Gate-count estimates for performing quantum chemistry on small quantum computers

As quantum computing technology improves and quantum computers with a small but nontrivial number of $N\ge 100$ qubits appear feasible in the near future the question of possible applications of small quantum computers gains importance. One frequently mentioned application is Feynman's original proposal of simulating quantum systems and, in particular, the electronic structure of molecules and materials. In this paper, we analyze the computational requirements for one of the standard algorithms to perform quantum chemistry on a quantum computer. We focus on the quantum resources required to find the ground state of a molecule twice as large as what current classical computers can solve exactly. We find that while such a problem requires about a 10-fold increase in the number of qubits over current technology, the required increase in the number of gates that can be coherently executed is many orders of magnitude larger. This suggests that for quantum computation to become useful for quantum chemistry problems, drastic algorithmic improvements will be needed.

Figure 1

This figure shows the energy of the water molecule as a function of bond angle and bond length for an STO-3G basis obtained from a restricted HF calculation.

Figure 2

Discretization error due to the Trotter decomposition for a water molecule at bond length $a=0.957213\AA$ and bond angle $\theta =104.{5225}^{\circ }$.

Figure 3

(i) Number of rotations, and number of gates in (ii) parallel and (iii) sequential mode. Dashed and dotted lines indicate the expected scaling, which is ${\left(N_{\mathrm{SO}}\right)}^4$ (dotted lines) for the number of rotations and the parallel gate count and ${\left(N_{\mathrm{SO}}\right)}^5$ (dashed line) for the sequential gate count. These data are also listed in Table 2.

Figure 4

Scaling of the number of necessary Trotter steps, $1/\Delta t$, with the number of terms in the Hamiltonian $m$. This is extracted from a series of artificial, but statistically appropriate, Hamiltonians with 12, 16, 18, 20, and 24 spin orbitals. The two curves correspond to different values of the inverse filling $r=N_{\mathrm{SO}}/N_{e^{-}}$. The dashed lines indicate fits to $N_{\mathrm{step}}\sim m^{\alpha }$, with exponents $\alpha (r=3)=1.27$ and $\alpha (r=2)=1.08$.

Figure 5

Circuit representations of Hamiltonian terms $H_{pp}$ and $H_{pq}$.

Figure 6

Circuit representation of Hamiltonian terms $H_{pqqp}$.

Figure 7

Circuit representations of Hamiltonian terms (i) $H_{pqqr},p<q<r$, and (ii) $H_{pqqr},q<p$ or $q<r$.

Figure 8

Circuit representation of Hamiltonian terms $H_{pqrs}$.