Product-formula Hamiltonian simulation

e^{- i t (A + B)}

by alternating exponentials of

A

and

B

, which are assumed easier to implement separately.

First-order two-local Hamiltonian simulation ( $O (M + M^{3} t^{2} / ϵ)$ )

For a sum of

M

Hermitian two-local terms of norm below one, repeat a first-order Lie-Trotter product formula step

r

t^{2} \sum_{j < k} ∥ [H_{j}, H_{k}] ∥/ (2 r) < M (M - 1) t^{2} / (2 r)

for

M > 1

. Taking

r = max (1, ⌈ M (M - 1) t^{2} / (2 ϵ)⌉)

M r

arbitrary two-qubit gates. At unit time and

M = O (n^{2})

this is a sufficient

O (n^{6} / ϵ)

bound; additional commutation structure can improve it.

The symmetric second-order formula, also called Strang splitting, is

e^{- i δ A /2} e^{- i δ B} e^{- i δ A /2} = e^{- i δ (A + B)} + O (δ^{3})

for bounded operators, with the constant controlled by nested commutators.

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