Double-bracket quantum algorithms

Qibo features a model implementing double-bracke quantum algorithms (DBQAs) which are helpful for approximating eigenstates based on the ability to run the evolution under the input Hamiltonian.

More specifically, given an input Hamiltonian $H_0$, there are several techniques in quantum computing to compute its eigenstates.

A standard way is to run variational quantum circuits. For example, Qibo already features the VQE model [2] which provides the implementation of the variational quantum eigensolver framework. DBQAs allow to go beyond VQE in that they take a different approach to compiling the quantum circuit approximating the eigenstate [3].

What is the unitary of DBQA?

Given $H_0$ we begin by assuming that we were given a diagonal and Hermitian operator $D_0$ and a time $s_0$. The dbi module provides numerical strategies for selecting them. For any such choice we define the bracket $$W_0 = [D_0, H_0]$$ and the double-bracket rotation (DBR) of the input Hamiltonian to time $s$ $$H_1(s) = e^{sW} H_0 e^{- s W}$$

Why are double-bracket rotations useful?

We can show that the magnitude of the off-diagonal norms will decrease. For this let us set the notation that $\sigma(A)$ is the restriction to the off-diagonal of the matrix A. In numpy this can be implemented by \sigma(A) = A-np.diag(A).

With this notation we next use the Hilbert-Schmidt scalar product and norm to measure the progress of diagonalization $$||\sigma(H_0(s))||^2- ||\sigma (H_0 )||^2= -2s \langle W, [H,\sigma(H)\rangle+O(s^2)$$ This equation tells us that as long as the scalar product $\langle W, [H,\sigma(H)\rangle$ is positive then after the DBR the magnitude of the off-diagonal couplings in $H_0(s)$ is less than in $H_0$.

How to choose $D$?

For theoretical considerations the canonical bracket is useful. For this we need the notation of the dephasing channel $\Delta(H)$ which is equivalent to np.diag(h). $M = [\Delta(H),\sigma(H)]= [H,\sigma(H)]= [\Delta(H),H]$ The canonical bracket appears on its own in the monotonicity relation above and gives an unconditional reduction of the magnitude of the off-diagonal terms $$||\sigma(H_0(s))||^2- ||\sigma (H_0 )||^2= -2s ||M||^2+O(s^2)$$

• the multi qubit Pauli Z generator with $Z(\mu) = (Z_1)^{\mu_1}\ldots (Z_L)^{\mu_L}$ where we optimize over all binary strings $\mu\in {0,1}^L$
• the magnetic field $D = \sum_i B_i Z_i$
• the two qubit Ising model $D = \sum_i B_i Z_i + \sum_{i,j} J_{i,j} Z_i Z_j$, please follow the tutorial by Matteo and use the QIBO ising model for that with $h=0$

How to choose s?

The theory above shows that in generic cases the DBR will have a linear diagonalization effect (as quantified by $||\sigma(H_0(s))||$). This can be further expanded with Taylor expansion and the Qibo implementation comes with methods for fitting the first local minimum. Additionally a grid search for the optimal step is provided for an exhaustive evaluation and hyperopt can be used for a more efficient 'unstructured' optimization; additionally simulated annealing is provided which sometimes outperforms hyperopt (and grid search), see example notebooks. The latter methods may output DBR durations $s_k$ which correspond to secondary local minima.

Installation

The package can be installed by source after cloning the repository:

cd qibo-dbqa
pip install .

References

[1] https://arxiv.org/abs/2206.11772

[2] https://qibo.science/qibo/stable/code-examples/advancedexamples.html#how-to-write-a-vqe

[3] https://arxiv.org/abs/2408.07431