Subspace-search variational quantum eigensolver for excited states

The variational quantum eigensolver (VQE), a variational algorithm to obtain an approximated ground state of a given Hamiltonian, is an appealing application of near-term quantum computers. To extend the framework to excited states, we here propose an algorithm, the subspace-search variational quantum eigensolver (SSVQE). This algorithm searches a low-energy subspace by supplying orthogonal input states to the variational ansatz and relies on the unitarity of transformations to ensure the orthogonality of the output states. The $k\mathrm{th}$ excited state is obtained as the highest-energy state in the low-energy subspace. The proposed algorithm consists only of two parameter optimization procedures and does not employ any ancilla qubits. The avoidance of the estimation of the inner product and the small number of procedures required are considerable improvements from the existing proposals for excited states, making our proposal an improved near-term quantum algorithm. We further generalize the SSVQE to obtain all excited states up to the $k\mathrm{th}$ by only a single optimization procedure. From numerical simulations, we verify the proposed algorithms. This work extends the applicable domain of the VQE to excited states and their related properties as a transition amplitude without sacrificing any of its feasibility. Moreover, the proposed variational subspace search, which generalizes the state search problem to the search of a unitary mapping to a specific subspace, itself would be useful for various quantum information processing methods such as finding a protected subspace or a good variational quantum error correction code.

Figure 1

Variational quantum circuit used in the simulations of Sec. 5. These parameters $\mathit{\varphi },\mathit{\theta }$ are optimized to to minimize $\mathcal{L}$. $D_1$ and $D_2$ denote the number of repetitions of a circuit in each bracket. Note that, in the explanation of weighted SSVQE, $\mathit{\theta }$ denotes $\left\{\mathit{\varphi },\mathit{\theta }\right\}$ in this figure.

Figure 2

Step (1) of the SSVQE to find the third excited state of a transverse Ising model. Black dashed line: $\mathrm{Avg}.\left(E_{0,1,2,3}\right)=\frac{1}{4}{\sum }_{k=0}^3{E}_k$, which is the globally optimal value of ${\mathcal{L}}_1/4$ in this case. Red solid lines: The evolution of ${\mathcal{L}}_1/4$ and the fidelity (see the main text for the definition) during the optimization process.

Figure 3

Step (2) of the SSVQE to find the third excited state of a transverse Ising model. Red solid lines: The evolution of ${\mathcal{L}}_2$ and the fidelity (see the main text for the definition) during the optimization process.

Figure 4

The weighted SSVQE to find the third excited state of a transverse Ising model. In the energy diagram, SSVQE($E_3$) (red solid line) is $\langle {\phi }_3\left|U^{†}\left(\mathit{\theta }\right)\mathcal{H}U\left(\mathit{\theta }\right)\right|{\phi }_3\rangle$ at each iteration.

Figure 5

The weighted SSVQE to find excited states of a transverse Ising model up to the third. In the energy diagram, SSVQE($E_k$) ($k=0,1,2,3$) are $\langle {\phi }_k\left|U^{†}\left(\mathit{\theta }\right)\mathcal{H}U\left(\mathit{\theta }\right)\right|{\phi }_k\rangle$ for each $k$ at each iteration.

Figure 6

The energy levels of the Hamiltonian of $\mathrm{HeH}$ and the calculated energy of the third excited state using the SSVQE.

Figure 7

The energy levels of the Hamiltonian of $\mathrm{HeH}$ and the predicted energy of the third excited state using the weighted SSVQE.

Figure 8

The energy levels of the Hamiltonian of $\mathrm{HeH}$ and the predicted energy of the excited states up to the third using the weighted SSVQE. Upper left: Ground state; upper right: first excited state; lower left: second excited state; lower right: third excited state.