Paper Implementation Challenge - An adaptive variational algorithm for exact molecular simulations on a quantum computer.

[gzemlevskiy17]

We (@gzemlevskiy17, @ethanpucylowski, @calebpucylowski, @Zeinork) aim to implement the ADAPT-VQE algorithm outlined in this paper. This work improves upon the VQE algorithm mainly by reducing the circuit depth, using fewer parameters, and returning a more accurate ground state electronic energy for a molecule (compared to the exact result).

The ADAPT-VQE algorithm accomplishes this by dynamically choosing and adding the operators used to construct the ansatz. This means that the ansatz is not constant like in vanilla VQE. This algorithm trades low shot count for reduced circuit depth, a tradeoff that the authors claim is beneficial because circuit depth determines whether an algorithm can run.

Approach

The algorithm will proceed as follows:

Step 1:

Define a Hamiltonian representing the electronic energy of the desired molecule.

Step 2:

Define an 'Operator Pool' of fermionic operators that could potentially be used to construct the ansatz.

Step 3:

Initialize qubits to an appropriate reference state (the HF state is recommended by the authors).
Initialize the ansatz to the identity operator.

Step 4:

Prepare the state initialized above on a quantum computer.

Step 6:

Measure the expectation value of the Hamiltonian to find the energy of the state.

Step 5:

Compute the gradient of each operator in the operator pool with respect to the current ansatz. This is done by measuring the commutator of the Hamiltonian with each gradient in the pool, and repeating that multiple times and averaging the result.

Step 6:

Add the operator with the largest gradient to the left of the ansatz, but exit if the norm of the gradient vector is smaller than a previously defined threshold. This operator is not removed from the operator pool.

Step 7:

Use a classical optimization algorithm to re-optimize the parameters, as in vanilla VQE, and go to Step 4.

Examples:

We will simulate the LiH and BeH_2 molecules, and possible the H_6 molecule, checking our results against those of the paper, as the authors also simulate those molecules.

[gzemlevskiy17]

The step labeled first "Step 6" (finding the energy of the state) is actually performed in Step 7, just before the re-optimization of the parameters. The 3rd member of our group is @CalebPuc.

[TomerGoldfriend]

Thank you @gzemlevskiy17 , it seems like this issue has an overlap with this one. lf your implementation is similar, we will not be able to accept it.

[gzemlevskiy17]

@TomerGoldfriend Can we still submit another proposal?