Suzuki Trotter

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The suzuki_trotter function produces the Suzuki-Trotter product for a given order and repetitions. Given a Hamiltonian as a sum of Pauli strings

H = \sum_{k = 1}^{L} α_{k} H_{k},

the Suzuki-Trotter formula of order $o$ and repetitions $r$ approximates the Hamiltonian simulation $U = e^{- i H t}$ according to the following:

Each order $S T^{(o)} (H, t)$ is defined recursively:
- The first order is : $S T^{(1)} (H, t) = Π_{k = 1}^{L} e^{- i H_{k} t}$
- The second order is : $S T^{(2)} (H, t) = Π_{k = 1}^{L} e^{- i H_{k} t / 2} Π_{k = L}^{1} e^{- i H_{k} t / 2}$
- Recursion formula for order $2 m$ with $m > 1$ is given in Eq. (5) of Ref. [2]
For a given order, repetitions refers to

S T^{(o, r)} (H, t) = [S T^{(o)} (H, t / r)]^{r} .

Function: suzuki_trotter

Arguments:

pauli_operator: CArray[PauliTerm]
evolution_coefficient: CReal
order: CInt,
repetitions: CInt,
qbv: QArray[QBit]

Example

from classiq import *


@qfunc
def main(a: CReal, x: CReal, qba: Output[QArray[QBit]]):
    allocate(3, qba)
    suzuki_trotter(
        [
            PauliTerm(pauli=[Pauli.X, Pauli.X, Pauli.Z], coefficient=a),
            PauliTerm(pauli=[Pauli.Y, Pauli.X, Pauli.Z], coefficient=0.5),
        ],
        evolution_coefficient=x,
        order=1,
        repetitions=1,
        qbv=qba,
    )


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

References

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

[2] Childs, et al., Toward the first quantum simulation with quantum speedup, (2018). https://arxiv.org/abs/1711.10980