Feedback-based ALgorithm for Quantum OptimizatioN - FALQON#

Code at:

Quantum Approximate Optimisation Algorithm (QAOA) is considered as one of the most important algorithms for optimisation in Quantum Computers, see arXiv:1411.4028 by Farhi, Goldstone and Gutmann for more information.

In this QAOA algorithm, the aim is to have a problem Hamiltonian H_P and a mixer Hamiltonian H_B. Then, starting with the ground state of H_B, the goal is to repeatedly apply e^(i H_P c) and e^(i H_B b), where c and b are tunable parameters. The values of such parameters are to be found via classical optimization

In the FALQON algorithm, arXiv:2103.08619 by Magann, Rudinger, Grace and Sarovan, they propose a similar although conceptually different algorithm. The proposal consists in evolving the initial state using the Schrödinger equation


This equation satisfies that the expectation value of H_P is monotonically decreasing. This feature is used to create a Hamiltonian evolution with 1 layer using e^(i H_P c) and e^(i H_B b). In the first layer, b=0, and c is a parameter to be defined. Then, the quantity A = i[H_P, H_B] is measured. Its expectation value is then taken to be the parameter b for the next layer. As more layers are added, the approximation to the ground state of the problem Hamiltonian H_P is more and more accurate.


Running the code#

This example contains just one file

  • is the file where the algorithm is run. The main class FALQON is now introduced in QIBO

The FALQON class behaves similarly to the QAOA one. It admits the following parameters:

  • hamiltonian: problem Hamiltonian whose ground state is sought.

  • mixer: mixer Hamiltonian. If None, qibo.hamiltonians.X is used.

  • solver: solver used to apply the exponential operators. Default solver is ‘exp’.

  • callbacks: List of callbacks to calculate during evolution.

  • accelerators: Dictionary of devices to use for distributed execution. See qibo.core.distcircuit.DistributedCircuit for more details. This option is available only when hamiltonian is a qibo.abstractions.hamiltonians.TrotterHamiltonian.

  • memory_device: Name of device where the full state will be saved. Relevant only for distributed execution (when accelerators is given).

When performing the execution of the problem, the following variables are to be set:

  • delta_t: initial guess for the time step. A too large delta_t will make the algorithm fail.

  • max_layers: maximum number of layers allowed for the FALQON.

  • initial_state: initial state vector of the FALQON.

  • tol: Tolerance of energy change. If not specified, no check is done.

  • callback: Called after each iteration for scipy optimizers.

  • options: a dictionary with options for the different optimizers.

  • compile: whether the TensorFlow graph should be compiled.

  • processes: number of processes when using the paralle BFGS method.

The attached example provides an easy implementation of the FALQON method for a Heisenberg XXZ model.