Adiabatic quantum algorithm for artificial graphene

We devise a quantum-circuit algorithm to solve the ground state and ground energy of artificial graphene. The algorithm implements a Trotterized adiabatic evolution from a purely tight-binding Hamiltonian to one including kinetic, spin-orbit, and Coulomb terms. The initial state is obtained efficiently using Gaussian-state preparation, while the readout of the ground energy is organized into seventeen sets of measurements, irrespective of the size of the problem. The total depth of the corresponding quantum circuit scales polynomially with the size of the system. A full simulation of the algorithm is performed and ground energies are obtained for lattices with up to four hexagons. Our results are benchmarked with exact diagonalization for systems with one and two hexagons. For larger systems we use the exact state vector and approximate matrix product state simulation techniques. The latter allows us to systematically trade off precision with memory and therefore to tackle larger systems. We analyze adiabatic and Trotterization errors, providing estimates for optimal periods and time discretizations given a finite accuracy. In the case of large systems we also study approximation errors.

Figure 1

Graphene lattice used in this work, defined by the number of hexagons in each dimension $(N_x,N_y)$. Sites are numbered following the red (gray) shadow. The regular hexagons are composed of the unitary vectors connecting sites ${\vec{d}}_i$, on the right side, defined in Eq. (5).

Figure 2

Swap operations performed on the qubits belonging to the same row. First, the pairs $(2k,2k+1)$, $k\in \mathbb{N}$, are swapped. Then, the pairs $(2k+1,2k+2)$, $k\in \mathbb{N}$, are swapped. This permutation is cyclic and after as many steps as qubits in the chain the original state is recovered. Along the path, all operations in ${\mathcal{H}}_{\mathrm{FH}}$ can be applied between adjacent qubits at least once.

Figure 3

Schematic description of an adiabatic step of the circuit, for the case of the Coulomb and hopping terms, for a single hexagon. In the Coulomb part, on the left, the gates $U_{\mathrm{C}}^{(q_1,q_2)}$ perform the evolution between qubits $q_1$ and $q_2$, corresponding to the same site in the lattice. The parameter $\theta$, not explicitly written, in this gate as defined in Eq. (20) will depend on the exact step of the evolution. In the hopping term, on the right, the fermionic-swapping layers are interspersed with operations $U_{\mathrm{Hh}}^{(q_1,q_2)}$ and $U_{\mathrm{Hv}}^{(q_1,q_2)}$ connecting qubits $q_1$ and $q_2$ through a horizontal (h) or vertical (v) edge. The operations are labeled according to the representation qubits, and not according to the circuit qubits, which are always the same ones. In the hopping case, the parameter $\theta$ of $U_{\mathrm{H}}$ as defined in Eq. (22) is constant for all adiabatic steps.

Figure 4

Measurement scheme for the lattice. All terms from $H_{\mathrm{FH}}$ coming from edges in the same color (i.e., alternating vertical and horizontal links) can be measured simultaneously. Horizontal terms can be measured directly since they do not commute. In the case of vertical terms, they are measured using swapping in a common substring in the full Jordan-Wigner mapping for all edges in the same row. Each of the four sets has two possible spins and regular and spin-orbit terms, amounting to a total of 16 independent measurements. Together with the Coulomb terms, the complete measurement can be done in only 17 steps, irrespective of the problem size.

Figure 5

Comparison between adiabatic evolutions, as described in Eq. (15), and Trotter evolutions, from Eq. (17), for different values of $C$ and ${\lambda }_R$ detailed in each subcaption. All systems are composed of one hexagon. For each example, comparison for two different periods $T$ and number of steps $N$ are displayed. The top figures represent the distance to the ground energy of the evolved state, while the bottom figures represent the fidelity of the evolved state with respect to the target one.

Figure 6

Error appearing in the circuit evolution for problems with $C=1/2$, ${\lambda }_R=0$ (top row) and $C=1/2$, ${\lambda }_R=1$ (bottom row) considering grids of different sizes: (a) $\mathrm{grid}=1\times 1$, (b) $\mathrm{grid}=2\times 1$, (c) $\mathrm{grid}=1\times 2$, (d) $\mathrm{grid}=1\times 1$, and (e) $\mathrm{grid}=2\times 1$. Evolution only contributes with the Coulomb term. The marked $\times$ in panels (a) and (d) correspond to the $T=1$, $N=40$ circuit lines in Fig. 5. Errors above $100\%$ have been discarded from the plot. The error is measured taking as reference the difference between the ground energy and the starting one as defined in Eq. (25). This difference is typically 0.1, while the target energy is $\sim 10$, depending on the size of the system. Thus, a $10\%$ error in the plot above corresponds to $1\%$ in the total energy. In general, different grids for the same problem parameters give rise to quantitatively similar error structures.

Figure 7

MPS fidelity as a function of the maximum bond dimension ${\chi }_{\mathrm{M}}$ for an adiabatic circuit with $N_s=40$, $T=1$, $U=0.5$, and ${\lambda }_R=0$ for different lattices. Fidelity of the state vector is found to decay abruptly as ${\chi }_M$ is decreased past a threshold. This threshold increases with the dimensions of the lattice, with the number of hexagons in the horizontal dimension playing the most relevant role.