Iterative quantum-assisted eigensolver

The task of estimating the ground state of Hamiltonians is an important problem in physics with numerous applications ranging from solid-state physics to combinatorial optimization. We provide a hybrid quantum-classical algorithm for approximating the ground state of a Hamiltonian that builds on the powerful Krylov subspace method in a way that is suitable for current quantum computers. Our algorithm systematically constructs the ansatz by using any given choice of the initial state and the unitaries describing the Hamiltonian. The only task of the quantum computer is to measure overlaps and no feedback loops are required. The measurements can be performed efficiently on current quantum hardware without requiring any complicated measurements such as the Hadamard test. Finally, a classical computer solves a well-characterized quadratically constrained optimization program. Our algorithm can reuse previous measurements to calculate the ground state of a wide range of Hamiltonians without requiring additional quantum resources. Further, we demonstrate our algorithm for solving a class of problems with thousands of qubits. The algorithm works for almost every random choice of the initial state and circumvents the barren plateau problem.

Figure 1

The IQAE algorithm systematically builds the ansatz using some efficiently implementable quantum state and the unitaries which describe the Hamiltonian. The ansatz is represented as a linear combination of the fine-grained Krylov subspace basis or cumulative $K$-moment states which are inspired by the Krylov subspace method (see Definition 1). The second step computes overlap matrices on a quantum computer. Finally, the optimization program (3) is executed on a classical computer. The approximation of the ground state improves with increasing $K$.

Figure 2

(a) Ground-state energy found by IQAE plotted against order $K$ of the fine-grained Krylov subspace for the transverse Ising model with $N=8$ qubits and $h=\frac{1}{2}$. As an initial state, we compare a random variational circuit with 200 layers, a product state, and a QAOA circuit ${\prod }_{k=1}^{p}exp(-i{\beta }_k{\sum }_j{\sigma }_j^x)exp(-i{\gamma }_{k}H){|+\rangle }^{\otimes N}$ with depth $p=1$ and parameters $\gamma ,\beta$ that were minimized using the COBYLA algorithm. Shaded lines indicate the standard deviation of the energy averaged over the random instances of the variational circuit. The dashed-dotted black line is the exact ground state. (b) Learning the ground state of the Ising Hamiltonian for all magnetic fields $h$ with a single run of IQAE. We plot fidelity $F=|\langle {\varphi }_1|{\psi }_{\text{IQAE}}\rangle {|}^2$ with the exact ground state against the transverse field strength $h$. From the 2-moment of the fine-grained Krylov subspace, we select a subset of 100 quantum states for our ansatz. Then, we run IQAE once and use the measured overlaps to determine the ground state for all values of $h$, as $h$ can be varied for free during postprocessing. We show the fidelity with the exact ground state for a finite number of measurement shots $N_{\text{shots}}=8192$ and an infinite $N_{\text{shots}}=\infty$. The initial state used for the $K$-moment expansion is a $p=2$ QAOA circuit that minimizes the energy for $h=\frac{1}{2}$ and its fidelity is shown in the figure as well. To evaluate the overlaps including the Hamiltonian expectation value, 6358 Pauli strings have to be calculated, which we measured by sampling 1098 different measurement settings. (c) IQAE to calculate the ground state of the Hamiltonian consisting of $r=8$ random Pauli strings and many qubits $N$. We show the difference between IQAE and the exact ground-state energy $\mathrm{\Delta }E$ as a function of the number of ansatz states $M$.