Manuscript VII-J-ST@1 · VII-J-ST.tex · proposition · f1f1d82c

Proposition: Duhamel and product-formula bounds

Approximate simulation and the Hamiltonian comparison › Constructive finite simulation estimates · lines 1196–1235 · proof included or linked

VII-J-ST@1#prop:hamiltonian-errors, VII-J-ST@1#eq:trotter

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For bounded self-adjoint generators on a fixed finite space,

d_{⋄} (Ad_{U}, Ad_{U}) \leq U - U \leq \frac{1}{ℏ} \int_{s}^{t} H (r) - H (r) d r .

For time-independent $H = \sum_{j = 1}^{m} H_{j}$ , put

V_{n} (t) = (e^{- i t H_{m} / (n ℏ)} \dots e^{- i t H_{1} / (n ℏ)})^{n}, C_{H} = j < k \sum ∥ [H_{j}, H_{k}] ∥ .

Then

e^{- i t H /ℏ} - V_{n} (t) \leq \frac{t ^{2} C _{H}}{2 n ℏ ^{2}}, d_{⋄} (Ad_{e^{- i t H /ℏ}}, Ad_{V_{n} (t)}) \leq min {1, \frac{t ^{2} C _{H}}{2 n ℏ ^{2}}} .

Proof. Differentiate the mixed propagator $U (t, r) U (r, s)$ , integrate, and use unitarity. The channel comparison follows by expanding $U X U^{†} - V X V^{†}$ into two terms, also after amplification.

For two summands, the double integral comparing the joint exponential and ordered product has norm at most $δ^{2} ∥ [H_{1}, H_{2}] ∥ / (2 ℏ^{2})$ . Iterating over the finite sum gives one-step error $δ^{2} C_{H} / (2 ℏ^{2})$ . Telescoping $n$ unitary steps with $δ = t / n$ proves (eq:trotter).

Displayed formulas in this passage

VII-J-ST@1#eq:trotter · line 1212
$e^{- i t H /ℏ} - V_{n} (t) \leq \frac{t ^{2} C _{H}}{2 n ℏ ^{2}}, d_{⋄} (Ad_{e^{- i t H /ℏ}}, Ad_{V_{n} (t)}) \leq min {1, \frac{t ^{2} C _{H}}{2 n ℏ ^{2}}} .$
unlabeled · line 1199
$d_{⋄} (Ad_{U}, Ad_{U}) \leq U - U \leq \frac{1}{ℏ} \int_{s}^{t} H (r) - H (r) d r .$
unlabeled · line 1205
$V_{n} (t) = (e^{- i t H_{m} / (n ℏ)} \dots e^{- i t H_{1} / (n ℏ)})^{n}, C_{H} = j < k \sum ∥ [H_{j}, H_{k}] ∥ .$

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section: Constructive finite simulation estimates