SlowGuess/NeurIPS_2023_OCR huggingface dataset:
If the quantum gate  $U_{\mu}$  in the QNN satisfies the parameter-shift rule, the explicit form of the factor  $g(\pmb{\theta})$  could be known along the axis of  $\theta_{\mu}$  passing through  $\pmb{\theta}^{*}$ , which is summarized in Corollary S12. We use  $\theta_{\bar{\mu}}$  to represent the other components except for  $\theta_{\mu}$ , namely  $\theta_{\bar{\mu}} = \{\theta_{\nu}\}_{\nu \neq \mu}$ .
Corollary S12 For QNNs satisfying the parameter-shift rule by  $\Omega_{\mu}^{2} = I$ , the expectation and variance of the fidelity loss function  $\mathcal{L}$  restricted by only varying the parameter  $\theta_{\mu}$  from  $\pmb{\theta}^{*}$  with respect to the target state ensemble  $\mathbb{T}$  can be exactly calculated as
$$
\begin{array}{l} \mathbb {E} _ {\mathbb {T}} \left[ \mathcal {L} | _ {\theta_ {\bar {\mu}} = \theta_ {\bar {\mu}} ^ {*}} \right] = 1 - p ^ {2} + \frac {d p ^ {2} - 1}{d - 1} g (\theta_ {\mu}), \\ \left. \operatorname {V a r} _ {\mathbb {T}} \left[ \mathcal {L} | _ {\theta_ {\bar {\mu}} = \theta_ {\bar {\mu}} ^ {*}} \right] = \frac {1 - p ^ {2}}{d - 1} g \left(\theta_ {\mu}\right) \left[ 4 p ^ {2} - \left(2 p ^ {2} - \frac {(d - 2) (1 - p ^ {2})}{d (d - 1)}\right) g \left(\theta_ {\mu}\right) \right], \right. \tag {S79} \\ \end{array}
$$
where  $g(\theta_{\mu}) = \frac{1}{2}\mathcal{F}_{\mu \mu}^{*}\sin^{2}\left(\theta_{\mu} - \theta_{\mu}^{*}\right)$
Proof According to Proposition 3, we only need to calculate the factor  $g(\theta)|_{\theta_{\bar{\mu}} = \theta_{\bar{\mu}}^{*}}$ . We simply denote this factor as  $g(\theta_{\mu})$ , the explicit expression of which could be calculated by just substituting the parameter-shift rule. Alternatively, the expression of  $g(\theta_{\mu})$  can be directly written down by considering the following facts. The parameter-shift rule ensures that  $g(\theta_{\mu})$  must take the form of linear combinations of 1,  $\cos (2\theta_{\mu})$  and  $\sin (2\theta_{\mu})$  since  $U_{\mu}(\theta_{\mu}) = e^{-i\Omega_{\mu}\theta_{\mu}} = \cos \theta_{\mu}I - i\sin \theta_{\mu}\Omega_{\mu}$  and  $g(\theta_{\mu})$  takes the form of  $U_{\mu}(\cdot)U_{\mu}^{\dagger}$ . Furthermore,  $g(\theta_{\mu})$  takes its minimum at  $\theta_{\mu}^{*}$  so that it is an even function relative to  $\theta_{\mu} = \theta_{\mu}^{*}$ . Combined with the fact that  $g(\theta_{\mu})$  also takes zero at  $\theta_{\mu}^{*}$ , we know  $g(\theta_{\mu}) \propto [1 - \cos (2(\theta_{\mu} - \theta_{\mu}^{*}))]$ . The coefficient can be determined by considering that the second order derivative of  $g(\theta_{\mu})$  equals to the QFI matrix element  $\mathcal{F}_{\mu \mu}^{*}$  by definition, so that
$$
g \left(\theta_ {\mu}\right) = g (\boldsymbol {\theta}) | _ {\theta_ {\bar {\mu}} = \theta_ {\bar {\mu}} ^ {*}} = \frac {1}{4} \mathcal {F} _ {\mu \mu} ^ {*} [ 1 - \cos \left(2 \left(\theta_ {\mu} - \theta_ {\mu} ^ {*}\right)\right) ] = \frac {1}{2} \mathcal {F} _ {\mu \mu} ^ {*} \sin^ {2} \left(\theta_ {\mu} - \theta_ {\mu} ^ {*}\right). \tag {S80}
$$
The expressions of the expectation and variance of the loss function can be obtained by directly substituting Eq. (S80) into Proposition 3.
C Generalization to the local loss function
In the main text, we focus on the fidelity loss function, also known as the "global" loss function [36], where the ensemble construction and calculation are preformed in a clear and meaningful manner. However, there is another type of loss function called "local" loss function [36], such as the energy expectation in the variational quantum eigensolver (VQE) which aims to prepare the ground state of a physical system. The local loss function takes the form of
$$
\mathcal {L} (\boldsymbol {\theta}) = \langle \psi (\boldsymbol {\theta}) | H | \psi (\boldsymbol {\theta}) \rangle , \tag {S81}
$$
where  $H$  is the Hamiltonian of the physical system as a summation of Pauli strings. Eq. (S81) can formally reduce to the fidelity loss function by taking  $H = I - |\phi \rangle \langle \phi|$ . In this section, we generalize the results of the fidelity loss function to the local loss function and show that the conclusion keeps the same, though the ensemble construction and calculation are more complicated.
The ensemble we used in the main text decomposes the unknown target state into the learnt component  $|\psi^{*}\rangle$  and the unknown component  $|\psi^{\perp}\rangle$ , and regards  $|\psi^{\perp}\rangle$  as a Haar random state in the orthogonal complement of  $|\psi^{*}\rangle$ . This way of thinking seems to be more subtle in the case of the local loss function since the Hamiltonian is usually already known in the form of Pauli strings and hence it is unnatural to assume an unknown Hamiltonian. However, a known Hamiltonian does not imply a known target state, i.e., the ground state of the physical system. One needs to diagonalize the Hamiltonian to find the ground state, which requires an exponential cost in classical computers. That is to say, what one really does not know is the unitary used in the diagonalization, i.e., the relation between the learnt state  $|\psi^{*}\rangle$  and the eigen-basis of the Hamiltonian. We represent this kind of uncertainty by a unitary  $V$  from the ensemble  $\mathbb{V}$ , where  $\mathbb{V}$  comes from the ensemble  $\mathbb{U}$  mentioned in Appendix A.1 by specifying  $\bar{P} = |\psi^{*}\rangle \langle \psi^{*}|$ . Such an ensemble  $\mathbb{V}$  induces an ensemble of loss functions via
$$
\mathcal {L} (\boldsymbol {\theta}) = \langle \psi (\boldsymbol {\theta}) | V ^ {\dagger} H V | \psi (\boldsymbol {\theta}) \rangle , \tag {S82}
$$
similar with the loss function ensemble induced by the unknown target state in the main text.  $\mathbb{V}$  can be interpreted as all of the possible diagonalizing unitaries that keeps the loss value  $\mathcal{L}(\theta^{*})$  constant, denoted as  $\mathcal{L}^*$ . In the following, similar with those for the global loss function, we calculate the expectation and variance of the derivatives of the local loss function in Lemma S13 and bound the probability of avoiding local minima in Theorem S14. Hence, the results and relative discussions in the main text could generalize to the case of local loss functions.
Lemma S13 The expectation and variance of the gradient  $\nabla \mathcal{L}$  and Hessian matrix  $H_{\mathcal{L}}$  of the local loss function  $\mathcal{L}(\pmb {\theta}) = \langle \psi (\pmb {\theta})|H|\psi (\pmb {\theta})\rangle$  at  $\pmb {\theta} = \pmb{\theta}^{*}$  with respect to the ensemble  $\mathbb{V}$  satisfy
$$
\mathbb {E} _ {\mathbb {V}} \left[ \nabla \mathcal {L} ^ {*} \right] = 0, \quad \operatorname {V a r} _ {\mathbb {V}} \left[ \partial_ {\mu} \mathcal {L} ^ {*} \right] = f _ {1} (H, d) \mathcal {F} _ {\mu \mu} ^ {*},
$$
$$
\mathbb {E} _ {\mathbb {V}} \left[ H _ {\mathcal {L}} ^ {*} \right] = \frac {\operatorname {t r} H - d \mathcal {L} ^ {*}}{d - 1} \mathcal {F} ^ {*}, \quad \operatorname {V a r} _ {\mathbb {V}} \left[ \partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*} \right] \leq f _ {2} (H, d) \| \Omega_ {\mu} \| _ {\infty} ^ {2} \| \Omega_ {\nu} \| _ {\infty} ^ {2}, \tag {S83}
$$
where  $\mathcal{F}$  denotes the QFI matrix.  $f_{1}$  and  $f_{2}$  are functions of the Hamiltonian  $H$  and the Hilbert space dimension  $d$ , i.e.,
$$
f _ {1} (H, d) = \frac {\left\langle H ^ {2} \right\rangle_ {*} - \left\langle H \right\rangle_ {*} ^ {2}}{d - 1}, \quad f _ {2} (H, d) = 3 2 \left(\frac {\left\langle H ^ {2} \right\rangle_ {*} - \left\langle H \right\rangle_ {*} ^ {2}}{d - 1} + \frac {2 \| H \| _ {2} ^ {2}}{d (d - 2)}\right), \tag {S84}
$$
where we introduce the notation  $\langle \cdot \rangle_{*} = \langle \psi^{*}|\cdot |\psi^{*}\rangle$
Proof Using Lemma S3, the expectation of the local loss function can be directly calculated as
$$
\mathbb {E} _ {\mathbb {V}} [ \mathcal {L} (\boldsymbol {\theta}) ] = \mathcal {L} ^ {*} + \frac {\operatorname {t r} H - d \mathcal {L} ^ {*}}{d - 1} g (\boldsymbol {\theta}). \tag {S85}
$$
where  $g(\pmb{\theta}) = 1 - \langle \psi^{*}|\varrho (\pmb {\theta})|\psi^{*}\rangle$  denotes the fidelity distance between the output states at  $\pmb{\theta}$  and  $\pmb{\theta}^{*}$ . By definition,  $g(\pmb {\theta})$  takes the global minimum at  $\pmb {\theta} = \pmb{\theta}^{*}$ . Thus the commutation between the expectation and differentiation gives
$$
\mathbb {E} _ {\mathbb {V}} [ \nabla \mathcal {L} ^ {*} ] = \nabla \left(\mathbb {E} _ {\mathbb {V}} [ \mathcal {L} ]\right) | _ {\boldsymbol {\theta} = \boldsymbol {\theta} ^ {*}} = \frac {\operatorname {t r} H - d \mathcal {L} ^ {*}}{d - 1} \nabla g (\boldsymbol {\theta}) | _ {\boldsymbol {\theta} = \boldsymbol {\theta} ^ {*}} = 0, \tag {S86}
$$
$$
\mathbb {E} _ {\mathbb {V}} \left[ H _ {\mathcal {L}} ^ {*} \right] = \frac {\operatorname {t r} H - d \mathcal {L} ^ {*}}{d - 1} \left. H _ {g} (\boldsymbol {\theta}) \right| _ {\boldsymbol {\theta} = \boldsymbol {\theta} ^ {*}} = \frac {\operatorname {t r} H - d \mathcal {L} ^ {*}}{d - 1} \mathcal {F} ^ {*}.
$$
By definition,  $H_{g}(\pmb{\theta})|_{\pmb{\theta} = \pmb{\theta}^{*}}$  is actually the QFI matrix  $\mathcal{F}^*$  of  $|\psi (\pmb {\theta})\rangle$  at  $\pmb {\theta} = \pmb{\theta}^{*}$  (see Appendix A.4), which is always positive semidefinite. To estimate the variance, we need to calculate the expression of derivatives first due to the non-linearity of the variance. The first order derivative of the local loss function can be expressed by
$$
\partial_ {\mu} \mathcal {L} = \operatorname {t r} \left[ V ^ {\dagger} H V D _ {\mu} \right] = 2 \operatorname {R e} \langle \psi | V ^ {\dagger} H V | \partial_ {\mu} \psi \rangle , \tag {S87}
$$
where  $D_{\mu} = \partial_{\mu}\varrho$  is a traceless Hermitian operator since  $\mathrm{tr}D_{\mu} = \partial_{\mu}(\mathrm{tr}\varrho) = 0$ . By definition, we know that  $|\psi \rangle^{*}$  is not changed by  $V$ , i.e.,  $V|\psi^{*}\rangle = |\psi^{*}\rangle$ , which leads to the reduction  $\partial_{\mu}\mathcal{L}^{*} = 2\operatorname {Re}(\langle \psi^{*}|HV|\partial_{\mu}\psi^{*}\rangle)$ . Hence, the variance of the first order derivative at  $\pmb {\theta} = \pmb{\theta}^{*}$  is
$$
\begin{array}{l} \operatorname {V a r} _ {\mathbb {V}} \left[ \partial_ {\mu} \mathcal {L} ^ {*} \right] = \mathbb {E} _ {\mathbb {V}} \left[ \left(\partial_ {\mu} \mathcal {L} ^ {*} - \mathbb {E} _ {\mathbb {V}} \left[ \partial_ {\mu} \mathcal {L} ^ {*} \right]\right) ^ {2} \right] = \mathbb {E} _ {\mathbb {V}} \left[ \left(\partial_ {\mu} \mathcal {L} ^ {*}\right) ^ {2} \right] \\ = \mathbb {E} _ {\mathbb {V}} \left[ (2 \operatorname {R e} \langle \psi^ {*} | H V | \partial_ {\mu} \psi^ {*} \rangle) ^ {2} \right] \\ = \mathbb {E} _ {\mathbb {V}} \left[ \langle \psi^ {*} | H V | \partial_ {\mu} \psi^ {*} \rangle^ {2} \right] + \mathbb {E} _ {\mathbb {V}} \left[ \langle \partial_ {\mu} \psi^ {*} | V ^ {\dagger} H | \psi^ {*} \rangle^ {2} \right] \tag {S88} \\ + 2 \mathbb {E} _ {\mathbb {V}} \left[ \langle \psi^ {*} | H V | \partial_ {\mu} \psi^ {*} \rangle \langle \partial_ {\mu} \psi^ {*} | V ^ {\dagger} H | \psi^ {*} \rangle \right]. \\ \end{array}
$$
Utilizing Lemmas S2 and S3, we obtain
$$
\begin{array}{l} \mathbb {E} _ {\mathbb {V}} \left[ \langle \psi^ {*} | H V | \partial_ {\mu} \psi^ {*} \rangle^ {2} \right] = \langle H \rangle_ {*} ^ {2} \langle \psi^ {*} | \partial_ {\mu} \psi^ {*} \rangle^ {2}, \\ \mathbb {E} _ {\mathbb {V}} \left[ \langle \partial_ {\mu} \psi^ {*} | V ^ {\dagger} H | \psi^ {*} \rangle^ {2} \right] = \langle H \rangle_ {*} ^ {2} \langle \partial_ {\mu} \psi^ {*} | \psi^ {*} \rangle^ {2}, \\ \mathbb {E} _ {\mathbb {V}} \left[ \langle \psi^ {*} | H V | \partial_ {\mu} \psi^ {*} \rangle \langle \partial_ {\mu} \psi^ {*} | V ^ {\dagger} H | \psi^ {*} \rangle \right] \tag {S89} \\ = \frac {\langle H ^ {2} \rangle_ {*} - \langle H \rangle_ {*} ^ {2}}{2 (d - 1)} \mathcal {F} _ {\mu \mu} ^ {*} + \langle H \rangle_ {*} ^ {2} \langle \psi^ {*} | \partial_ {\mu} \psi^ {*} \rangle \langle \partial_ {\mu} \psi^ {*} | \psi^ {*} \rangle , \\ \end{array}
$$
where we introduce the notation  $\langle \cdot \rangle_{*} = \langle \psi^{*}|\cdot |\psi^{*}\rangle$  and hence  $\mathcal{L}^* = \langle H\rangle_*$ . The  $1 / 2$  factor in the third line arises from the definition of the QFI matrix. Note that there are three terms above canceling each other due to the fact
$$
2 \operatorname {R e} [ \langle \partial_ {\mu} \psi | \psi \rangle ] = \langle \partial_ {\mu} \psi | \psi \rangle + \langle \psi | \partial_ {\mu} \psi \rangle = \partial_ {\mu} (\langle \psi | \psi \rangle) = 0, \tag {S90}
$$
$$
\langle \psi | \partial_ {\mu} \psi \rangle^ {2} + \langle \partial_ {\mu} \psi | \psi \rangle^ {2} + 2 \langle \psi | \partial_ {\mu} \psi \rangle \langle \partial_ {\mu} \psi | \psi \rangle = (2 \operatorname {R e} [ \langle \partial_ {\mu} \psi | \psi \rangle ]) ^ {2} = 0.
$$
Therefore, the variance of the first order derivative at  $\theta = \theta^{*}$  equals to
$$
\operatorname {V a r} _ {\mathbb {V}} \left[ \partial_ {\mu} \mathcal {L} ^ {*} \right] = \mathbb {E} _ {\mathbb {V}} \left[ \left(2 \operatorname {R e} \langle \psi^ {*} | H V | \partial_ {\mu} \psi^ {*} \rangle\right) ^ {2} \right] = \frac {\left\langle H ^ {2} \right\rangle_ {*} - \left\langle H \right\rangle_ {*} ^ {2}}{d - 1} \mathcal {F} _ {\mu \mu} ^ {*}. \tag {S91}
$$
The second-order derivative can be expressed by
$$
\partial_ {\mu} \partial_ {\nu} \mathcal {L} = \left(H _ {\mathcal {L}}\right) _ {\mu \nu} = \operatorname {t r} \left[ V ^ {\dagger} H V D _ {\mu \nu} \right], \tag {S92}
$$
where  $D_{\mu \nu} = \partial_{\mu}\partial_{\nu}\varrho$  is a traceless Hermitian operator since  $\mathrm{tr}D_{\mu \nu} = \partial_{\mu}\partial_{\nu}(\mathrm{tr}\varrho) = 0$ . By direct expansion, the variance of the second order derivative at  $\pmb{\theta} = \pmb{\theta}^{*}$  can be expressed as
$$
\operatorname {V a r} _ {\mathbb {V}} \left[ \partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*} \right] = \mathbb {E} _ {\mathbb {V}} \left[ \left(\partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*}\right) ^ {2} \right] - \left(\mathbb {E} _ {\mathbb {V}} \left[ \partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*} \right]\right) ^ {2}, \tag {S93}
$$
where the second term is already obtained in Eq. (S86). Lemma S7 directly implies
$$
\begin{array}{l} \mathbb {E} _ {\mathbb {V}} \left[ (\partial_ {\mu} \partial_ {\nu} \mathcal {L}) ^ {2} \right] = \mathbb {E} _ {\mathbb {V}} \left[ \operatorname {t r} \left(V ^ {\dagger} H V D _ {\mu \nu}\right) \operatorname {t r} \left(V ^ {\dagger} H V D _ {\mu \nu}\right) \right] \\ = \langle H \rangle_ {*} ^ {2} \langle D _ {\mu \nu} \rangle_ {*} ^ {2} + \frac {2 \langle H \rangle_ {*} \langle D _ {\mu \nu} \rangle_ {*}}{d - 1} (\operatorname {t r} H - \langle H \rangle_ {*}) (\operatorname {t r} D _ {\mu \nu} - \langle D _ {\mu \nu} \rangle_ {*}) \\ + \frac {2}{d - 1} \left(\langle H ^ {2} \rangle_ {*} - \langle H \rangle_ {*} ^ {2}\right) \left(\langle D _ {\mu \nu} ^ {2} \rangle_ {*} - \langle D _ {\mu \nu} \rangle_ {*} ^ {2}\right) \\ + \frac {1}{d (d - 2)} (\operatorname {t r} H - \langle H \rangle_ {*}) ^ {2} (\operatorname {t r} D _ {\mu \nu} - \langle D _ {\mu \nu} \rangle_ {*}) ^ {2} \tag {S94} \\ + \frac {1}{d (d - 2)} \left(\operatorname {t r} \left(H ^ {2}\right) - 2 \langle H ^ {2} \rangle_ {*} + \langle H \rangle_ {*} ^ {2}\right) \left(\operatorname {t r} \left(D _ {\mu \nu} ^ {2}\right) - 2 \langle D _ {\mu \nu} ^ {2} \rangle_ {*} + \langle D _ {\mu \nu} \rangle_ {*} ^ {2}\right) \\ - \frac {1}{d (d - 1) (d - 2)} \left[ (\operatorname {t r} (H ^ {2}) - 2 \langle H ^ {2} \rangle_ {*} + \langle H \rangle_ {*} ^ {2}) (\operatorname {t r} D _ {\mu \nu} - \langle D _ {\mu \nu} \rangle_ {*}) ^ {2} \right] \\ - \frac {1}{d (d - 1) (d - 2)} \left[ (\operatorname {t r} H - \langle H \rangle_ {*}) ^ {2} (\operatorname {t r} (D _ {\mu \nu} ^ {2}) - 2 \langle D _ {\mu \nu} ^ {2} \rangle_ {*} + \langle D _ {\mu \nu} \rangle_ {*} ^ {2}) \right]. \\ \end{array}
$$
According to Eq. (S86),  $\langle D_{\mu \nu}^{*}\rangle_{*} = -\mathcal{F}_{\mu \nu}^{*}$  in Eq. (S51) and  $\mathcal{L}^* = \langle H\rangle_*$  , we have
$$
\left(\mathbb {E} _ {\mathbb {V}} \left[ \partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*} \right]\right) ^ {2} = \left(\frac {\operatorname {t r} H - d \mathcal {L} ^ {*}}{d - 1} \mathcal {F} _ {\mu \nu} ^ {*}\right) ^ {2} = \left(\frac {\operatorname {t r} H - \langle H \rangle_ {*}}{d - 1} - \langle H \rangle_ {*}\right) ^ {2} \langle D _ {\mu \nu} ^ {*} \rangle_ {*} ^ {2}. \tag {S95}
$$
Combining Eqs. (S94) and (S95) together with the condition  $\operatorname{tr} D_{\mu \nu} = 0$ , we obtain
$$
\begin{array}{l} \mathrm {V a r} _ {\mathbb {V}} [ \partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*} ] = \frac {2}{d - 1} (\langle H ^ {2} \rangle_ {*} - \langle H \rangle_ {*} ^ {2}) (\langle D _ {\mu \nu} ^ {2 *} \rangle_ {*} - \langle D _ {\mu \nu} ^ {*} \rangle_ {*} ^ {2}) \\ + \frac {1}{d (d - 2)} (\operatorname {t r} (H ^ {2}) - 2 \langle H ^ {2} \rangle_ {*} + \langle H \rangle_ {*} ^ {2}) (\operatorname {t r} (D _ {\mu \nu} ^ {2 *}) - 2 \langle D _ {\mu \nu} ^ {2 *} \rangle_ {*} + \langle D _ {\mu \nu} ^ {*} \rangle_ {*} ^ {2}) \\ - \frac {1}{d (d - 1) (d - 2)} \left[ \left(\operatorname {t r} \left(H ^ {2}\right) - 2 \langle H ^ {2} \rangle_ {*} + \langle H \rangle_ {*} ^ {2}\right) \langle D _ {\mu \nu} ^ {*} \rangle_ {*} ^ {2} \right] \tag {S96} \\ - \frac {1}{d (d - 1) (d - 2)} \left[ (\mathrm {t r} H - \langle H \rangle_ {*}) ^ {2} (\mathrm {t r} (D _ {\mu \nu} ^ {2 *}) - 2 \langle D _ {\mu \nu} ^ {2 *} \rangle_ {*} + \frac {d - 2}{d - 1} \langle D _ {\mu \nu} \rangle_ {*}) \right]. \\ \end{array}
$$
Note that we always have  $d \geq 2$  in qubit systems. If  $\text{rank } H \geq 2$ , then it holds that
$$
\left. \operatorname {t r} \left(H ^ {2}\right) - 2 \langle H ^ {2} \rangle_ {*} + \langle H \rangle_ {*} ^ {2} \geq \| H \| _ {2} ^ {2} - 2 \| H \| _ {\infty} ^ {2} \geq (\operatorname {r a n k} H) \| H \| _ {\infty} ^ {2} - 2 \| H \| _ {\infty} ^ {2} \geq 0. \right. \tag {S97}
$$
Otherwise if  $\operatorname{rank} H = 1$  (the case of  $\operatorname{rank} H = 0$  is trivial), then we assume  $H = \lambda |\phi\rangle\langle\phi|$  and it holds that
$$
\left. \right. \operatorname {t r} \left(H ^ {2}\right) - 2 \langle H ^ {2} \rangle_ {*} + \langle H \rangle_ {*} ^ {2} = \lambda^ {2} - 2 \lambda^ {2} | \langle \psi^ {*} | \phi \rangle | ^ {2} + \lambda^ {2} | \langle \psi^ {*} | \phi \rangle | ^ {4} = \lambda^ {2} \left(1 - | \langle \psi^ {*} | \phi \rangle | ^ {2}\right) ^ {2} \geq 0. \tag {S98}
$$
Hence we conclude that it always holds that
$$
\operatorname {t r} \left(H ^ {2}\right) - 2 \left\langle H ^ {2} \right\rangle_ {*} + \left\langle H \right\rangle_ {*} ^ {2} \geq 0. \tag {S99}
$$
Similarly, because  $\operatorname{tr} D_{\mu \nu} = 0$ , we know  $\operatorname{rank} D_{\mu \nu} \geq 2$  and thus
$$
\operatorname {t r} \left(D _ {\mu \nu} ^ {2}\right) - 2 \left\langle D _ {\mu \nu} ^ {2} \right\rangle_ {*} \geq \left(\operatorname {r a n k} D _ {\mu \nu}\right) \| D _ {\mu \nu} \| _ {\infty} ^ {2} - 2 \| D _ {\mu \nu} \| _ {\infty} ^ {2} \geq 0. \tag {S100}
$$
Therefore, we can upper bound the variance by just discarding the last two terms in Eq. (S96)
$$
\begin{array}{l} \operatorname {V a r} _ {\mathbb {V}} \left[ \partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*} \right] \leq \frac {2}{d - 1} \left(\langle H ^ {2} \rangle_ {*} - \langle H \rangle_ {*} ^ {2}\right) \left(\langle D _ {\mu \nu} ^ {2 *} \rangle_ {*} - \langle D _ {\mu \nu} ^ {*} \rangle_ {*} ^ {2}\right) \tag {S101} \\ + \frac {1}{d (d - 2)} (\operatorname {t r} (H ^ {2}) - 2 \langle H ^ {2} \rangle_ {*} + \langle H \rangle_ {*} ^ {2}) (\operatorname {t r} (D _ {\mu \nu} ^ {2 *}) - 2 \langle D _ {\mu \nu} ^ {2 *} \rangle_ {*} + \langle D _ {\mu \nu} ^ {*} \rangle_ {*} ^ {2}). \\ \end{array}
$$
$$
\operatorname {t r} \left(H ^ {2}\right) - 2 \left\langle H ^ {2} \right\rangle_ {*} + \left\langle H \right\rangle_ {*} ^ {2} = \operatorname {t r} \left(H ^ {2}\right) - \left\langle H ^ {2} \right\rangle_ {*} - \left(\left\langle H ^ {2} \right\rangle_ {*} - \left\langle H \right\rangle_ {*} ^ {2}\right) \leq \operatorname {t r} \left(H ^ {2}\right), \tag {S102}
$$
since  $\langle H^2\rangle_* - \langle H\rangle_*^2 = \langle H(I - |\psi^*\rangle \langle \psi^* |)H\rangle_*\geq 0$  . A similar inequality also holds for  $D_{\mu \nu}$  . Thus the variance can be further bounded by
$$
\operatorname {V a r} _ {\mathbb {V}} \left[ \partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*} \right] \leq \frac {2}{d - 1} \left(\langle H ^ {2} \rangle_ {*} - \langle H \rangle_ {*} ^ {2}\right) \| D _ {\mu \nu} ^ {*} \| _ {\infty} ^ {2} + \frac {4 \| H \| _ {2} ^ {2} \| D _ {\mu \nu} ^ {*} \| _ {\infty} ^ {2}}{d (d - 2)}, \tag {S103}
$$
where we have used the properties in Eq. (S58). Using the inequality in Eq. (S60) associated with the gate generators, the variance of the second order derivative at  $\theta = \theta^{*}$  can be ultimately upper bounded by
$$
\operatorname {V a r} _ {\mathbb {V}} \left[ \partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*} \right] \leq f _ {2} (H, d) \| \Omega_ {\mu} \| _ {\infty} ^ {2} \| \Omega_ {\nu} \| _ {\infty} ^ {2}, \tag {S104}
$$
The factor  $f_{2}(H,d)$  reads
$$
f _ {2} (H, d) = 3 2 \left(\frac {\langle H ^ {2} \rangle_ {*} - \langle H \rangle_ {*} ^ {2}}{d - 1} + \frac {2 \| H \| _ {2} ^ {2}}{d (d - 2)}\right), \tag {S105}
$$
which vanishes at least of order  $\mathcal{O}(\mathrm{poly}(N)2^{-N})$  with the qubit count  $N = \log_2d$  if  $\| H\|_{\infty}\in \mathcal{O}(\mathrm{poly}(N))$
Theorem S14 If  $\mathcal{L}^* < \frac{\mathrm{tr}H}{d}$ , the probability that  $\pmb{\theta}^{*}$  is not a local minimum of the local cost function  $\mathcal{L}$  up to a fixed precision  $\epsilon = (\epsilon_{1},\epsilon_{2})$  with respect to the ensemble  $\mathbb{V}$  is upper bounded by
$$
\Pr_ {\mathbb {V}} [ \neg \operatorname {L o c a l M i n} (\boldsymbol {\theta} ^ {*}, \epsilon) ] \leq \frac {2 f _ {1} (H , d) \| \boldsymbol {\omega} \| _ {2} ^ {2}}{\epsilon_ {1} ^ {2}} + \frac {f _ {2} (H , d) \| \boldsymbol {\omega} \| _ {2} ^ {4}}{\left(\frac {\operatorname {t r} H - d \mathcal {L} ^ {*}}{d - 1} e ^ {*} + \epsilon_ {2}\right) ^ {2}}, \tag {S106}
$$
where  $e^*$  denotes the minimal eigenvalue of the QFI matrix at  $\theta = \theta^{*}$ .  $f_{1}$  and  $f_{2}$  are defined in Eq. (S84) which vanish at least of order  $\mathcal{O}(\mathrm{poly}(N)2^{-N})$  with the qubit count  $N = \log_2d$  if  $\| H\|_{\infty}\in \mathcal{O}(\mathrm{poly}(N))$ .
Proof Utilizing Lemma S13, the proof is exactly the same as that of Theorem 2 up to the different hessian expectation  $\mathbb{E}_{\mathbb{V}}[H_{\mathcal{L}}^{*}]$  and coefficient functions  $f_{1}$  and  $f_{2}$ .