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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=∑k=1LαkHk,H = \sum^L_{k=1} \alpha_k H_k, the Suzuki-Trotter formula of order oo and repetitions rr approximates the Hamiltonian simulation U=e−iHtU = e^{-iHt} according to the following:
  • Each order ST(o)(H,t)ST^{(o)}(H,t) is defined recursively:
  • The first order is : ST(1)(H,t)=Πk=1Le−iHktST^{(1)}(H,t) = \Pi^L_{k=1} e^{-iH_k t}
  • The second order is : ST(2)(H,t)=Πk=1Le−iHkt/2Πk=L1e−iHkt/2ST^{(2)}(H,t) = \Pi^L_{k=1} e^{-iH_k t/2} \Pi^1_{k=L} e^{-iH_k t/2}
  • Recursion formula for order 2m2m with m>1m>1 is given in Eq. (5) of Ref. [2]
  • For a given order, repetitions refers to
ST(o,r)(H,t)=[ST(o)(H,t/r)]r.ST^{(o,r)}(H,t) = [ST^{(o)}(H,t/r)]^r. Function: suzuki_trotter Arguments:
  • pauli_operator: CArray[PauliTerm]
  • evolution_coefficient: CReal
  • order: CInt,
  • repetitions: CInt,
  • qbv: QArray[QBit]

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Example

from classiq import *


@qfunc
def main(x: CReal, qba: Output[QArray[QBit]]):
    allocate(3, qba)
    suzuki_trotter(
        [
            PauliTerm(pauli=[Pauli.X, Pauli.X, Pauli.Z], coefficient=1.5),
            PauliTerm(pauli=[Pauli.Y, Pauli.X, Pauli.Z], coefficient=0.5),
        ],
        evolution_coefficient=x,
        order=1,
        repetitions=1,
        qbv=qba,
    )


qmod = create_model(main)
qprog = synthesize(qmod)

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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