qrisp.operators.qubit.QubitOperator.trotterization#

QubitOperator.trotterization(method='commuting_qw', forward_evolution=True)[source]#

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

H = (O + O^{†}) / 2

Parameters:

methodstr, optional: The method for grouping the QubitTerms. Available are commuting (groups such that all QubitTerms mutually commute) and commuting_qw (groups such that all QubitTerms mutually commute qubit-wise). The default is commuting_qw.
forward_evolutionbool, optional: If set to False $U (t)^{†} = e^{i t H}$ 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 $H = H_{1} + \dots + H_{m}$ the unitary evolution $e^{- i t H}$ is approximated by

e^{- i t H} \approx U (t, N) = {(e^{- i H_{1} t / N} \dots e^{- i H_{m} t / N})}^{N}

This function receives the following arguments:

qargQuantumVariable
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 QubitOperator.

>>> from sympy import Symbol
>>> from qrisp.operators import A,C,Z,Y
>>> from qrisp import QuantumVariable
>>> O = A(0)*C(1)*Z(2) + Y(3)
>>> U = O.trotterization()
>>> qv = QuantumVariable(4)
>>> t = Symbol("t")
>>> U(qv, t = t)
>>> print(qv.qs)
QuantumCircuit:
---------------
      ┌───┐                      ┌───┐┌────────────┐┌───┐          ┌───┐
qv.0: ┤ X ├──────────────────────┤ X ├┤ Rz(-0.5*t) ├┤ X ├──────────┤ X ├
      └─┬─┘     ┌───┐     ┌───┐  └─┬─┘├───────────┬┘└─┬─┘┌───┐┌───┐└─┬─┘
qv.1: ──■───────┤ H ├─────┤ X ├────■──┤ Rz(0.5*t) ├───■──┤ X ├┤ H ├──■──
                └───┘     └─┬─┘       └───────────┘      └─┬─┘└───┘     
qv.2: ──────────────────────■──────────────────────────────■────────────
      ┌────┐┌───────────┐┌──────┐                                       
qv.3: ┤ √X ├┤ Rz(2.0*t) ├┤ √Xdg ├───────────────────────────────────────
      └────┘└───────────┘└──────┘                                       
Live QuantumVariables:
----------------------
QuantumVariable qv

Execute a simulation:

>>> print(qv.get_measurement(subs_dic = {t : 0.5}))
{'0000': 0.77015, '0001': 0.22985}