require a high-overlap guiding state. It proposes a dissipative evolution protocol that transports a warm-start ground state along a discretized reaction path using Procrustes-aligned orbital gauges and engineered cooling primitives, with performance guaranteed under an ETH-inspired rapid-mixing assumption. The key contributions include a polynomial-time complexity bound $\widetilde{O}(C_{DK}^2 N_o/ε_E)$ (or $\widetilde{O}(\|H\|/Δ_{\min}^3 · 1/ε_E · N_o^3)$) for ground-state preparation at transition geometries, along with practical resource estimates and a numerical demonstration on H$_4$ illustrating rapid convergence. The approach offers a principled route to quantum advantage in chemically relevant, strongly correlated regimes by exploiting path smoothness and open-system cooling, with potential integration into hybrid quantum-classical workflows and future fault-tolerant implementations.

Abstract

Simulating chemical reactions is a central challenge in computational chemistry, characterized by an uneven difficulty profile: while equilibrium reactant and product geometries are often classically tractable, intermediate transition states frequently exhibit strong correlation that defies standard approximations. We present a protocol for dissipative ground state preparation that exploits this structure by treating the reaction path itself as a computational primitive. Our protocol uses an approach where a state prepared at a tractable geometry is propagated along a discretized reaction coordinate using Procrustes-aligned orbital rotations and stabilized by engineered dissipative cooling. We show that for reaction paths satisfying a localized Eigenstate Thermalization Hypothesis (ETH) drift condition in the strongly correlated regime, the algorithm prepares ground states of chemical systems with $N_o$ orbitals to an energy error $ε_E$ with a total gate complexity scaling as $\widetilde{O}(N_o^{3}/ε_E)$. We provide logical resource estimates for benchmark systems including FeMoco, Cytochrome P450, and Ru-based carbon capture catalysts.

Figures (5)

• Figure 1: (a) The transition digraph associated with the Lindbladian-induced Markov process $P$ depicting the longest canonical path between an excited state and the ground state; this path length indicates that the Markov process will mix rapidly. The underlying transition graph is a numerical example of H$_4$ molecule at its multi-reference square geometry (see \ref{['fig:h4_pes', 'fig:h4_numerical']}); the transition graph has the same number of nodes as the FCI dimension $D = \binom{2N_o}{N_e} = 70$ for a $\text{CAS}(4,4)$ active space in the STO-3G orbtial basis. The values above each arrow are the transition probabilities from $|E_j\rangle \to |E_k\rangle$, $P_{j \to k}$ along the longest path; in this case the longest path is $|E_{D-1}\rangle \to |E_{0}\rangle$ with length $\ell = 4$ whose transition probabilities are bounded below by $P_{j \to k} \geq p_{\min} = 0.01$. (b) An example of a Ru-based carbon-capture catalyst reaction VIII $\to$ VIII–IX (transition state) $\to$ IX of the multi-step reaction mechanism for CO$_2$ reduction vonBurg2021; the 2D potential energy surface (PES) is representative of a more complex, multi-dimensional mechanism underlying the reaction. (c) The logical quantum circuit for implementing the purely dissipative map $e^{t \mathcal{L}}$ that sends the approximate ground state $\rho_i$ and geometry $\mathbf{R}(s_i)$ to the next ground state along the reaction path $\rho_{i+1}$ where circuit $U_H$ denotes the block encoding of a chemical Hamiltonian which is used to implement the logical circuit described in \ref{['sec:logical_circuit']}.
• Figure 2: (a) Energy error $\Delta E(N_T) = \left|\mathrm{tr}\left(H (e^{\tau \mathcal{L}})^{N_T}[\rho^{(0)}]\right)-E_0\right|$ (a.u.) after $N_T$ application of the dissipative time step at each successive Hamiltonian along a discretized path, beginning from the ground state of $H(0)$, $\rho^{(0)} = |E_0^{(0)}\rangle\langle E_0^{(0)}|$, on the H$_4$ potential energy surface (STO-3G) where we take $\tau=0.01$ in these numerical simulations; the final chemical geometry is the transition state (square H$_4$ molecule) corresponding to $s=0.5$ and the 32nd Hamiltonian along the reaction path. Each marker shows the final energy estimate after $N_T$ dissipative time steps per Hamiltonian along the reaction path, with $N_T\in\{1,5,10,50\}$ and Hamiltonian indices starting at $H(s_1)$. (b) Corresponding infidelity $1-\langle E^{(i)}_0|\rho^{(i)}|E^{(i)}_0\rangle$ on a logarithmic scale for the same runs, showing that larger $N_T$ improves preparation fidelity across the path where $|E_0^{(i)}\rangle$ is the ground state of the $i$-th reaction path Hamiltonian $H(s_i)$ and $\rho^{(i)}$ is the approximate ground state prepared via dissipative evolution.
• Figure 3: H$_4$ potential energy surface (STO-3G) over rectangular geometries parameterized by $(X,Y)$, with the reaction-path segment connecting $s=0$ weakly correlated starting point to the target transition state (TS) point $s=0.5$. The path follows a rectangle distortion of H$_4$ at fixed area $XY=a^2$, with atoms at $(\pm X/2,\pm Y/2,0)$ and aspect ratio swept as $\lambda(s)=\lambda_{\min}^{1-s}\lambda_{\max}^{s}$ with $\lambda_{\min}=0.6$, $\lambda_{\max}=1.6$, and $X=a/\sqrt{\lambda}$, $Y=a\sqrt{\lambda}$ for $s\in[0,1]$; the square geometry with side length $a=1.2$ Å occurs at $s=1/2$ and exhibits strong multi-reference character.
• Figure 4: Quantum logical resource estimates for a single dissipative time step operator $W(\sqrt{\tau}) \approx e^{\tau \mathcal{L}}$ along with the cost of doing QPE across benchmark chemical systems Low_2025. The resource estimates are calculated from resource estimates provided in Low_2025 for QPE and the block encoding $U_H$ using the "DFTHC-BLISS-SA" SOTA electronic structure compression method to minimize $T$/Toffoli gate counts. Each point corresponds to a row in \ref{['tab:block-enc-lindblad-qpe']}, with Toffoli count on the vertical axis and logical qubits on the horizontal axis. Regions of current and projected hardware capabilities are shaded to illustrate which systems and algorithms may be accessible on near- or medium-term fault-tolerant devices camps2025quantumcomputingtechnologyroadmaps.
• Figure 5: Circuit for realizing $A_k(\sqrt{\tau}) = \exp(- i \sqrt{\tau} \sigma_k \otimes A_a/ 2) = \exp(- i \sqrt{\tau} \mathrm{Im}(c_k)\sigma_y \otimes A_a/ 2)$ for $A_a = (X_iX_j+Y_i Y_j)/2$ using the square wave filter function which only has purely imaginary Fourier coefficients for $\ell > 0$, when $k = 0$, $A_0(\sqrt{\tau})=\exp(- i \sqrt{\tau} \sigma_x \otimes A_a/ 4)$, where $i, j$ are corresponding fermionic modes. The circuit costs enough $T$ gates to synthesize $1$$R_z$ gates, 2 acting on the ancilla and the gate decomposition of $i\mathrm{SWAP}(\theta)$ requires only a single $R_z$ leading to a tiny $T$-gate overhead for implementing local jump operators.

Theorems & Definitions (25)

• Definition 1: Reaction path Hamiltonian
• Definition 2: Lipschitz reaction path
• Definition 3: Davis-Kahan constant
• Definition 4: Dissipative Evolution Mixing time
• Theorem 1: Reaction-path ground-state preparation; informal
• Lemma 1: Bounding Successive Ground State Overlap; informal
• Lemma 2: ETH-motivated rapid cooling bound; informal
• Lemma 3: Linear Cooling Time; informal
• Definition 5
• Theorem 2: Time support scaling for Gevrey vs. periodic square-wave filters; informal