Quantum algorithms for electronic structure calculations: Particle-hole Hamiltonian and optimized wave-function expansions

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 picture, which offers a better starting point for the expansion of the system wave function. The state of the molecular system at study is parametrized so as to constrain the sampling of the corresponding wave function within the sector of the molecular Fock space that contains the desired solution. To this end, we explore different mapping schemes to encode the molecular ground state wave function in a quantum register. 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 in the expansion of the system wave function can accurately and efficiently reproduce the ground-state energy of simple molecular systems.

Figure 1

Circuits for the exponentiation of the (a) single and (b) double excitation operators $({\widehat{a}}_p^{†}{\widehat{a}}_r-\text{H.c.})$ and $({\widehat{a}}_p^{†}{\widehat{a}}_q^{†}{\widehat{a}}_r{\widehat{a}}_s-\text{H.c.})$, which contribute to ${\widehat{T}}_1$ and ${\widehat{T}}_2$, 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 pp2 (Fig. 8).

Figure 2

Definition of the three entangler blocks: (a) $U_{\mathrm{ent}}^{\left(1\right)}$, (b) $U_{\mathrm{ent}}^{\left(2\right)}$, and (c) $U_{\mathrm{ent}}^{\left(3\right)}$, composed by the $U_{1,\mathrm{ex}}$, see Eq. (19), $U_{2,\mathrm{ex}}$, see Eq. (20), and CNOT gates, respectively. The repeated units across several qubits are shown in dotted boxes (see Appendix pp2).

Figure 3

Top: Dissociation profile of the ${\mathrm{H}}_2$ molecule for different definitions of the active space (AS). AS 4 (orange triangles): only two occupied and two virtual orbitals are considered in the definition of the ${\widehat{T}}_1$ and ${\widehat{T}}_2$ operators; AS 6 (green squares): two occupied and four virtual orbitals; AS 8 (blue circles): two occupied and six virtual orbitals. The red (gray, no symbols) 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 (eight-qubit) space. Bottom: Corresponding energy errors along the dissociation profile. The blue (gray) shaded area corresponds to the energy range within chemical accuracy.

Figure 4

Top: Dissociation profile of the ${\mathrm{H}}_2\mathrm{O}$ molecule for different definitions of the active space (AS). AS 8 (orange triangles): four HF orbitals (starting form the highest occupied one, see inset) and all virtual orbitals are considered in the definition of the ${\widehat{T}}_1$ and ${\widehat{T}}_2$ operators; AS 10 (green squares): six occupied and all virtual orbitals; AS 12 (blue circles): eight occupied and all virtual orbitals. The red (gray, no symbols) 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. Bottom: Corresponding energy errors along the dissociation profile. The blue (gray) 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. (25) for the UCCSD energy of ${\mathrm{H}}_2$ at a bond length of 0.592 Å. The reference energy, $E_{\text{exact}}$, corresponds to $E_{\mathrm{diag}}$ from Table 2. Green circles: analytic dependence of the Trotter error [$E_{\mathrm{UCCSD}}^{\mathrm{opt}/n}$ in Eq. (26)]. Red triangles: Trotter errors obtained after the optimization of the angles $\vec{\theta }$ at each value of $n$ using the VQE approach $[E_{\mathrm{UCCSD}}^{\mathrm{circ}/n}\left(n\right)$ in Eq. (27)].

Figure 6

Dissociation profiles of the ${\mathrm{H}}_2$ molecule computed using the p-h Hamiltonian and the heuristic Ansatz for the trial wave function. The blue crosses and green dots (mostly overlapping) are obtained using the particle-conserving entanglers $U_{\mathrm{ent}}^{\left(1\right)}$ and $U_{\mathrm{ent}}^{\left(2\right)}$, respectively. The cyan triangles are computed using the non-particle-conserving operator circuit $U_{\mathrm{ent}}^{\left(3\right)}$. The blue (gray) shaded area corresponds to the chemical accuracy energy range. For all entangler types, the number of repeated blocks, $D$ in Eq. (17), was set to 8.

Figure 7

Dissociation profiles of the ${\mathrm{H}}_2\mathrm{O}$ molecule computed using the p-h Hamiltonian and the heuristic Ansatz for the trial wave function. The blue crosses and green dots are obtained using the particle-conserving entanglers $U_{\mathrm{ent}}^{\left(1\right)}$ and $U_{\mathrm{ent}}^{\left(2\right)}$, respectively. The cyan triangles are computed using the non-particle-conserving operator circuit $U_{\mathrm{ent}}^{\left(3\right)}$. The blue (gray) shaded area corresponds to the chemical accuracy energy range. For all entangler types, the number of repeated blocks, $D$ in Eq. (17), was set to 20.

Figure 8

Definition of single- and two-qubit gate blocks (dashed lines) that span multiple qubits.

Figure 9

Decomposition of an exchange gate ($i=1,2$) between qubit $n$ and $m$ into their elementary gates.