How to train quantum circuit-based models using gradient based methods ?

The goal in the case of classical models is to minimize the cost or loss function which is the objective function via the vector of parameters $\vec{\theta}$. In the case where we train a parameterized quantum circuit model, the function to minimize is the expected value $$\langle\Psi(\vec{\theta})|\hat{H}|\Psi(\vec{\theta})\rangle$$ with $\hat{H}$ the Hamiltonian operator defined by the Schrödinger equation $$i\hbar \frac{d|\Psi(\vec{\theta})\rangle}{d\theta} = \hat{H} |\Psi(\vec{\theta})\rangle$$

The expectation value is what we’d expect to measure if we averaged out a large number of results.

The quantum approach has two fundamental steps :

1. Prepare the quantum state $|\Psi(\vec{\theta})\rangle$ called the ansatz
2. Measure the expectation value $\langle\Psi(\vec{\theta})|\hat{H}|\Psi(\vec{\theta})\rangle$

We define our ansatz $|\Psi(\vec{\theta})\rangle = U(\vec{\theta})|0\rangle$ with $U(\vec{\theta})$ the parametrized quantum circuit.

We can see that the different quantum natural gradients approach the target faster than vanilla gradient descent.

• Simultaneous Perturbation Stochastic Approximation (SPSA)
• The Quantum Natural SPSA (QN-SPSA)
• Powerlaw SPSA : automatically calibrate the learning rate

1. Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac and Nathan Killoran, Evaluating analytic gradients on quantum hardware, Physical Revview A 99, 032331 (2019), doi:10.1103/PhysRevA.99.032331, arXiv:1811.11184.

2. James Stokes, Josh Izaac, Nathan Killoran and Giuseppe Carleo, Quantum Natural Gradient, Quantum 4, 269 (2020), doi:10.22331/q-2020-05-25-269, arXiv:1909.02108.

3. Julien Gacon, Christa Zoufal, Giuseppe Carleo and Stefan Woerner, Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information, arXiv:2103.09232.

4. Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush and Hartmut Neven, Barren plateaus in quantum neural network training landscapes, Nature Communications, Volume 9, 4812 (2018), doi:10.1038/s41467-018-07090-4, arXiv:1803.11173.

5. M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio and Patrick J. Coles, Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits, Nature Communications 12, 1791 (2021), doi:10.1038/s41467-021-21728-w, arXiv:2001.00550.