Toward convergence of effective-field-theory simulations on digital quantum computers

We report results for simulating an effective field theory to compute the binding energy of the deuteron nucleus using a hybrid algorithm on a trapped-ion quantum computer. Two increasingly complex unitary coupled-cluster ansatze have been used to compute the binding energy to within a few percent for successively more complex Hamiltonians. By increasing the complexity of the Hamiltonian, allowing more terms in the effective field theory expansion, and calculating their expectation values, we present a benchmark for quantum computers based on their ability to scalably calculate the effective field theory with increasing accuracy. Our result of $E_4=-2.220\pm 0.179$ MeV may be compared with the exact deuteron ground-state energy $-2.224$ MeV. We also demonstrate an error mitigation technique using Richardson extrapolation on ion traps. The error mitigation circuit represents a record for deepest quantum circuit on a trapped-ion quantum computer.

Figure 1

Expectation values of Hamiltonian terms in $H_3$ as a function of noise parameter $r=2M+1$ which is the number of repetitions of noisy two-qubit identity gates before a given two-qubit gate. $M$ is defined in the Results section. Various colored, solid symbols are the expectation values of individual terms in $H_3$. Black crosses are $H_3$, computed according to Eq. (3). Colored, solid lines are the linear fits to the corresponding individual Hamiltonian terms in $H_3$. The black solid line is the linear fit to $H_3$. We use the linear fits to extrapolate to the zero noise limit. The error bars in the figure are statistical errors based on finite sampling and a binomial distribution. The binding energy is determined as $-2.030\pm 0.034$ MeV.

Figure 2

Expectation values of Hamiltonian terms in $H_4$ as a function of noise parameter $r=2M+1$ which is the number of repetitions of noisy two qubit identity gates before a given two-qubit gate. $M$ is defined in the Results section. Various colored, solid symbols are the expectation values of individual terms in $H_4$. Black crosses are $H_4$, computed according to Eq. (3). Colored, solid lines are the linear fits to the corresponding individual Hamiltonian terms in $H_4$. The black solid line is the linear fit to $H_4$. We use the linear fits to extrapolate to the zero noise limit. The error bars in the figure are statistical errors based on finite sampling and a binomial distribution. The binding energy is determined as $-2.220\pm 0.179$ MeV.

Figure 3

Expectation value $\langle H_4\rangle (r=0)$ as a function of a parameter chosen from the set $\{{\lambda }_0,{\lambda }_1,{\lambda }_2\}$. The plot symbols denote the zero-noise-limit extrapolated from the experimental data, also given in Table 1, and the solid lines denote the theoretical values.

Figure 4

Aggregate results on the deuteron simulation performed across different quantum computing platforms. Open symbols denote the experimental results. Star symbols denote the exact UCCS results. The black solid line denotes the exact deuteron ground-state energy. We note that the results on other platform has been determined variationally on the hardware.