Basic example of use of simulated annealing

Let us consider the one-dimensional Ising model of an antiferromagnet described by the Hamiltonian

$$ H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j$$

where $\langle i, j \rangle$ denotes nearest neighbour sites, magnetic coupling J > 0 for antiferromagnetism and $\sigma_i$ is the respective Pauli spin matrix acting on site $i$.

We want to use the simulated annealing implemented on the QLM to solve this problem. The following example presents how one can do so.

Encoding and solving the problem

import numpy as np
from qat.qpus import SimulatedAnnealing
from qat.core import Observable, Schedule, Variable, Term

# Define the the problem
n_spins = 100
mag_coupling = 1  # >0 is antiferromagnetic, <0 is ferromagnetic
    
# Create a temperature function
t = Variable("t", float)
temp_t = 2 * (1 - t) + 0.001  # annealing requires going from a high to a very low temperature

# Create a QPU
n_steps = 500
qpu = SimulatedAnnealing(temp_t=temp_t,
                         n_steps=n_steps,
                         seed=817)

# Create an observable
observable = Observable(n_spins, pauli_terms=[Term(mag_coupling, "ZZ", [i, i + 1])
                                              for i in range(n_spins - 1)])

# Create a schedule for the annealing, a job, and send it to the QPU
drive = [(1, observable)]
schedule = Schedule(drive=drive)
job = schedule.to_job(nbshots=1)
result = qpu.submit(job)

# Show the solution
for sample in result:
    print(sample.state)

|0101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101>

Solution

We can see that as expected for an antiferromagnet, the resulting spin configuration is alternating spins up and spins down which translates to alternating $|0>$ and $|1>$ qubit states.