Variational quantum algorithms for discovering Hamiltonian spectra

Calculating the energy spectrum of a quantum system is an important task, for example to analyze reaction rates in drug discovery and catalysis. There has been significant progress in developing algorithms to calculate the ground state energy of molecules on near-term quantum computers. However, calculating excited state energies has attracted comparatively less attention, and it is currently unclear what the optimal method is. We introduce a low depth, variational quantum algorithm to sequentially calculate the excited states of general Hamiltonians. Incorporating a recently proposed technique [O. Higgott, D. Wang, and S. Brierley, arXiv:1805.08138], we employ the low depth swap test to energetically penalize the ground state, and transform excited states into ground states of modified Hamiltonians. We use variational imaginary time evolution as a subroutine, which deterministically propagates toward the target eigenstate. We discuss how symmetry measurements can mitigate errors in the swap test step. We numerically test our algorithm on Hamiltonians which encode 3SAT optimization problems of up to 18 qubits, and the electronic structure of the lithium hydride molecule. As our algorithm uses only low depth circuits and variational algorithms, it is suitable for use on near-term quantum hardware.

Figure 1

Using symmetry measurements to detect and mitigate errors in the conventional swap test.

Figure 2

The expected energy as variational imaginary time evolution discovers some low-lying states of 18 Boolean (left) and 16 Boolean (right) 3SAT Hamiltonians, using the compact Ansatz (with 126 and 112 parameters, respectively). Vertical dashed lines indicate iterations when the Hamiltonian was excited and the parameters rerandomized. Horizontal dashed and colored lines indicate the true eigenvalues and those found by our method, respectively. Labels indicate the degeneracies of the states.

Figure 3

Variational simulations discovering then exciting several low-lying energy eigenstates of the reduced six-qubit LiH Hamiltonian, using the low depth Ansatz with 56 parameters. The top plot shows the expected energy in red, as the states reached at the vertical dashed lines are excited in the Hamiltonian. Horizontal dashed and colored lines indicate the true and discovered energy eigenvalues, respectively. The bottom plot shows the population of the eigenstates as found by numerically diagonalizing the Hamiltonian. The spectrum discovered in the long term is included in Fig. 4. (a) Imaginary time. Regions I, II, and III converge to orthogonal superpositions of the three degenerate first-excited states, which are themselves eigenstates. (b) Gradient descent. The green (labeled 1, 2, and 3), orange (labeled 6 and 7), and blue (labeled 8 and 9) states are degenerate. Regions I, II, and III show gradient descent converging to noneigenstates.

Figure 4

The 6-qubit LiH spectra discovered by a single process of parameter evolution using imaginary time and gradient descent, compared to that found by direct numerical diagonalization of the Hamiltonian. The low depth and compact Ansätze use 56 and 42 parameters, respectively. The low depth Ansatz simulations extend those in Figs. 3 and 3. Energies closer than $5\times {10}^{-3}$ apart are combined and their degeneracy labeled.

Figure 5

The lowest-lying states discovered by imaginary time evolution of the ten-qubit LiH Hamiltonian over varying bond length. The gray lines indicate the true spectrum as found by diagonalization, and the line labels indicate degeneracy in both the true and discovered states. The compact Ansatz is used with 70 parameters and for 10 000 iterations at every bond length. The low depth Ansatz uses 145 parameters for 40 000 iterations.

Figure 6

An example of variational imaginary time evolution whereby energy plateaued during a stage of continued parameter change. This is the low depth Ansatz of 56 parameters exploring the reduced six-qubit LiH Hamiltonian, without parameter rerandomization.

Figure 7

A quantum circuit which evaluates $\text{Re}\left(e^{i\varphi }\langle \overline{0}|V|\overline{0}\rangle \right)$. $H$ is the Hadamard gate. The first qubit is measured in the computational $\{|0\rangle ,|1\rangle \}$ basis, and the average value $\langle Z\rangle$ (Pauli) of the second qubit equals $\text{Re}\left(e^{i\varphi }\langle \overline{0}|V|\overline{0}\rangle \right)$.

Figure 8

Schematic of the shallow swap test circuit from Ref. [19]. This circuit evaluates the overlap of two input density matrices $\rho$ and $\sigma$.