VQE
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The Variational Quantum Eigensolver (VQE) is a prominent quantum algorithm designed to estimate the ground-state energy of quantum systems. This algorithm plays a critical role in the field of quantum computing because it addresses problems that are practically intractable for classical computers, particularly in simulating the behavior of molecules and materials at the quantum level.
VQE combines classical and quantum computation in a hybrid approach. It leverages a quantum computer to perform part of the computation, specifically, evaluating the expectation value of a Hamiltonian operator representing the system of interest. To do this, VQE uses a quantum variational circuit with tunable parameters, which effectively encodes a quantum state. Classical optimization algorithms then adjust these parameters to minimize the energy expectation value, which converges to the ground-state energy of the system.
VQE has garnered significant attention in quantum chemistry and material science due to its potential for solving complex problems, such as molecular structure prediction and materials optimization. However, it’s important to note that VQE is sensitive to errors and noise in current quantum hardware, and its true power may be fully realized as quantum devices continue to improve. Researchers are actively working to enhance VQE’s performance and applicability for real-world problems, making it a promising avenue for quantum computation.
Ansatz and initial trial function
The choice of ansatz state depends on the system of interest. In gate based quantum computing, the ansatz is given by a parametrized qc, whose parameters can be updated after each run. The ansatz has to be adaptable enough to not miss the desired state. A common method to obtain a valid ansatz is given by the unitary coupled cluster (UCC) framework and its extensions.
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If the ansatz is not chosen adequately the procedure may halt at suboptimal parameters that do not correspond to a minima. In this situation, the algorithm is said to have reached a ‘barren plateau’.
The ansatz can be set to an initial trial function to start the algorithm. For example, for a molecular system, one can use the Hartree Fock Method to provide a starting state that is close to the real ground state.
Variational method and optimization
Given a parametrized ansatz for the ground state eigenstate, with parameters that can be modified, one is sure to find the parametrized state that is closest to the ground state based on the variational method of quantum mechanics . Using classical algorithms in a digital computer, the parameters of the ansatz can be optimized. For this minimization, it is necessary to find the minima of a multivariable function. Classical optimizers using gradient descent can be used for this purpose.
By running the circuit many times and constantly updating the parameters to find the global minima of the expectation value of the desired observable, one can approach the ground state of the given system and store it in a quantum processor as a series of quantum gate instructions.
Next day we will try to follow these instructions on qiskit .
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