Variational quantum algorithms (VQAs) are a type of hybrid classical-quantum algorithms consisting of a parametrized quantum circuit and a classical optimization loop to update the parameters of the quantum circuit.
VQAs are often seen as the quantum analog of classical neural networks. The depth of the parametrized quantum circuit is often shallow, and these algorithms can thus often be run on near-term devices.
In a VQA, the parametrized quantum circuit is often called an ansatz. It can be represented by a unitary operation $U(\theta)$ where the $\theta$ are the parameters of the circuit. Given an input state and choice of measurements, the circuit will produce some outputs which will be used to evaluate a cost function $C(\theta)$. The optimization loop aims to find the best parameters for which the cost function is minimized:
$\theta^* = \text{arg min}_\theta C(\theta)$
There has been extensive research on the choice of ansatz and expressivity of the resulting model, on the optimization methods that can be used and their scaling, on the trainability of VQAs and associated issues such as barren plateaus.
Examples of VQAs and applications
– The Variational Quantum Eigensolver (VQE) can be used to estimate the ground state energy of a molecule.
– The Quantum Approximate Optimization Algorithm (QAOA) can be used to obtain approximate solutions for combinatorial optimization problems such as Max-Cut.
– VQAs as quantum neural networks for various tasks such as classification or generative learning. For example, quantum generative adversarial networks (QGANs) can be defined and trained using the VQA paradigm.
Frequently asked questions about variational quantum algorithms
- What will happen to VQAs after the NISQ era?
While VQAs can be run on near-term devices – and this has certainly contributed to their popularity — they can also be implemented using logical qubits on fault-tolerant quantum computers. They are thus not limited to the NISQ era.
- Is it true that VQAs cannot be trained due to barren plateaus?
While there are challenges with the optimization of variational quantum circuits, there also existed challenges in classical machine learning, some of which have been overcome through clever training strategies. Barren plateaus for instance relate to statements about the loss function on average across the whole optimization landscape. It may thus be possible to avoid them with smart initialization strategies and warm starts.