Project // Barone Francesco, Coppi Alberto, Zinesi Paolo

In this work, we explore the paradigmatic perceptron model for binary classification in a quantum computing context. The minimization of the cost function is performed using Quantum Annealing (QA), which is one of the most promising frameworks for quantum optimization.

Following the work of G. Lami et al (2022), we address the minimization of the complex classical cost function of a quantum perceptron model, i.e. we focus on the Hamiltonian $H = \sum_\mu \Theta(-\xi^\mu\cdot\sigma)\frac{-\xi^\mu\cdot\sigma}{\sqrt{N}}$, where $\lbrace \xi^\mu : \mu = 1,...,N_\xi \rbrace$ is a set of $N_\xi$ binary patterns $\xi^\mu \in \lbrace-1,+1\rbrace^N$.

As shown in G. Lami et al, the adiabatic time evolution of QA can be efficiently represented as Matrix Product Operators in a Tensor Network framework. Such representation allows for simple classical simulations, well-beyond small sizes amenable to exact diagonalization techniques. Indeed, discretizing the QA dynamics (dQA) into $P$ steps, we can evolve the initial state towards an optimized Matrix Product State (MPS).

We compute the energy density $\varepsilon(s)$ as a function of the interpolation parameter $s \in [0,1]$. The residual energy density at the end of the annealing schedule, $\varepsilon(1)$, can be regarded as a figure of merit of dQA effectiveness.

We validate the results as shown in the reference paper, for various dataset sizes $N$ in the range $8-21$.

Eventually, we propose a circuit to execute the discretized QA on a quantum computer. This circuit is probed using a quantum tensor-network emulator developed in Padua, Quantum Tea Leaves.

├── data/            dataset and benchmarks
├── exact-diag/      QA with exact diagonalization
├── img/             plots
├── quimb-dqa/       first implementation of MPS dQA (with Quimb)
│
├─ dQA_mps.py        implementation of dQA with MPS (jax)
└─ dQA_circuit.py    implementation of dQA with Matcha/Qiskit

Python requirements are listed in requirements.txt.

Quantum Tea Leaves and Quantum Matcha Tea are required to run the dQA_circuit implementation. To setup a custom environment, we suggest to take a look at the script setup-env.sh, as it should work in most cases.

Our software has been tested with the following software version:

qiskit==0.38.0
qmatchatea==0.4.7
qtealeaves==0.4.15

Furthermore, in order to use the dQA circuit simulation via Matcha in step-by-step mode, it is necessary to hot-fix the Matcha library itself.

#   fix line 840 of file  qtealeaves.emulator.mps_simulator.py  as
obj = cls(len(tensor_list), 0, conv_params, local_dim)

• G. Lami et al. "Quantum Annealing for Neural Network optimization problems: a new approach via Tensor Network simulations", in SciPost Physics, 2022 (arxiv:2208.14468)