Hybrid quantum/classical algorithms (HQCA)

Hybrid quantum/classical algorithms (HQCA) are a class of computational methods that leverage the unique capabilities of both quantum and classical computers to solve complex problems more efficiently. These algorithms use quantum computing to perform certain computationally intensive tasks, while relying on classical computing for tasks better suited to classical hardware. By combining the strengths of both quantum and classical approaches, HQCA have the potential to address a wide range of problems, particularly in optimization, chemistry, and machine learning. The general structure of hybrid quantum/classical algorithms involves the following components:

1. Quantum state preparation: The quantum computer prepares a parameterized quantum state, often referred to as the ansatz state. This state typically depends on a set of parameters, which are optimized during the algorithm.

2. Quantum computation: The quantum computer manipulates the ansatz state using quantum operations (gates) to perform a specific computation, such as finding the energy expectation value of a molecule or solving an optimization problem.

3. Measurement: After the quantum computation is performed, the results are measured and extracted from the quantum computer. These measurements provide information about the system being studied, such as energy values or optimization solutions.

4. Classical optimization: Using the information obtained from the quantum computer, a classical optimization algorithm is employed to update the parameters of the ansatz state. This iterative process continues until a predetermined stopping criterion is met, such as reaching a specified accuracy or performing a maximum number of iterations.

5. Final output: Once the optimization process is complete, the optimized ansatz state is used to extract the desired solution or approximation.

HQCA integrate quantum state preparation and measurement (|ψ⟩) with classical optimization techniques to find the eigenvector (|v⟩) and eigenvalue (λ) of a Hermitian operator (H) such that H|v⟩ = λ|v⟩

QAOA: Quantum Approximate Optimization Algorithm

The QAOA is a model for quantum annealing utilized to solve graph theory problems . The algorithm employs classical optimization of quantum operations to maximize an objective function (F):

F(γ, β) = ⟨ψ(γ, β)|H|ψ(γ, β)⟩,

where γ and β are classical parameters optimized to find the best solution.

VQE: Variational Quantum Eigensolver

The VQE algorithm uses classical optimization to minimize the energy expectation (E) of an ansatz state (|ψ(θ)⟩) to determine the ground state energy of a molecule:

E(θ) = ⟨ψ(θ)|H|ψ(θ)⟩,

This method can also be extended to find excited energies of molecules.

CQE: Contracted Quantum Eigensolver

The CQE algorithm aims to minimize the residual (R) of a contraction (C) or projection of the Schrödinger equation (H|ψ⟩ = E|ψ⟩) onto the space of two (or more) electrons to find the ground or excited-state energy (E) and two-electron reduced density matrix (2-RDM) of a molecule:

R(ψ) = ||HC|ψ⟩ - E|ψ⟩||₂,

It is based on classical methods for solving energies and 2-RDM directly from the anti-Hermitian contracted Schrödinger equation.

Notable examples of hybrid quantum/classical algorithms include the Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA), and the Contracted Quantum Eigensolver (CQE). These algorithms are designed to tackle various problems, such as finding the ground state energy of molecules, solving combinatorial optimization problems, and approximating excited states of molecules. By effectively combining the strengths of quantum and classical computing, HQCA have the potential to outperform purely classical algorithms in solving a diverse range of problems.