SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The function  $t \mapsto g_t$  as the diffusion coefficient in the forward Fokker-Planck equation (1), in fact, plays a role as time re-scaling as long as  $g_t > 0$  for any  $t$ . Indeed, if we have an SDE in the following
form (the VP-SDE in [30])
$$
\mathrm {d} X _ {t} = - \frac {g _ {t} ^ {2}}{2} X _ {t} \mathrm {d} t + g _ {t} \mathrm {d} W _ {t}, \qquad t \in [ 0, T ],
$$
then by introducing  $\tau: \mathbb{R}^+ \to \mathbb{R}^+$  via the ODE  $\tau'(t) = (g_{\tau(t)})^{-2}$ ,  $\tau(0) = 0$ , we know that  $\underline{\mathbf{X}}_t := X_{\tau(t)}$  satisfies the following SDE [20]:
$$
\mathrm {d} \underline {{\mathbf {X}}} _ {t} = - \frac {1}{2} \underline {{\mathbf {X}}} _ {t} \mathrm {d} t + \mathrm {d} W _ {t}, \qquad t \in [ 0, \theta ],
$$
where  $\theta \coloneqq \tau^{-1}(T)$  means the inverse function of  $\tau$  at time  $T$ . The Brownian motion  $W$  in  $\underline{\mathbf{X}}_t$  is not the same Brownian motion in  $X_{t}$ , i.e., the driven-noise in the last two equations are not the same technically; we slightly abuse the notation for the simplicity of notations. By Lem. B.1, this SDE has the backward process as follows:
$$
\begin{array}{l} \mathrm {d} \underline {{\mathbf {Y}}} _ {t} \stackrel {(3)} {=} \left(\frac {1}{2} \underline {{\mathbf {Y}}} _ {t} + \frac {1 + \underline {{\mathbf {h}}} _ {t} ^ {\leftarrow 2}}{2} \nabla \log \mathfrak {p} _ {t} ^ {\leftarrow} (\underline {{\mathbf {Y}}} _ {t})\right) \mathrm {d} t + \underline {{\mathbf {h}}} _ {t} ^ {\leftarrow} \mathrm {d} W _ {t} \quad t \in [ 0, \theta ] \tag {20} \\ = \left(\frac {1}{2} \underline {{\mathbf {Y}}} _ {t} + \frac {1 + \underline {{\mathbf {h}}} _ {\theta - t} ^ {2}}{2} \nabla \log \mathbf {p} _ {\theta - t} (\underline {{\mathbf {Y}}} _ {t})\right) \mathrm {d} t + \underline {{\mathbf {h}}} _ {\theta - t} \mathrm {d} W _ {t}, \\ \end{array}
$$
where  $\mathfrak{p}_t$  is the density function of  $\underline{\mathbf{X}}_t$  by notation conventions in  $\S 2$ . By Lem. B.1, so far we know that
$$
\operatorname {l a w} \left(\underline {{\mathrm {Y}}} _ {\theta - t}\right) \stackrel {\text {L e m . B . 1}} {=} \operatorname {l a w} \left(\underline {{\mathrm {X}}} _ {t}\right) = \operatorname {l a w} \left(X _ {\tau (t)}\right).
$$
Let  $f_{s} \coloneqq \theta - (\tau^{-1})(T - s)$  and  $Y_{s} \coloneqq \underline{Y}_{f_{s}}$ , where  $s \in [0, T]$ . By straightforward computation, we know
$$
\begin{array}{l} \mathrm {d} Y _ {s} = f _ {s} ^ {\prime} \Big (\frac {1}{2} \underline {{\mathbf {Y}}} _ {f _ {s}} + \frac {1 + \underline {{\mathbf {h}}} _ {\theta - f _ {s}} ^ {2}}{2} \nabla \log \underline {{\mathbf {p}}} _ {\theta - f _ {s}} (\underline {{\mathbf {Y}}} _ {f _ {s}}) \Big) \mathrm {d} t + \sqrt {f _ {s} ^ {\prime} \underline {{\mathbf {h}}} _ {\theta - f _ {s}} ^ {2}} \mathrm {d} W _ {s} \\ = f _ {s} ^ {\prime} \left(\frac {1}{2} Y _ {s} + \frac {1 + \underline {{\mathrm {h}}} _ {\theta - f _ {s}} ^ {2}}{2} \nabla \log p _ {T - s} (Y _ {s})\right) \mathrm {d} t + \sqrt {f _ {s} ^ {\prime} \underline {{\mathrm {h}}} _ {\theta - f _ {s}} ^ {2}} \mathrm {d} W _ {s}. \\ \end{array}
$$
To get the second line, we used the fact that
$$
\mathbf {p} _ {\theta - f _ {s}} = \mathrm {l a w} \big (\underline {{\mathbf {X}}} _ {\theta - f _ {s}} \big) = \mathrm {l a w} \big (\underline {{\mathbf {X}}} _ {\tau^ {- 1} (T - s)} \big) = \mathrm {l a w} \big (X _ {T - s} \big) \equiv p _ {T - s}.
$$
By chain rules, it is easy to compute that
$$
f _ {s} ^ {\prime} = \frac {1}{\tau^ {\prime} (\tau^ {- 1} (T - s))} = \frac {1}{(g _ {T - s}) ^ {- 2}} = g _ {T - s} ^ {2} > 0.
$$
Therefore, the above SDE for  $Y_{s}$  has the form
$$
\mathrm {d} Y _ {s} = \left(\frac {g _ {T - s} ^ {2}}{2} Y _ {s} + g _ {T - s} ^ {2} \frac {1 + \underline {{\mathbf {h}}} _ {\theta - f _ {s}} ^ {2}}{2} \nabla \log p _ {s} ^ {\leftarrow} (Y _ {s})\right) \mathrm {d} t + g _ {T - s} \underline {{\mathbf {h}}} _ {\theta - f _ {s}} \mathrm {d} W _ {s}.
$$
This matches the form in Lem. B.1 by choosing
$$
h _ {s} ^ {\leftarrow} = g _ {T - s} \underline {{\mathrm {h}}} _ {\theta - f _ {s}} \equiv g _ {s} ^ {\leftarrow} \underline {{\mathrm {h}}} _ {\tau^ {- 1} (T - s)}.
$$
Hence, if we simply pick  $\underline{\mathbf{h}} = c$  as a constant function in (20), it has the same effect as choosing  $h_s^{\leftarrow} = cg_s^{\leftarrow}$  where  $c\in \mathbb{R}^{+}$ . The former  $(\underline{\mathbf{h}} = c)$  is used in § 3 for simplicity in theoretical analysis, and the latter  $(h_s^{\leftarrow} = cg_s^{\leftarrow})$  is used in numerical experiments in § 4 to match previous literature in practice (namely, a general  $g$ ). In conclusion, the above discussion justifies the consistency of notation and set-up between our theoretical analysis and numerical experiments.
B.3 Existing analysis of sample generation quality
In practice, one simulates the following SDE:
$$
\mathrm {d} Z _ {t} = \left(- f _ {t} ^ {\leftarrow} \left(Z _ {t}\right) + \frac {\left(g _ {t} ^ {\leftarrow}\right) ^ {2} + \left(h _ {t} ^ {\leftarrow}\right) ^ {2}}{2} \mathfrak {S} _ {t} ^ {\leftarrow} \left(Z _ {t}\right)\right) \mathrm {d} t + h _ {t} ^ {\leftarrow} \mathrm {d} W _ {t}, \quad \text {l a w} \left(Z _ {0}\right) = \mathcal {N} \left(0, I _ {d}\right). \tag {21}
$$
From the analysis by Chen et al. [7, Theorem 2.1] for the case  $h^{\leftarrow} = g^{\leftarrow}$ , one has
$$
\operatorname {K L} \left(p _ {0} | | \widehat {q}\right) \leq \operatorname {K L} \left(p _ {T} | | \mathcal {N} (0, I _ {d})\right) + \mathcal {O} \left(T \epsilon^ {2}\right) + \mathcal {O} \left(T ^ {2} d / N\right),
$$
where  $\widehat{q}$  is the distribution of  $Z_{T}$  after applying the exponential integrator to (21) and  $N$  is the number of time-discretization steps. The first term  $\mathrm{KL}\big(p_T||\mathcal{N}(0,I_d)\big)\leq (\mathcal{M}_2 + d)e^{-T}$ , where  $\mathcal{M}_2 = \mathbb{E}_{p_0}|x|^2$  is the second moment of data distribution.
Therefore, to ensure that  $\mathrm{KL}\big(p_0||\widehat{q}\big)\leq \delta$  , it is sufficient to choose
$$
T = \mathcal {O} \left(\log \left((\mathcal {M} _ {2} + d) / \delta\right)\right).
$$
The time-discretization error can be eliminated by choosing  $N \to \infty$ . What is so far less clear in literature is the term  $\mathcal{O}\big(T\epsilon^2\big)$ .
B.4 Existing error bounds appear to fail to characterize the optimal  $h^{\leftarrow}$
Note that the target dynamics  $q_{t}$  in (2) and the approximated dynamics  $\widetilde{q}_{t}$  in (7) only differ in the drift term. Recall that  $q_{T} = p_{0}$ , and to estimate  $\mathrm{KL}\big(p_0||\widetilde{q}_T\big)\equiv \mathrm{KL}\big(q_T||\widetilde{q}_T\big)$ , we can simply estimate how the quantity  $\mathrm{KL}\big(q_t||\widetilde{q}_t\big)$  changes for  $t\in (0,T)$ . By [7, Lemma C.1], for any  $t\in \mathbb{R}$
$$
\partial_ {t} \mathrm {K L} \left(q _ {t} \| \widetilde {q} _ {t}\right) = - \frac {h _ {t} ^ {\leftarrow 2}}{2} \mathcal {J} \left(q _ {t} \| \widetilde {q} _ {t}\right) + \frac {g _ {t} ^ {\leftarrow 2} + h _ {t} ^ {\leftarrow 2}}{2} \mathbb {E} \left[ \left\langle - \epsilon \mathcal {E} _ {t} ^ {\leftarrow} \left(Y _ {t}\right), \nabla \log \frac {q _ {t} \left(Y _ {t}\right)}{\widetilde {q} _ {t} \left(Y _ {t}\right)} \right\rangle \right], \tag {22}
$$
where the Fisher information  $\mathcal{J}(p||q) \coloneqq \int \mathrm{d}p \left| \nabla \log \frac{p}{q} \right|^2$ . By Cauchy-Schwarz inequality,
$$
\begin{array}{l} \partial_ {t} \mathrm {K L} (q _ {t} | | \widetilde {q} _ {t}) \\ \leq - \frac {h _ {t} ^ {\leftarrow 2}}{2} \mathcal {J} \left(q _ {t} \| \widetilde {q} _ {t}\right) + \frac {g _ {t} ^ {\leftarrow 2} + h _ {t} ^ {\leftarrow 2}}{2} \left(\frac {c ^ {2}}{2} \mathbb {E} \left[ \| \epsilon \mathcal {E} _ {t} ^ {\leftarrow} (Y _ {t}) \| ^ {2} \right] + \frac {1}{2 c ^ {2}} \mathbb {E} \left[ \left\| \nabla \log \frac {q _ {t} (Y _ {t})}{\widetilde {q} _ {t} (Y _ {t})} \right\| ^ {2} \right]\right) \tag {23} \\ = \frac {\left(h _ {t} ^ {\leftarrow 2} + g _ {t} ^ {\leftarrow 2}\right) ^ {2}}{8 h _ {t} ^ {\leftarrow 2}} \epsilon^ {2} \mathbb {E} \left[ \left\| \mathcal {E} _ {t} ^ {\leftarrow} \left(Y _ {t}\right) \right\| ^ {2} \right], \\ \end{array}
$$
where we chose  $c^2 = \frac{h_t^{-2} + g_t^{-2}}{2h_t^{-2}}$  in the last line. This bound captures the scaling extremely well when  $h^{\leftarrow} = g^{\leftarrow}$ , which helps to establish the effectiveness of score-based diffusion method in [7]. However, this bound is not directly applicable for a general  $h^{\leftarrow}$ , as it is clear that this bound fails to provide useful information when  $h^{\leftarrow} \approx 0$  (namely, the probability flow), as well as the large diffusion case ( $h^{\leftarrow} \rightarrow \infty$ ). From directly optimizing the upper bound, i.e., minimizing  $\frac{(h_t^{-2} + g_t^{-2})^2}{8h_t^{-2}}$ , one ends up with the choice  $h^{\leftarrow} = g^{\leftarrow}$ , which has been used in many literature. We acknowledge that  $h^{\leftarrow} = g^{\leftarrow}$  is an effective choice; however, as we show in §4, this is not really the optimal case in general, and the above argument cannot justify choosing the diffusion model over the probability flow.
C Discussion for § 3 and proof of Prop. 3.2
C.1 The operator  $\Phi_{s,t}^{(h^{\leftarrow})}$
-  $\Phi_{(\cdot),(\cdot)}^{(h\leftarrow)}$  satisfies the following property, i.e., for any  $s,t,r\in \mathbb{R}$
$$
\Phi_ {t, r} ^ {(h ^ {\leftarrow})} \circ \Phi_ {s, t} ^ {(h ^ {\leftarrow})} = \Phi_ {s, r} ^ {(h ^ {\leftarrow})}. \tag {24}
$$
Moreover,  $\Phi_{t,t}^{(h^{\leftarrow})} = Id$  is an identity operator for any  $t$ .
- For any  $s, t \in \mathbb{R}$ ,
$$
\partial_ {s} \left(\Phi_ {s, t} ^ {(h ^ {\leftarrow})} (\mu)\right) = - \Phi_ {s, t} ^ {(h ^ {\leftarrow})} \left(\mathcal {L} _ {s} ^ {(h ^ {\leftarrow})} \mu\right) \quad \partial_ {t} \left(\Phi_ {s, t} ^ {(h ^ {\leftarrow})} (\mu)\right) = \mathcal {L} _ {t} ^ {(h ^ {\leftarrow})} \left(\Phi_ {s, t} ^ {(h ^ {\leftarrow})} (\mu)\right). \tag {25}
$$
Proof. The structure in (24) is easy to imagine and is thus omitted herein. Next, we shall only prove the first result in (25) for illustration:
$$
\begin{array}{l} \partial_ {s} \big (\Phi_ {s, t} ^ {(h ^ {\leftarrow})} (\mu) \big) = \lim _ {\delta s \rightarrow 0 ^ {+}} \frac {\Phi_ {s + \delta s , t} ^ {(h ^ {\leftarrow})} (\mu) - \Phi_ {s , t} ^ {(h ^ {\leftarrow})} (\mu)}{\delta s} \\ \stackrel {(2 4)} {=} \lim  _ {\delta s \to 0 ^ {+}} \frac {\Phi_ {s , t} ^ {(h ^ {\leftarrow})} \Phi_ {s + \delta s , s} ^ {(h ^ {\leftarrow})} (\mu) - \Phi_ {s , t} ^ {(h ^ {\leftarrow})} (\mu)}{\delta s} \\ = \lim  _ {\delta s \rightarrow 0 ^ {+}} \frac {\Phi_ {s , t} ^ {(h ^ {\leftarrow})} \left(\Phi_ {s + \delta s , s} ^ {(h ^ {\leftarrow})} - \operatorname {I d}\right) (\mu)}{\delta s} \\ = \lim  _ {\delta s \rightarrow 0 ^ {+}} \frac {\Phi_ {s , t} ^ {(h ^ {\leftarrow})} \left(\operatorname {I d} - \delta s \mathcal {L} _ {s} ^ {(h ^ {\leftarrow})} - \operatorname {I d}\right) (\mu)}{\delta s} \\ = - \Phi_ {s, t} ^ {(h ^ {\leftarrow})} \bigl (\mathcal {L} _ {s} ^ {(h ^ {\leftarrow})} (\mu) \bigr). \\ \end{array}
$$
Lemma C.2. Suppose  $(t,x)\mapsto \mu_t(x)$ ,  $(t,x)\mapsto \theta_t(x)$  are time-dependent functions. Suppose  $\partial_t\mu_t = \mathcal{L}_t^{(h^{\leftarrow})}(\mu_t) + \theta_t$  with  $\mu_0 = 0$ , then
$$
\mu_ {T} = \int_ {0} ^ {T} \Phi_ {t, T} ^ {(h ^ {\leftarrow})} (\theta_ {t})   \mathrm {d} t.
$$
$$
\partial_ {t} \left(\Phi_ {t, 0} ^ {(h ^ {\leftarrow})} \mu_ {t}\right) \stackrel {{(2 5)}} {{=}} \Phi_ {t, 0} ^ {(h ^ {\leftarrow})} (- \mathcal {L} _ {t} ^ {(h ^ {\leftarrow})} \mu_ {t}) + \Phi_ {t, 0} ^ {(h ^ {\leftarrow})} (\mathcal {L} _ {t} ^ {(h ^ {\leftarrow})} \mu_ {t} + \theta_ {t}) = \Phi_ {t, 0} ^ {(h ^ {\leftarrow})} (\theta_ {t}).
$$
$$
\Phi_ {T, 0} ^ {(h \leftarrow)} (\mu_ {T}) = \int_ {0} ^ {T} \Phi_ {t, 0} ^ {(h \leftarrow)} (\theta_ {t})   \mathrm {d} t.
$$
By applying the operator  $\Phi_{0,T}^{(h\leftarrow)}$  to both sides, we have
$$
\mu_ {T} = \int_ {0} ^ {T} \Phi_ {0, T} ^ {(h \leftarrow)} \circ \Phi_ {t, 0} ^ {(h \leftarrow)} (\theta_ {t})   \mathrm {d} t \stackrel {{(2 4)}} {{=}} \int_ {0} ^ {T} \Phi_ {t, T} ^ {(h \leftarrow)} (\theta_ {t})   \mathrm {d} t.
$$
C.2 Time re-scaling of  $\mathcal{L}^{(h\leftarrow)}$  and connections to almost quasi-static Langevin process
By time re-scaling, it is immediate to obtain the following result:
Lemma C.3. Suppose  $h_t^{\leftarrow} = \mathfrak{h}$  for any  $t \in [0, T]$  for simplicity. For any probability distribution  $\mu$ , the probability distribution  $\Phi_{s,t}^{(h^{\leftarrow})}(\mu)$  (with  $s < t$ ) is the final state of the following PDE on the time interval  $\widetilde{r} \in \left[0, (t - s)\mathfrak{h}^2 / 2\right]$ :
$$
\partial_ {\tilde {r}} \widetilde {\mu} _ {\tilde {r}} = \triangle \widetilde {\mu} _ {\tilde {r}} + \boldsymbol {\nabla} \cdot \left(\nabla V _ {s + 2 \tilde {r} / h ^ {2}} ^ {\leftarrow} \widetilde {\mu} _ {\tilde {r}}\right), \quad \widetilde {\mu} _ {0} = \mu . \tag {26}
$$
It corresponds to the following Langevin with time-dependent potential:
$$
\mathrm {d} X _ {\tilde {r}} = - \nabla V _ {s + 2 \tilde {r} / \mathrm {h} ^ {2}} ^ {\leftarrow} \left(X _ {\tilde {r}}\right) \mathrm {d} \tilde {r} + \sqrt {2} \mathrm {d} W _ {\tilde {r}}, \quad l a w \left(X _ {0}\right) = \mu .
$$
Proof. To obtain  $\Phi_{s,t}^{(h^{\leftarrow})}(\mu)$ , we simply solve the following PDE:
$$
\partial_ {r} \mu_ {r} \stackrel {(1 1)} {=} \frac {\mathrm {h} ^ {2}}{2} \left(\triangle \mu_ {r} + \nabla \cdot \left(\nabla V _ {r} ^ {\leftarrow} \mu_ {r}\right) \quad \mu_ {s} = \mu . \right.
$$
By change of variables  $\widetilde{r} = (r - s)^{\mathrm{h}^2 /2}$  and  $\widetilde{\mu}_{\widetilde{r}}\coloneqq \mu_r\equiv \mu_{s + 2\widetilde{r} /\mathrm{h}^2}$ , we immediately have (26), and the corresponding Langevin dynamics easily follows.
When  $h \to \infty$ , the potential in the Langevin dynamics  $\widetilde{r} \mapsto V_{s+2\widetilde{r}/h^2}^{\leftarrow}$  evolves extremely slowly. From (11), we also know that  $V_{s+2\widetilde{r}/h^2}^{\leftarrow} \approx U_s^{\leftarrow}$  when  $\widetilde{r} = 0$  and is approximately  $U_t^{\leftarrow}$  when  $\widetilde{r} = \frac{(t - s)h^2}{2}$ . Therefore, the Langevin dynamics can be viewed as an almost quasi-static process [6] approximately transforming the state  $p_s^{\leftarrow}$  to the state  $p_t^{\leftarrow}$  over an extremely long time period though.
C.3 Proof of Prop. 3.2
$$
v _ {t} (x) := \partial_ {\epsilon} q _ {t} ^ {\epsilon} (x) | _ {\epsilon = 0}, \quad \zeta_ {t} (x) := \partial_ {\epsilon} ^ {2} q _ {t} ^ {\epsilon} (x) | _ {\epsilon = 0}.
$$
Namely, we expand  $\widetilde{q}_t^\epsilon$  via the following
$$
\widetilde {q} _ {t} ^ {\epsilon} (x) = \underbrace {\widetilde {q} _ {t} ^ {0} (x)} _ {\equiv q _ {t} (x)} + \epsilon v _ {t} (x) + \frac {\epsilon^ {2}}{2} \zeta_ {t} (x) + \mathcal {O} \big (\epsilon^ {3} \big).
$$
Recall that when  $\epsilon = 0$ ,  $\widetilde{q}_t^0 \equiv q_t$ ; in particular,  $\widetilde{q}_T^0 = q_T^{\mathrm{Lem.B.1}} = p_0$ . The cost function can be easily expanded via Taylor's formula:
$$
\mathrm {K L} \left(p _ {0} | | \widetilde {q} _ {T} ^ {\epsilon}\right) = - \epsilon \int_ {\mathbb {R} ^ {d}} v _ {T} (x) \mathrm {d} x - \frac {\epsilon^ {2}}{2} \int_ {\mathbb {R} ^ {d}} \left(\zeta_ {T} (x) - \frac {v _ {T} ^ {2} (x)}{p _ {0} (x)}\right) \mathrm {d} x + \mathcal {O} \left(\epsilon^ {3}\right).
$$
Since  $\int \widetilde{q}_T^\epsilon \equiv 1$ , it is easy to know that  $\int v_{T} = \int \zeta_{T} = 0$  and thus
$$
\mathrm {K L} \left(p _ {0} | | \widetilde {q} _ {T} ^ {\epsilon}\right) = \frac {\epsilon^ {2}}{2} \int_ {\mathbb {R} ^ {d}} \frac {v _ {T} ^ {2} (x)}{p _ {0} (x)} \mathrm {d} x + \mathcal {O} \left(\epsilon^ {3}\right).
$$
Next we need to study  $v_{T}$ . Recall from (7) that the Fokker-Planck equation of  $\widetilde{q}_t$  is
$$
\partial_ {t} \widetilde {q} _ {t} ^ {\epsilon} = \mathcal {L} _ {t} ^ {(h ^ {\leftarrow})} (\widetilde {q} _ {t} ^ {\epsilon}) - \epsilon \boldsymbol {\nabla} \cdot \big (\frac {1 + h _ {t} ^ {\leftarrow 2}}{2} \mathcal {E} _ {t} ^ {\leftarrow} \widetilde {q} _ {t} ^ {\epsilon} \big).
$$
By taking derivatives with respect to  $\epsilon$  on both sides of this equation and then passing  $\epsilon \to 0$ ,
$$
\partial_ {t} v _ {t} = \mathcal {L} _ {t} ^ {(h ^ {\leftarrow})} (v _ {t}) - \boldsymbol {\nabla} \cdot \left(\frac {1 + h _ {t} ^ {\leftarrow 2}}{2} \mathcal {E} _ {t} ^ {\leftarrow} q _ {t}\right).
$$
By Lem. C.2,
$$
v _ {T} = - \frac {1}{2} \int_ {0} ^ {T} \left(1 + h _ {t} ^ {\leftarrow 2}\right) \Phi_ {t, T} ^ {\left(h ^ {\leftarrow}\right)} \left(\boldsymbol {\nabla} \cdot \left(\underbrace {q _ {t}} _ {\equiv p _ {t} ^ {\leftarrow}} \mathcal {E} _ {t} ^ {\leftarrow}\right)\right) \mathrm {d} t. \tag {27}
$$
D Preliminary results
We collect two lemmas which will be useful in proving Prop. 3.5 and Prop. 3.6 later.
1. The operator  $\mathcal{K}(f) \coloneqq \triangle f + \nabla \cdot (\nabla V \cdot f)$  is a Hermitian/self-adjoint operator in the space  $\langle \cdot, \cdot \rangle_{L^2(\rho^{-1})}$  where  $\rho \propto e^{-V}$ . Therefore,  $\mathcal{K}$  has the eigen-decomposition in this weighted  $L^2(\rho^{-1})$  space.
2. Assume that  $\nabla^2 V(x) \geq m\mathbf{I}_d$  for any  $x \in \mathbb{R}^d$  for some  $m > 0$ . Then the operator  $-\mathcal{K}$  is a positive operator on the space  $\{f : \int f = 0\}$  with spectrum gap at least  $m$ .
Proof. For any  $f, g$ , note that
$$
\langle f, \mathcal {K} g \rangle_ {L ^ {2} (\rho^ {- 1})} = - \int_ {\mathbb {R} ^ {d}} \nabla (f / \rho) \cdot \nabla (g / \rho) \rho .
$$
Therefore,  $\mathcal{K}$  is a Hermitian operator. In the space  $\{f:\int f = 0\}$ , we know
$$
\begin{array}{l} - \langle f, \mathcal {K} f \rangle_ {L ^ {2} (\rho^ {- 1})} = \int_ {\mathbb {R} ^ {d}} | \nabla (f / \rho) | ^ {2} \rho \\ \overset {\text {P o i n c a r e i n e q .}} {\geq} m \int_ {\mathbb {R} ^ {d}} \left(f ^ {2} / \rho^ {2}\right) \rho = m \langle f, f \rangle_ {L ^ {2} \left(\rho^ {- 1}\right)}. \\ \end{array}
$$
The validity of Poincaré inequality under the strong convexity assumption is well-known in literature; see, e.g., [2, 24] and [3, Corollary 4.8.2].
Lemma D.2. Suppose the operator  $\mathcal{K}$  is defined as  $\mathcal{K}(f)\coloneqq \triangle f + \nabla \cdot (\nabla V\cdot (f))$  with  $\nabla^2 V(x)\geq m\pmb {I}_d$  for any  $x\in \mathbb{R}^d$ . Then for any  $c,a\in \mathbb{R}^{+}$ , and any function  $\varphi$  with  $\int \varphi = 0$ ,
$$
\begin{array}{l} \int_ {\mathbb {R} ^ {d}} \frac {1}{\rho} \Big (\int_ {0} ^ {a} \exp (c t \mathcal {K}) (\varphi) \mathrm {d} t \Big) ^ {2} = \sum_ {k = 1} ^ {\infty} \left(\frac {1 - e ^ {- a c \lambda_ {k}}}{c \lambda_ {k}}\right) ^ {2} \alpha_ {k} ^ {2} \\ \leq \left(\frac {1 - e ^ {- a c m}}{c m}\right) ^ {2} \int_ {\mathbb {R} ^ {d}} \frac {\varphi^ {2}}{\rho}, \\ \end{array}
$$
where  $\rho \propto e^{-V}$ ,  $\{(\lambda_k,\phi_k)\}_{k = 1}^{\infty}$  are eigen pairs for the operator  $-\mathcal{K}$ , and  $\varphi = \sum_{k = 1}^{\infty}\alpha_{k}\phi_{k}$ .
Remark. Note that as  $\mathcal{K}$  is an operator,  $\exp (ct\mathcal{K})\coloneqq \sum_{k = 0}^{\infty}\frac{(ct\mathcal{K})^k}{k!}$  is the operator exponential.
Proof. Let us first denote the eigenvalue decomposition of  $\mathcal{K}$  as  $\mathcal{K}(\phi_k) = -\lambda_k\phi_k$  where  $\lambda_{k}\geq m$  for  $k\in \mathbb{N}$  and  $\langle \phi_j,\phi_k\rangle_{L^2 (\rho^{-1})} = \delta_{j,k}$  for any  $j,k\in \mathbb{N}$  by Lem. D.1. Then we can decompose  $\varphi$  by  $\varphi = \sum_{k}\alpha_{k}\phi_{k}$ . It is not hard to verify that
$$
\exp \left(c t \mathcal {K}\right) (\varphi) = \sum_ {k = 1} ^ {\infty} e ^ {- c t \lambda_ {k}} \alpha_ {k} \phi_ {k}.
$$
$$
\int_ {0} ^ {a} \exp (c t \mathcal {K}) (\varphi) d t = \sum_ {k = 1} ^ {\infty} \frac {1 - e ^ {- a c \lambda_ {k}}}{c \lambda_ {k}} \alpha_ {k} \phi_ {k}.
$$
$$
\begin{array}{l} \int_ {\mathbb {R} ^ {d}} \frac {1}{\rho} \left(\int_ {0} ^ {a} \exp (c t \mathcal {K}) (\varphi) d t\right) ^ {2} = \int_ {\mathbb {R} ^ {d}} \frac {1}{\rho} \left(\sum_ {k = 1} ^ {\infty} \frac {1 - e ^ {- a c \lambda_ {k}}}{c \lambda_ {k}} \alpha_ {k} \phi_ {k}\right) ^ {2} \\ = \sum_ {k = 1} ^ {\infty} \left(\frac {1 - e ^ {- a c \lambda_ {k}}}{c \lambda_ {k}}\right) ^ {2} \alpha_ {k} ^ {2} \\ \leq \left(\frac {1 - e ^ {- a c m}}{c m}\right) ^ {2} \sum_ {k = 1} ^ {\infty} \alpha_ {k} ^ {2} \\ = \left(\frac {1 - e ^ {- a c m}}{c m}\right) ^ {2} \int_ {\mathbb {R} ^ {d}} \frac {\varphi^ {2}}{\rho}. \\ \end{array}
$$
Proof of Prop. 3.4, Prop. 3.5, and a discussion on the pulse-shape error
E.1 Proof of Prop. 3.4
For convenience, we summarize some notations below; see also Appx. A.
- Denote the global minimum of  $U_{t}^{\leftarrow}$  as  $\mathcal{X}_t^{\leftarrow}$  and denote the global minimum of  $V_{t}^{\leftarrow}$  as  $\mathcal{Y}_t^{\leftarrow}$ . When  $h\to \infty$ , we know
$$
\lim_{\mathfrak{h}\to \infty}\mathcal{Y}_{t}\to \mathcal{X}_{t}.
$$
- Denote the normalizing constant  $Z_V \coloneqq \int e^{-V}$  for an arbitrary potential  $V$ .
- Recall that the distribution of the exact dynamics  $Y_{t}$  is  $q_{t} \equiv p_{T - t} \equiv p_{t}^{\leftarrow}$  and the distribution of the approximated dynamics  $\widetilde{Y}_{t}$  is  $\widetilde{q}_{t}$ .
- Recall that  $p_t \coloneqq e^{-U_t}$  and  $\rho_t \propto e^{-V_t}$  in (12).
Lemma E.1. Under Assumption 3.3, we have:
(i) Assumption 3.1 (3) is valid, i.e., the existence of  $c_{U}$ .
(ii) For any  $t \in [0, T]$ , the probability distribution  $\rho_t$  defined in (12) satisfies the Poincaré inequality with constant  $\kappa_t$  (16):
$$
\int_ {\mathbb {R} ^ {d}} \left| \nabla \varphi \right| ^ {2} \mathrm {d} \rho_ {t} \geq \kappa_ {t} \int_ {\mathbb {R} ^ {d}} \varphi^ {2} \mathrm {d} \rho_ {t}, \quad \forall \varphi \text {w i t h} \int_ {\mathbb {R} ^ {d}} \varphi \rho_ {t} = 0. \tag {28}
$$
(iii)  $Z_{V_0} \coloneqq \int_{\mathbb{R}^d} e^{-V_0}$  is both upper and lower bounded: for any  $\delta \in (0,1)$ , there exists  $h_0(\delta) > 0$  such that whenever  $h \geq h_0(\delta)$ ,
$$
1 - \delta \leq Z _ {V _ {0}} \leq e ^ {- h ^ {- 2} c _ {U}}.
$$
$$
\frac {\rho_ {0}}{p _ {0}} \leq \frac {e ^ {- h ^ {- 2} c _ {U}}}{1 - \delta}.
$$
(iv) If we pick  $\delta = 1/2$ , for  $h \geq \max \{h_0(1/2), \sqrt{\max\{0, -\frac{c_U}{\ln(2)}\}}\}$ , the function  $\rho_0/p_0$  is uniformly bounded:
$$
\frac {\rho_ {0}}{p _ {0}} \leq 4. \tag {29}
$$
Proposition E.2. For any  $h \geq 1/2$ , we have
$$
\left| \int_ {\mathbb {R} ^ {d}} \varphi^ {2} \frac {\mathrm {d}}{\mathrm {d} t} \rho_ {t} ^ {\leftarrow} \right| \leq C _ {t} ^ {\leftarrow , (1)} \int_ {\mathbb {R} ^ {d}} \varphi^ {2} \rho_ {t} ^ {\leftarrow} + C _ {t} ^ {\leftarrow , (2)} \int_ {\mathbb {R} ^ {d}} \left| \nabla \varphi \right| ^ {2} \rho_ {t} ^ {\leftarrow}, \quad \forall \varphi \in C _ {0} ^ {2} (\mathbb {R} ^ {d}).
$$
$$
\left\{ \begin{array}{l} C _ {t} ^ {\leftarrow , (1)} := \xi_ {t} ^ {\leftarrow} + 2 \varsigma_ {t} ^ {\leftarrow} \left| \mathcal {Y} _ {t} ^ {\leftarrow} - \mathcal {X} _ {t} ^ {\leftarrow} \right| ^ {2} + \frac {2 0 d M _ {t} ^ {\leftarrow} \varsigma_ {t} ^ {\leftarrow}}{m _ {t} ^ {\leftarrow 2}}, \\ C _ {t} ^ {\leftarrow , (2)} := \frac {8 \varsigma_ {t} ^ {\leftarrow}}{m _ {t} ^ {\leftarrow 2}} \equiv \frac {2 0 + 3 0 M _ {t} ^ {\leftarrow 2}}{m _ {t} ^ {\leftarrow 2}}, \\ \xi_ {t} ^ {\leftarrow} := \left| \partial_ {t} \log Z _ {V _ {t} ^ {\leftarrow}} \right| + \frac {5 d \left(1 + M _ {t} ^ {\leftarrow}\right)}{2} + \frac {5}{2} \left| \mathcal {X} _ {t} ^ {\leftarrow} \right| ^ {2}, \\ \varsigma_ {t} ^ {\leftarrow} := 5 \left(\frac {1}{2} + \frac {3 M _ {t} ^ {\leftarrow 2}}{4}\right). \end{array} \right. \tag {30}
$$
$Z_{V_t^{\leftarrow}} = \int e^{-V_t^{\leftarrow}}, M_t^{\leftarrow}$  and  $m_t^{\leftarrow}$  are shown in Assumption 3.3, and  $\mathcal{X}_t^{\leftarrow}$  and  $\mathcal{Y}_t^{\leftarrow}$  are the global minimum points of functions  $x \mapsto U_t^{\leftarrow}(x)$  and  $x \mapsto V_t^{\leftarrow}(x)$ , respectively.
Remark. A similar result holds for any  $h > 0$ ; we choose  $h \geq 1/2$  in order to simplify the above constants (30).