Quantum State Preparation Of Multiconfigurational States For Quantum Chemistry

Abstract

The ability to prepare states for quantum chemistry is a promising feature of quantum computers, and efficient techniques for chemical state preparation is an active area of research. In this paper, we implement and investigate two methods of quantum circuit preparation for multiconfigurational states for quantum chemical applications. It has previously been shown that controlled Givens rotations are universal for quantum chemistry. To prepare a selected linear combination of Slater determinants (represented as occupation number configurations) using Givens rotations, the gates that rotate between the reference and excited determinants need to be controlled on qubits outside the excitation (external controls), in general. We implement a method to automatically find the external controls required for utilizing Givens rotations to prepare multiconfigurational states on a quantum circuit. We compare this approach to an alternative technique that exploits the sparsity of the chemical state vector and find that the latter can outperform the method of externally controlled Givens rotations; highly reduced circuits can be obtained by taking advantage of the sparse nature (where the number of basis states is significantly less than 2$^{n_q}$ for $n_q$ qubits) of chemical wavefunctions. We demonstrate the benefits of these techniques in a range of applications, including the ground states of a strongly correlated molecule, matrix elements of the Q-SCEOM algorithm for excited states, as well as correlated initial states for a quantum subspace method based on quantum computed moments and quantum phase estimation.

Figures (15)

• Figure 1: Circuit resources obtained for UCCSD and CISD, where the latter are prepared using the GR (Sec. \ref{['sec:gr_method']}) or the SSP gleinig21 (Sec. \ref{['sec:ssp_method']}) methods. Number of qubits $n_q \in \{4, 6, 8, 10, 12, 14, 16\}$. For a given $n_q$ (equal to number of spin orbitals), the number of electrons is $\frac{n_q}{2}$ if $\frac{n_q}{2}$ is even or $\frac{n_q - 1}{2}$ if $\frac{n_q}{2}$ is odd, for which the HF reference is a closed shell singlet configuration. For a given $n_q$ and number of electrons, the number of configurations corresponds to the number of single and double excitations plus the HF reference. Circuits are compiled to the standard gate set using the Qiskit qiskit extension of TKET tket20.
• Figure 2: Circuit corresponding to state $c_1|1100\rangle + c_2|1001\rangle+c_3|0110\rangle+c_4|0011\rangle$ prepared using the GR method (see Sec. \ref{['sec:gr_method']}). Algorithm \ref{['alg:gr']} found that externally controlling $\mathcal{G}_3^2$ on the second qubit is required. Substituting the gate parameters for 90° torsion as an example and compiling to the H-series gate set h11, this circuit can be represented using 61 PhasedX gates, 4 Rz gates, and 44 2-qubit ZZMax gates.
• Figure 3: Circuit corresponding to state $c_1|1100\rangle + c_2|1001\rangle+c_3|0110\rangle+c_4|0011\rangle$ prepared using the SSP method gleinig21 (see Sec. \ref{['sec:ssp_method']}). Substituting the gate parameters for 90° torsion as an example and compiling to the H-series gate set h11, this circuit can be represented using 10 PhasedX gates, 4 Rz gates, and 5 2-qubit ZZMax gates.
• Figure 4: Energies obtained from VQE-optimized multiconfigurational states for the 4-qubit (2 electrons in 4 spin orbitals) active space of C$_2$H$_4$. Simulated measurements correspond to 10$^4$ shots per circuit, and the Hamiltonian consists of 14 Pauli operators. Top graph: GR method (see Sec. \ref{['sec:gr_method']}). Bottom graph: SSP method (see Sec. \ref{['sec:ssp_method']}). C$_2$H$_4$ structure at torsion angles 0°, 90°, and 180° shown above graphs. H11E corresponds to emulations of hardware experiments with a noise model calibrated to the H1 trapped ion device h11.
• Figure 5: Ideal QCM4 and CMX2 energies calculated using a single configuration HF (top, $D=1$) and VQE-optimized multiconfigurational input states (bottom, $D=4$) for the 8-qubit (4 electrons in 8 spin orbitals) active space of C$_2$H$_4$. VQE energies correspond to expectation values of $\hat{H}$ taken with respect to the optimized multiconfigurational state. For ideal simulations, externally controlled GRs (Sec. \ref{['sec:gr_method']}) and the SSP method (Sec. \ref{['sec:ssp_method']}) yield identical results.