Exponentiation

View on GitHub Experiment in the IDE

The exponentiation function produces a quantum gate that approximates the exponentiation, $\exp (- i H t)$ , of any input Hermitian operator, $H$ . The Classiq engine automatically generates an efficient higher-order Trotter-Suzuki quantum program [1] that minimizes the functional error and satisfies a given local constraint on the depth. The functional error is measured by the operator norm [2] and evaluated according to Ref. [3]

The Hamiltonian is given as any $n$ -qubit operator in its Pauli basis [4]:

H = \sum_{i} c_{i} [σ_{j_{1, i}} \otimes σ_{j_{2, i}} \otimes \dots \otimes σ_{j_{n, i}}],

where $σ_{0, 1, 2, 3} = I, X, Y, Z$ are the single-qubit Pauli operators, and $j \in {0, 1, 2, 3}$ , and $c_{i}$ are complex coefficients. For example, the operator $H = 0.1 \cdot I \otimes Z + 0.2 \cdot X \otimes Y$ is input as follows:

from classiq import Pauli, PauliTerm

pauli_list = [
    PauliTerm(pauli=[Pauli.I, Pauli.Z], coefficient=0.1),
    PauliTerm(pauli=[Pauli.X, Pauli.Y], coefficient=0.2),
]

Function: exponentiation_with_depth_constraint

Arguments:

pauli_operator: CArray[PauliTerm] - the Hamiltonian to exponentiate as described above,
evolution_coefficient: CReal - a global evolution coefficient (the parameter $t$ above),
max_depth: CInt - a maximal depth for the implementation,
qbv: QArray[QBit] - the quantum state on which we apply the evolution.

Example

from classiq import *


@qfunc
def main(qba: Output[QArray[QBit]]):

    allocate(4, qba)
    exponentiation_with_depth_constraint(
        [
            PauliTerm(pauli=[Pauli.X, Pauli.X, Pauli.I, Pauli.I], coefficient=0.1),
            PauliTerm(pauli=[Pauli.Y, Pauli.Y, Pauli.I, Pauli.I], coefficient=0.2),
            PauliTerm(pauli=[Pauli.Z, Pauli.Z, Pauli.Y, Pauli.X], coefficient=0.4),
            PauliTerm(pauli=[Pauli.I, Pauli.I, Pauli.I, Pauli.X], coefficient=0.4),
        ],
        evolution_coefficient=0.05,
        max_depth=50,
        qbv=qba,
    )


qmod = create_model(main, out_file="hamiltonian_evolution_exponentiation")
qprog = synthesize(qmod)

References

[1] N. Hatano and M. Suzuki, Finding Exponential Product Formulas of Higher Orders, https://arxiv.org/abs/math-ph/0506007 (2005).

[2] https://en.wikipedia.org/wiki/Operator_norm

[3] A. M. Childs et al, Toward the first quantum simulation with quantum speedup, https://arxiv.org/abs/1711.10980 (2017).

[4] https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices#Multi-qubit_Pauli_matrices_(Hermitian)