Exponential quantum speedups for near-term molecular electronic structure methods

TL;DR

The paper addresses whether near-term quantum circuits can offer provable speedups for molecular electronic-structure problems. It proves that particle-number conserving orbital-rotation circuits augmented with fermionic magic states are universal under post-selection, implying $\mathsf{GapP}$-hard strong simulation and $\mathsf{PH}$-hard weak simulation, and it shows that polynomial-depth generalized UCCSD circuits yield $\mathsf{BQP}$-complete hardness for output probabilities. These hardness results are then applied to quantum non-orthogonal multi-reference methods (TNQE and NOQE), revealing that estimating off-diagonal matrix elements and certain expectation values are intractable for classical algorithms under standard complexity assumptions, while still enabling practical near-term quantum-classical hybrids with linear-depth reference-state circuits. The work also discusses explicit candidate systems with static and dynamic electron correlation (e.g., transition-metal catalysis and Mn-containing complexes) where near-term quantum hardware could achieve meaningful quantum advantage. Overall, the results provide theoretical evidence for super-polynomial quantum speedups in quantum chemistry, particularly for strongly correlated regimes, and guide the design of near-term, low-depth quantum algorithms and reference-state architectures.

Abstract

We prove classical simulation hardness, under the generalized $\mathsf{P}\neq\mathsf{NP}$ conjecture, for quantum circuit families with applications in near-term chemical ground state estimation. The proof exploits a connection to particle number conserving matchgate circuits with fermionic magic state inputs, which are shown to be universal for quantum computation under post-selection, and are therefore not classically simulable in the worst case, in either the strong (multiplicative) or weak (sampling) sense. We apply this result to quantum non-orthogonal multi-reference methods designed for near-term hardware by ruling out certain dequantization strategies for computing the off-diagonal matrix elements between reference states. We demonstrate these quantum speedups for two choices of ansatz that incorporate both static and dynamic correlations to model the electronic eigenstates of molecular systems: linear combinations of orbital-rotated matrix product states, which are preparable in linear depth, and linear combinations of states prepared by generalized UCCSD circuits of polynomial depth, for which computing the expectation values of local fermionic observables up to a constant additive error is $\mathsf{BQP}$-complete. We discuss the implications for achieving practical quantum advantage in resolving the electronic structure of catalytic systems composed from multivalent transition metal atoms using near-term quantum hardware.

Figures (7)

• Figure 1: A logical single-qubit gate is decomposed into $R_y$ and $R_z$ rotations via Euler angles $(\varphi_1,\theta,\varphi_2)$ up to a global phase (top). Each logical rotation is mapped to a PN conserving matchgate in the dual-rail encoding (bottom). $G(\theta)$ denotes a Givens rotation gate (Eq. \ref{['eq:givens_matchgate']}) and $R(\varphi)=e^{-i\varphi/2}R_z(\varphi)$ is the generic phase rotation gate (Eq. \ref{['eq:phase_matchgate']}).
• Figure 2: A logical two-qubit gate is implemented in the dual-rail encoding via Givens rotations and phase rotation gates applied locally on each dual-rail qubit, and controlled-$Z$ gates applied between the dual-rail qubits (see Appendix \ref{['app:dualrail']}).
• Figure 3: A gadget to implement the controlled-$Z$ gate under post-selection using only Givens rotation gates $G(\pi/4)$ and FSWAP gates (linked crossed circles, Eq. \ref{['eq:fswap_gate']}), and a magic state $\ket{M}$ (Eq. \ref{['eq:magic_state']}). The C$Z$ gate is implemented on the top two qubits provided that the bottom four qubits are measured in the $\ket{1010}$ state, which occurs with probability $1/4$.
• Figure 4: Diatomic hydrogen dissociation in the STO-3G basis set (computed using PySCF sun_pyscf_2018). Near the equilibrium geometry ($r_0\approx0.74$Å), the ground state $\ket{\psi}$ is dominated by the RHF determinant, $\ket{x_b}$. The light gray line indicates the Coulson-Fisher point ($r_\text{CF}\approx1.25$Å), where the unrestricted Hartree-Fock solutions undergo spontaneous symmetry breaking, resulting in two degenerate UHF determinants with up and down spins localized around opposite atomic centers. In the shaded region, the ground state $\ket{\psi}$ is dominated by an equal superposition of these two non-orthogonal bonding determinants, meaning it is not qualitatively close to any single Slater determinant.
• Figure 5: A tensor network to compute the overlap matrix element $s_{ij}$ in Eqs. \ref{['eq:ovlp_el']} between matrix product states $\ket{\phi_i}$ and $\ket{\phi_j}$ expressed in different orbital bases. The orbital rotation operator $\hat{G}_{ij}$ has been factorized into a sequence of Givens rotation gates of depth $O(n)$ following Eq. \ref{['eq:givens_factorized']}. $s_{ij}$ can be obtained up to an additive error $\epsilon$ by a linear depth Hadamard test circuit with $O(1/\epsilon^2)$ circuit repetitions leimkuhler2025quantum. The Hamiltonian matrix elements $h_{ij}$ can be resolved up to error $\epsilon$ by $O(n^4)$ circuits of the same form with an inserted Pauli string, corresponding to the unique terms in the JW decomposition of $\hat{H}_i$, requiring $O(\lambda^2/\epsilon^2)$ circuit repetitions.

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 6
• Corollary 7
• Corollary 8
• Theorem 9
• Theorem 10