Quantum algorithms for electronic structure calculations: particle/hole Hamiltonian and optimized wavefunction expansions

TL;DR

The paper develops a particle-hole reformulation of the electronic-structure Hamiltonian to improve the efficiency of quantum-chemistry simulations on quantum hardware. It compares two trial-wavefunction families within VQE—a unitary CC (q-UCC) and a heuristic, hardware-efficient Ansatz—using both exact and approximate (Trotter) implementations, and demonstrates substantial resource savings and robust accuracy for H2 and H2O. Key contributions include showing that a single Trotter step can achieve chemical accuracy when variational optimization is applied, and that exchange-type, particle-conserving gates substantially reduce circuit depth for the heuristic approach. The findings provide practical guidance for near-term quantum devices, showing that p/h-based VQE with active spaces and ECPs can achieve reliable electronic-structure energies with significantly reduced gate counts and circuit depths.

Abstract

In this work we investigate methods to improve the efficiency and scalability of quantum algorithms for quantum chemistry applications. We propose a transformation of the electronic structure Hamiltonian in the second quantization framework into the particle-hole (p/h) picture, which offers a better starting point for the expansion of the trial wavefunction. The state of the molecular system at study is parametrized in a way to efficiently explore the sector of the molecular Fock space that contains the desired solution. To this end, we explore several trial wavefunctions to identify the most efficient parameterization of the molecular ground state. Taking advantage of known post-Hartree Fock quantum chemistry approaches and heuristic Hilbert space search quantum algorithms, we propose a new family of quantum circuits based on exchange-type gates that enable accurate calculations while keeping the gate count (i.e., the circuit depth) low. The particle-hole implementation of the Unitary Coupled Cluster (UCC) method within the Variational Quantum Eigensolver approach gives rise to an efficient quantum algorithm, named q-UCC , with important advantages compared to the straightforward 'translation' of the classical Coupled Cluster counterpart. In particular, we show how a single Trotter step can accurately and efficiently reproduce the ground state energies of simple molecular systems.

Figures (9)

• Figure 1: Circuits for the exponentiation of the single (a) and double (b) excitation operators $(\hat{a}_{p}^{\dag} \hat{a}_r - h.c.)$ and $(\hat{a}_p^{\dag} \hat{a}_q^{\dag} \hat{a}_r \hat{a}_s - h.c.)$, which contribute to $\hat{T}_{1}$ and $\hat{T}_{2}$, respectively. The $p$, $q$ indices refer to virtual and $r$,$s$ to occupied orbitals. The generic state $| . \rangle$ corresponds to $| 1 \rangle$ in case it is part of the occupied manifold and $| 0 \rangle$ otherwise. The repeated units across several qubits are shown in dashed lines. The definition of the gates that span more than two qubits (dashed lines) is given in Appendix \ref{['App:boxes']}.
• Figure 2: Definition of the three entangler blocks: (a) ${U}^{(1)}_{\rm ent}$, (b) ${U}^{(2)}_{\rm ent}$ and (c) ${U}^{(3)}_{\rm ent}$, composed by the ${U}_{1,\rm{ex}}$ , see Eq. \ref{['Eq:USWAP']}, ${U}_{2,\rm{ex}}$, see Eq. \ref{['Eq:UFLIP']} and CNOT gates, respectively. The repeated units across several qubits are shown in dotted boxes (see Appendix \ref{['App:boxes']}).
• Figure 3: Upper panel: Dissociation profile of the $\rm{H_2}$ molecule for different definitions of the active space (AS). AS 4 (orange): only 2 occupied and 2 virtual orbitals are considered in the definition of the $\hat{T}_1$ and $\hat{T}_2$ operators; AS 6 (green): 2 occupied and 4 virtual orbitals; AS 8 (blue): 2 occupied and 6 virtual orbitals. The red curve corresponds to the reference HF calculation and the black one is the analytic solution evaluated using the p/h Hamiltonian expanded in the full (12 qubit) space. Lower panel: Corresponding energy errors along the dissociation profile. The blue shaded area corresponds to the energy range within chemical accuracy.
• Figure 4: Upper panel: Dissociation profile of the $\rm{H_2O}$ molecule for different definitions of the active space (AS). AS 8 (orange): 4 HF orbitals (starting form the highest occupied one, see inset) and all virtual orbitals are considered in the definition of the $\hat{T}_1$ and $\hat{T}_2$ operators; AS 10 (green): 6 occupied and all virtual orbitals; AS 12 (blue): 8 occupied and all virtual orbitals. The red curve corresponds to the reference HF calculation and the black one is the analytic solution evaluated using the p/h Hamiltonian expanded in the full (12 qubit) space. Lower panel: Corresponding energy errors along the dissociation profile. The blue shaded area corresponds to the energy range within chemical accuracy.
• Figure 5: Convergence of the Trotter error as a function of the Trotter expansion coefficient $n$ in Eq. \ref{['eq:Trotter_approx']} for the UCCSD energy of $\rm H_2$ at a bond length of $0.592$ Å. The reference energy, $E_{\text{exact}}$, corresponds to $E_{\rm diag}$ from Table \ref{['table:UCCSD_simu']}. Green circles: analytic dependence of the Trotter error ($E_{\rm UCCSD}^{opt/n}$ in Eq. \ref{['eq:E_optn_UCCSD']}). Red triangles: Trotter errors obtained after the optimization of the angles $\vec{\theta}$ at each value of $n$ using the VQE approach ($E_{\rm UCCSD}^{{\rm circ}/n}(n)$ in Eq. \ref{['eq:E_circn_UCCSD']}).