Fermionic simulations

A fermionic Hamiltonian is described in a similar fashion to bosons and spins - with an Observable (or many of them) in a Schedule. The key difference lies within the operators used. The fermionic creation operator \(c^\dagger\) is encoded via the big letter C, while the annihilation operator \(c\) - through the small letter c.

Let us look at the following simple Hamiltonian:

\[H = c_0^\dagger c_1 + c_1^\dagger c_0\]

Imagine we want to evolve \(H\) for time \(\frac{\pi}{2}\) starting from the state \(\left|01\right>\) (i.e. only site \(1\) is occupied). If at the end we measure the occupation of site \(0\) via the \(n_0 = c^\dagger_0 c_0\) operator, we would expect that the ocupation of the two sites has swapped such that the state of the system is \(\left|10\right>\) and \(\left<n_0\right> = 1\). One way to simulate this behaviour would look like:

import numpy as np
from qat.qpus import QutipQPU
from qat.core import Observable, Term, Schedule
from qat.core.variables import Variable, sin

# Define a time Variable
t = Variable("t", float)

# Create the Schedule
drive = Observable(2, pauli_terms=[Term(1, "Cc", [0, 1]), Term(1, "Cc", [1, 0])])
schedule = Schedule(drive=drive,
                    tmax=np.pi / 2)

# Construct a job with an observable to measure
job = schedule.to_job(psi_0="01", # start from the 0th site unoccupied and the 1st - occupied
                      job_type="OBS",
                      observable=Observable(2, pauli_terms=[Term(1, 'Cc', [0, 0])]))

# Create a QPU and submit the job
qpu = QutipQPU(nsteps=200) # can specify the number of time steps for the integration
res = qpu.submit(job)
print("<n_0 (pi/2)> =", res.value)

<n_0 (pi/2)> = 0.9999999999845415

For a more involved Hamiltonian considering interactions between the fermions in a noisy setting, take a look at the following fermionic notebook.

On Qaptiva there is also the module qat.fermion specifically dedicated to spin and fermionic computations.

Note

Fermionic simulations are currently only supported for QutipQPU.