Iterative Quantum Assisted Eigensolver

Abstract

The task of estimating ground state and ground state energy 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 and ground state energy of a Hamiltonian. The description of the Hamiltonian is assumed to be a linear combination of unitaries. Our algorithm is iterative and systematically constructs the Ansatz using any given choice of the initial state and the unitaries describing the Hamiltonian. In a particular iteration, the task of the quantum computer remains to measure two overlap matrices. Using recent results in literature, this task can be performed efficiently on current quantum hardware without requiring any complicated measurements such as the Hadamard test. At the end of a particular iteration, the classical computer solves a quadratically constrained quadratic program. The algorithm terminates if the desired stopping criterion has been achieved, otherwise proceeds to the next iteration. Our algorithm works for almost every random choice of the initial state and provides an approach to circumvent the barren plateau problem.

Quantum Krylov subspace algorithms for ground and excited state energy estimation

“…For instance, McClean et al showed that by using the set of non-orthogonal basis states a † i a j |Ψ G , it is possible to find low-lying excited states based on the ground state |Ψ G ≈ |Ψ(θ) , which is found through the standard vari- ational quantum eigensolver [26,27]. K-moment states have also been proposed as an alternative way of constructing the non-orthogonal basis states, which becomes scalable when the K-moment unitaries are tensor products of Pauli operators [28,29].…”

Optimal training of variational quantum algorithms without barren plateaus

“…Further, the optimization routine of VQAs was shown to be NP-hard even for problems that are easy for classical computers [11]. Quantum algorithms that can avoid training PQCs to circumvent the barren plateau problem have been proposed [12][13][14][15][16][17]. To improve training of VQAs, one can use quantum geometric information via the quantum natural gradient (QNG) [18][19][20].…”

“…Adaptively chosen PQCs [45] or PQCs tailored to the specific problem could enhance the representation power. Hybrid states as linear combination of quantum states generated using the problem Hamiltonian could systematically create an ansatz suited for the problem [13][14][15]. We note that training VQAs without a lower bounded fidelity is expected to be difficult, as most likely training will be stuck in the barren plateau.…”

Natural parameterized quantum circuit

“…Then, one can immediately calculate the parameters for the corresponding superposition state θ s (see Supplemental materials). The power to create superposition states could help to prepare basis states for quantum assisted algorithms [48][49][50][51] or become an ingredient to implement algorithms based on linear combination of unitaries on NISQ quantum computers [52,53]. Python code for the numerical calculations are available at [54].…”