Manuscript Oai265-A@2 · Oai265-A.tex · proposition · c0a34a45

Proposition: Energy accuracy needed for vector accuracy

Stage 3: PEPS existence, algorithms, and approximation budgets › The separate algorithmic interface · lines 930–955 · proof included or linked

Oai265-A@2#prop:energy

For a normalized $ϕ$ and a Hamiltonian obeying (eq:gap), put $η = ⟨ ϕ, H ϕ ⟩ - E_{0}$ . Then

1 - ∣ ⟨ Ω, ϕ ⟩ ∣^{2} \leq η /Δ,

and

θ min ∥ ϕ - e^{i θ} Ω ∥^{2} \leq 2 η /Δ.

In particular, the sufficient total-energy tolerance for vector error $L^{- 1}$ is $η \leq Δ/ (2 L^{2})$ .

Proof. Take the expectation of (eq:gap). If $F = ∣ ⟨ Ω, ϕ ⟩ ∣$ , then

θ min ∥ ϕ - e^{i θ} Ω ∥^{2} = 2 (1 - F) \leq 2 (1 - F^{2}) .

Displayed formulas in this passage

unlabeled · line 936
$1 - ∣ ⟨ Ω, ϕ ⟩ ∣^{2} \leq η /Δ,$
unlabeled · line 940
$θ min ∥ ϕ - e^{i θ} Ω ∥^{2} \leq 2 η /Δ.$
unlabeled · line 951
$θ min ∥ ϕ - e^{i θ} Ω ∥^{2} = 2 (1 - F) \leq 2 (1 - F^{2}) .$