Quantum simulation approach to implementing nuclear density functional theory via imaginary time evolution

The quantum imaginary time evolution (QITE) algorithm is a direct implementation of the classical imaginary time evolution algorithm on quantum computer. We implement the QITE algorithm for the case of nuclear Hartree-Fock equations in a formalism equivalent to nuclear density functional theory. We demonstrate the algorithm in the case of the helium-4 nucleus with a simplified effective interaction of the Skyrme kind and demonstrate that the QITE, as implemented on simulated quantum computer, gives identical results to the classical algorithm.

Figure 1

Quantum circuit performing one iteration of ITE in the $N=1$ case.

Figure 2

Quantum circuit performing one iteration of ITE in the $N=2$ case.

Figure 3

Quantum subcircuit $SP\left(\mathfrak{N}\right)$ performing $\mathfrak{N}$-qubit real state preparation.

Figure 4

Quantum subcircuit $SP\left(1\right)$ performing one-qubit real state preparation.

Figure 5

Quantum subcircuit $SP\left(2\right)$ performing two-qubit real state preparation.

Figure 6

Classical imaginary time evolution for $N=1$. (a) Wave function as a function of iteration. (b) Potential as a function of iteration.

Figure 7

Quantum imaginary time evolution for $N=1$ carried out using a quantum simulator. (a) Wave function as a function of iteration. (b) Potential as a function of iteration.

Figure 8

Imaginary time evolution for $N=1$ case after 40 iterations.

Figure 9

Quantum imaginary time evolution for $N=2$ carried out using a quantum simulator. (a) Wave function as a function of iteration. (b) Potential as a function of iteration.

Figure 10

Imaginary time evolution for $N=2$ case after 60 iterations.

Figure 11

Per-iteration development of the $\beta$ coefficients for the terms in the Pauli expansion of the unitary operator in the quantum imaginary time evolution algorithm for $N=1$ case. (a) Full view. (b) Zoomed in view.

Figure 12

Per-iteration development of the $\beta$ coefficients for the terms in the Pauli expansion of the unitary operator in the quantum imaginary time evolution algorithm for $N=2$ case.

Figure 13

Quantum circuit for one time step in the $N=3$ case.

Figure 14

Quantum circuit for one time step in the $N=4$ case.