Concurrent quantum eigensolver for multiple low-energy eigenstates

We propose a quantum algorithm that diagonalizes a Hamiltonian by implementing an ansatz that satisfies the generalized Rayleigh-Ritz variational principle. This algorithm uses a purification technique to target many quantum states in one quantum circuit and allows multiple eigenstates to be optimized and determined simultaneously. Moreover, it requires a reasonable circuit depth compared to existing algorithms and enables flexible postprocessing on the accurately determined eigensubspace. Using the transverse-field Ising model, we confirm that the eigenvalues obtained with the algorithm converge efficiently and uniformly with the iteration steps, tested by both simulations and IBM platform measurements. As limited quantum resources are needed, this algorithm is promising for noise resilience, better performances, and versatile applications.

Figure 1

Sketch of a quantum circuit for implementing a variational optimization for $N_p=4$ and $N_a=1$ systems. An entanglement generator $U_{\mathrm{ent}}$ is applied between the ancillary and physical qubits to prepare a purified initial state. $U_l$ is a collection of all unitary gate operations, which act only on physical qubits, in the $l\mathrm{th}$ layer of the circuit. $H$ is the Hadamard gate. $V$ denotes a postprocessing operator, which applies to the ancillas only.

Figure 2

Four lowest eigenvalues obtained with a purified trial wave function using two ancillas for the eight-spin transverse Ising model with $h_x=0.5$ and $J=1$. The energy values plotted are in units of $J$. (a) shows the four lowest eigenvalues, $E_0$ (solid line), $E_1$ (dash-dotted line), $E_2$ (dotted line), and $E_3$ (dashed line), and (b) shows how their absolute errors converge with the iteration number of optimizations $m$. (c) Comparison between the error of the spectral gap $\delta {\mathrm{\Delta }}_1$ (lower line) and the errors of $E_0$ and $E_1$.

Figure 3

The ground-state energy $E_0$ (lower line), the first excited eigenenergy $E_1$ (upper line), and the loss function $\mathcal{L}$ (middle line) obtained with a purified trial wave function using one ancilla for the three-spin TFIM with $h_x=0.5$ and $J=1$. The energy values plotted are in units of $J$. The relative errors of the ground and first excited eigenenergies are $1.244\%$ and $2.924\%$ compared to the exact results (dashed horizontal lines), respectively.

Figure 4

Eight low-energy eigenvalues obtained using three ancillas for the eight-spin TFIM. The left panel shows how the eight eigenvalues converge with the optimization step $m$. The right panel shows the lowest 11 eigenvalues in ascending order from the bottom obtained by exact diagonalization. Among these 11 eigenvalues, the sixth, seventh, and ninth excited eigenvalues are skipped in the variational calculation. Six variational layers ($L=6$) are used.

Figure 5

Eight lowest eigenvalues obtained in ascending order from the bottom using four ancillas for the eight-spin TFIM. (a) and (b) show the $m$ dependence of the eight lowest eigenvalues $E_i$ and their absolute errors, $\delta E_i=E_i-E_i^{\text{ex}}$, respectively. Six variational layers ($L=6$) are used.

Figure 6

(a) Variational circuit for solving the lowest two eigenenergies of the three-spin TFIM with one ancilla. $\tilde{U}$ is the two-qubit rotation gate, whose structure is depicted in (b). $S$ is the phase gate, $S=R_Z(\pi /2)$.