qrisp.operators.fermionic.FermionicOperator.trotterization#

FermionicOperator.trotterization(forward_evolution=True)[source]#

Returns a function for performing Hamiltonian simulation, i.e., approximately implementing the unitary operator \(U(t) = e^{-itH}\) via Trotterization. Note that this method will always simulate the hermitized operator, i.e.

\[H = (O + O^\dagger)/2\]

Parameters:

forward_evolution, bool, optional: If set to False \(U(t)^\dagger = e^{itH}\) will be executed (usefull for quantum phase estimation). The default is True.

Returns:

Ufunction

A Python function that implements the first order Suzuki-Trotter formula. Given a Hamiltonian \(H=H_1+\dotsb +H_m\) the unitary evolution \(e^{-itH}\) is approximated by

\[e^{-itH}\approx U(t,N)=\left(e^{-iH_1t/N}\dotsb e^{-iH_mt/N}\right)^N\]

This function recieves the following arguments:

qargQuantumVariable or QuantumArray
The quantum argument.
tfloat, optional
The evolution time \(t\). The default is 1.
stepsint, optional
The number of Trotter steps \(N\). The default is 1.
iterint, optional
The number of iterations the unitary \(U(t,N)\) is applied. The default is 1.

Examples

We simulate a simple FermionicOperator.

>>> from sympy import Symbol
>>> from qrisp.operators import a,c
>>> from qrisp import QuantumVariable
>>> O = a(0)*a(1) + a(2)
>>> U = O.trotterization()
>>> qv = QuantumVariable(3)
>>> t = Symbol("t")
>>> U(qv, t = t)
>>> print(qv.qs)
QuantumCircuit:
---------------
            ┌───┐             ┌───┐┌────────────┐┌───┐     ┌───┐      
qv.0: ────■─┤ X ├─────────────┤ X ├┤ Rz(-0.5*t) ├┤ X ├─────┤ X ├─■────
          │ └─┬─┘    ┌───┐    └─┬─┘├────────────┤└─┬─┘┌───┐└─┬─┘ │    
qv.1: ─■──┼───■──────┤ H ├──────■──┤ Rz(-0.5*t) ├──■──┤ H ├──■───┼──■─
       │  │ ┌───┐┌───┴───┴───┐┌───┐└────────────┘     └───┘      │  │ 
qv.2: ─■──■─┤ H ├┤ Rz(1.0*t) ├┤ H ├──────────────────────────────■──■─
            └───┘└───────────┘└───┘                                   
Live QuantumVariables:
----------------------
QuantumVariable qv

Execute a simulation:

>>> print(qv.get_measurement(subs_dic = {t : 0.5}))
{'000': 0.9242, '001': 0.06026, '110': 0.01459, '111': 0.00095}