SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\Pr_ {\mathbb {T}} \left[ H _ {\mathcal {L}} ^ {*} \neq - \epsilon_ {2} I \right] \leq \sum_ {\mu , \nu = 1} ^ {M} \frac {\operatorname {V a r} _ {\mathbb {T}} \left[ \partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*} \right]}{\left(\frac {d p ^ {2} - 1}{d - 1} e ^ {*} + \epsilon_ {2}\right) ^ {2}} \leq \frac {f _ {2} (p , d) \| \boldsymbol {\omega} \| _ {2} ^ {4}}{\left(\frac {d p ^ {2} - 1}{d - 1} e ^ {*} + \epsilon_ {2}\right) ^ {2}}. \tag {11}
$$
Combining the bounds regarding the gradient and hessian matrix, one arrives at Eq. (9).
Theorem 2 directly points out that if the loss function takes a value lower than the critical threshold  $\mathcal{L}_c = 1 - 1 / d$ , then the surrounding landscape would be a local minimum for almost all of the target states, the proportion of which is exponentially close to 1 as the qubit count  $N = \log_2d$  grows. Note that  $\| \omega \| _2^2 = \sum_{\mu = 1}^{M}\| \Omega_\mu \|_\infty^2$  scales linearly with the number of parameters  $M$  and at most polynomially with the qubit count  $N$ . Because practically the operator norm  $\| \Omega_{\mu}\|_{\infty}$  is constant such as the generators of Pauli rotations  $\| X\|_{\infty} = \| Y\|_{\infty} = \| Z\|_{\infty} = 1$ , or grows polynomially with the system size such as the layer with globally correlated parameters [56] and the global evolution in analog quantum computing [69]. Here we focus on the former and conclude that the upper bound in Theorem 2 is of order  $\mathcal{O}(2^{-N}M^2 /\epsilon^2)$ , implying the exponential training cost. The conclusion also holds for noisy quantum states (see Appendix B). A similar result for the so-called local loss function, like the energy expectation used in variational quantum eigensolvers, is provided in Appendix C.
In principle, if one could explore the whole Hilbert space with exponentially many parameters,  $\theta^{*}$  was a saddle point at most instead of a local minimum since there must exist a unitary connecting the learnt state  $|\psi^{*}\rangle$  and the target state  $|\phi \rangle$ . This is also consistent with our bound by taking  $M \in \Omega(2^{N/2})$  to cancel the  $2^{-N}$  factor such that the bound is no more exponentially small. However, the number of parameters one can control always scales polynomially with the qubit count due to the memory constraint. This fact indicates that if the QNN is not designed specially for the target state using some prior knowledge so that the "correct" direction towards the target state is contained in the accessible tangent space, the QNN will have the same complexity as the normal quantum state tomography.
Dependence on the loss value  $\mathcal{L}^*$ . The dependence of the bound in Theorem 2 on the overlap  $p = \sqrt{1 - \mathcal{L}^*}$  shows that, as the loss function value becomes lower, the local minima becomes denser so that the training proceeds harder. This agrees with the experience that the loss curves usually decay fast at the beginning of a training process and slow down till the convergence. If  $e^* \neq 0$ , the second term in Eq. (9) becomes larger as  $p^2 \to 1 / d$ , suggesting that the local minima away from the critical point  $\mathcal{L}_c$  is more severe than that near  $\mathcal{L}_c$ . Moreover, if  $\epsilon_2 = 0$ , the bound diverges as the QFI minimal eigenvalue  $e^*$  vanishes, which reflects the fact that over-parameterized QNNs have many equivalent local minima connected by the redundant degrees of freedom of parameters.
By contrast, if  $\mathcal{L}^* > \mathcal{L}_c$ , the results could be established similarly by slightly modifying the proof yet with respect to local maxima, as depicted in Fig. 1(a). However, the critical point  $\mathcal{L}_c = 1 - 2^{-N}$
moves to 1 exponentially fast as  $N$  increases, i.e., the range of  $\mathcal{L}^*$  without severe local minima shrink exponentially. Hence for large-scale systems, even with a polynomially small fidelity, one would encounter a local minimum almost definitely if no more prior knowledge can be used.
3.3 Implication on the learnability of QNNs
In practical cases, if the QNN is composed of  $D$  repeated layers with  $z$  trainable parameters for each layer and each qubit, then the total number of trainable parameters becomes  $M = N D z$ , and hence the probability of avoiding local minima is of order  $\mathcal{O}(N^2 2^{-N} D^2 / \epsilon^2)$ , which increases quadratically as the QNN becomes deeper. This seems contrary to the conclusion from barren plateaus [35] where deep QNNs lead to poor trainability. But in fact, they are complementary to each other. The reason is that the ensemble here originates from the unknown target state instead of the random initialization. Similar to classical neural networks, a deeper QNN has stronger expressibility, which creates a larger accessible manifold to approach the unknown state and may turn a local minimum into a saddle point with the increased dimensions. But on the other hand, a deeper QNN with randomly initialized parameters leads to barren plateaus [36]. In short, the local minima here arise due to the limited expressibility together with a non-vanishing fidelity while barren plateaus stem from the strong expressibility together with the random initialization.
To solve this dilemma, our results suggest that a well-designed QNN structure taking advantage of prior knowledge of the target state is vitally necessary. Otherwise, a good initial guess (i.e., an initial state with high fidelity) solely is hard to play its role. An example of prior knowledge from quantum many-body physics is the tensor network states [70] satisfying the entanglement area law, which lives only in a polynomially large space but generally can not be solved in two and higher spatial dimensions by classical computers. Other examples include the UCCSD ansatz [71] in quantum chemistry and the QAOA ansatz [72] in combinatorial optimization, which all attempt to utilize the prior knowledge of the target states.
Finally, we remark that our results also place general theoretical limits for adaptive [27] or layerwise training methods [47]. Relevant phenomena are observed previously in special examples [55]. Adaptive methods append new training layers incrementally during the optimization instead of placing a randomly initialized determinate ansatz at the beginning, which is hence beyond the scope of barren plateaus [35]. Nevertheless, our results imply that for moderately large systems with  $\mathcal{L}^* < \mathcal{L}_c$ , every time a new training layer is appended, the learnt state  $|\psi^{*}\rangle$  would be a local minimum of the newly created landscape so that the training process starting near  $|\psi^{*}\rangle$  would go back to the original state  $|\psi^{*}\rangle$  without any effective progress more than applying an identity. Note that in adaptive methods, one usually initializes the new appended layer near the identity to preserve the historical learnt outcomes. Similar phenomena are also expected to occur in the initialization strategies where the circuit begins near the identity [43, 44, 46]. We emphasize that our results do not imply the ineffectiveness of all adaptive methods. Instead, they only suggest that simplistic brute-force adaptive methods provide no significant benefit in terms of enhancing learnability on average.
3.4 Concentration of training landscapes
Theorem 2 analyses the statistical properties of the vicinity of a certain point  $\theta^{*}$ , i.e., the probability distributions of the gradient and Hessian matrix of  $\theta^{*}$ . To characterize the training landscape beyond the vicinity, a pointwise result is established in Proposition 3 with the proof in Appendix B.
Proposition 3 The expectation and variance of the fidelity loss function  $\mathcal{L}$  with respect to the target state ensemble  $\mathbb{T}$  can be exactly calculated as
$$
\mathbb {E} _ {\mathbb {T}} [ \mathcal {L} (\boldsymbol {\theta}) ] = 1 - p ^ {2} + \frac {d p ^ {2} - 1}{d - 1} g (\boldsymbol {\theta}), \tag {12}
$$
$$
\operatorname {V a r} _ {\mathbb {T}} \left[ \mathcal {L} (\boldsymbol {\theta}) \right] = \frac {1 - p ^ {2}}{d - 1} g (\boldsymbol {\theta}) \left[ 4 p ^ {2} - \left(2 p ^ {2} - \frac {(d - 2) (1 - p ^ {2})}{d (d - 1)}\right) g (\boldsymbol {\theta}) \right],
$$
where  $g(\pmb {\theta}) = 1 - |\langle \psi^{*}|\psi (\pmb {\theta})\rangle |^{2}$
Since the factor  $g(\theta)$  takes its global minimum at  $\theta^{*}$  by definition, the exponentially small variance in Proposition 3 implies that the entire landscape concentrates exponentially in the qubit count to
Figure 3: Samples of training landscapes along randomly chosen directions as a function of the distance  $\|\pmb{\theta} - \pmb{\theta}^{*}\|$  for random target states from  $\mathbb{T}$  with the sample size 200, the qubit count  $N = 4$  and  $N = 10$  and the overlap (a)  $p = 0.2$  and (b)  $p = 0.8$ , respectively. The setup in (c) is the same as in (b) but fixes the target state and only samples the directions. A clear convex shape is present in the samples from  $N = 10$  but absent in the samples from  $N = 4$ . This phenomenon intuitively shows that the training landscapes from the ensemble  $\mathbb{T}$  are concentrated to a local minimum around  $\pmb{\theta}^{*}$  as the qubit count increases.
the expectation  $\mathbb{E}_{\mathbb{T}}[\mathcal{L}(\pmb {\theta})]$  with a pointwise convergence (not necessarily a uniform convergence), which takes its global minimum at  $\pmb{\theta}^{*}$  with respect to the training landscape as long as  $p^2 >1 / d$  . For QNNs satisfying the parameter-shift rule, the factor  $g(\pmb {\theta})$  along the Cartesian axis corresponding to  $\theta_{\mu}$  passing through  $\pmb{\theta}^{*}$  will take the form of a trigonometric function  $\frac{1}{2}\mathcal{F}_{\mu \mu}^{*}\sin^{2}(\theta_{\mu} - \theta_{\mu}^{*})$  , which is elaborated in Appendix B. Other points apart from  $\pmb{\theta}^{*}$  is allowed to have a non-vanishing gradient expectation in our setup, which leads to prominent local minima instead of plateaus [35].
Previous sections theoretically characterize the limitation of QNNs in state learning tasks considering the information of the loss value  $\mathcal{L}^*$ . In this section, we verify these results by conducting numerical experiments on the platform Paddle Quantum [73] and Tensorcircuit [74] from the following three perspectives. The codes for numerical experiments can be found in [75].
Comparison with loss curves. Firstly, we show the prediction ability of Theorem 2 by direct comparison with experimental loss curves in Fig. 1(c). We create 9 ALT circuits for qubit counts of 2, 6, 10 and circuit depth of 1, 3, 5 with randomly initialized parameters, denoted as  $\theta^{*}$ . For each circuit, we sample 10 target states from the ensemble  $\mathbb{T}$  with  $p = 0.2$  and then generate 10 corresponding loss curves using the Adam optimizer with a learning rate 0.01. We exploit the background color intensity to represent the corresponding bounds from Theorem 2 by assigning  $e^{*} = 0.1$  and  $\epsilon_{1} = \epsilon_{2} = 0.05$ . One can find that the loss curves decay fast at the beginning and then slow down till convergence, in accordance with the conclusion that the probability of encountering local minima is larger near the bottom. The convergent loss value becomes higher as the qubit count grows and can be partially reduced by increasing the circuit depth, which is also consistent with Theorem 2.
**Landscape profile sampling.** We visualize the existence of asymptotic local minima by sampling training landscape profiles in Fig. 3. Similar to the setup above, we create an ALT circuit with randomly initialized parameters  $\theta^{*}$ , sample 200 target states from the ensemble  $\mathbb{T}$  and compute the loss values near  $\theta^{*}$ . Figs. 3(a) and (b) are obtained by randomly choosing a direction for each landscape sample, while Fig. 3(c) is obtained by randomly sampling 200 directions for one fixed landscape sample. There is no indication of local minimum in the case of few qubits and small fidelity as shown by the blue curves in Fig. 3(a). However, as the qubit number grows, the landscape profiles concentrate into a clear convex shape centered at  $\theta^{*}$  for both  $p = 0.2$  and  $p = 0.8$ , where the curvature for  $p = 0.8$  is larger due to the factor  $(p^2 - 1/d)$  in Eq. (7). Fig. 3(c) further demonstrates that beyond the convexity along a specific direction,  $\theta^{*}$  is a highly probable local minimum in the case of large qubit counts.
Probability evaluation. Finally, we compute the gradients and diagonalize the Hessian matrices to directly verify the exponentially likely local minima proposed by Theorem 2 in Fig. 4. Similar to
Figure 4: Numerical evaluation for the probability that  $\theta$  is a local minimum up to a fixed precision  $\epsilon$ , i.e.,  $\mathrm{Pr}_{\mathbb{T}}[\mathrm{LocalMin}(\theta^{*},\epsilon)]$  for different qubit count, the number of trainable parameters and the overlap  $p$ , with the error bar representing the statistical uncertainty in experiments. (a) shows that the probability converges to 1 rapidly with the increasing qubit count. (b) shows that the probability is reduced by increasing the number of parameters, implying the local minimum phenomenon is mitigated. (c) illustrates that the probability of encountering local minima always converges to 1 for any fixed overlap  $p$ .  $p = 0.8$  for both (a) and (b) and the number of parameters in (c) is 6.
the setup above, we create ALT circuits for qubit count from  $N = 1$  to 11 with depth  $D = 5$  and sample 200 target states from  $\mathbb{T}$  for each circuit. After specifying a certain subset of parameters to be differentiated, we estimate the probability that  $\theta$  is a local minimum by the proportion of samples satisfying the condition LocalMin  $(\theta^{*},\epsilon)$ , where we assign  $\epsilon_{1} = \epsilon_{2} = 0.05$ . One can find that the probability of encountering local minima saturates to 1 very fast as the qubit count increases for arbitrary given values of  $p$ , and at the same time, it can be reduced by increasing the number of trainable parameters, which is consistent with the theoretical findings in Theorem 2.
5 Conclusion and outlook
In this paper, we prove that during the process of learning an unknown quantum state with QNNs, the probability of avoiding local minima is of order  $\mathcal{O}(N^2 2^{-N}D^2 /\epsilon^2)$  which is exponentially small in the qubit count  $N$  while increases polynomially with the circuit depth  $D$ . The curvature of local minima is concentrated to the QFI matrix times a fidelity-dependent constant which is positive at  $p^2 >1 / d$ . In practice, our results can be regarded as a quantum version of the no-free-lunch (NFL) theorem suggesting that no single QNN is universally the best-performing model for learning all target quantum states. We remark that compared to previous works, our findings first establish quantitative limits on good initial guesses and adaptive training methods for improving the learnability and scalability of QNNs.
In the technical part of our work, our ensemble arises from the unknown target state. Alternatively, if the QNN is sufficiently deep to form a subspace 2-design (cf. Appendix A.1) replacing the ensemble we used here, a different interpretation could be established with the same calculations: there are exponentially large proportion of local minima on some cross sections of the training landscape with a constant loss function value. However, it remains an open question what the scaling of the QNN depth is to constitute such a subspace 2-design, given that a local random quantum circuit of polynomially depth forms an approximate unitary  $t$ -design [76]. We would like to note that the case where the output and target states are mixed states is not covered due to the quantum nature of the hard-to-defining orthogonal ensemble of mixed states, which may be left for future research.
Future progress will necessitate more structured QNN architectures and optimization tools, where insights from the field of deep learning may prove beneficial. Our findings suggest that the unique characteristics and prior information of quantum systems must be thoughtfully encoded in the QNN in order to learn the state successfully, such as the low entanglement structure in the ground state [70], the local interactions in Hamiltonians [71, 77] and the adiabatic evolution from product states [72].
Acknowledgement. We would like to thank the helpful comments from the anonymous reviewers. Part of this work was done when H. Z., C. Z., M. J., and X. W. were at Baidu Research.
[79] Jay M. Gambetta, A. D. Corcoles, S. T. Merkel, B. R. Johnson, John A. Smolin, Jerry M. Chow, Colm A. Ryan, Chad Rigetti, S. Poletto, Thomas A. Ohki, Mark B. Ketchen, and M. Steffen. Characterization of Addressability by Simultaneous Randomized Benchmarking. Physical Review Letters, 109(24):240504, dec 2012.
Appendix for "Statistical Analysis of Quantum State Learning Process in Quantum Neural Networks"
A.1 Subspace Haar integration
The central technique used in our work is the subspace Haar integration, i.e., a series of formulas on calculating Haar integrals over a certain subspace of the given Hilbert space. In this section, we give a brief introduction to the common Haar integrals and then the basic formulas on subspace Haar integrals used in our work together with the proofs.
Haar integrals refer to the matrix integrals over the  $d$ -degree unitary group  $\mathcal{U}(d)$  with the Haar measure  $d\mu$ , which is the unique uniform measure on  $\mathcal{U}(d)$  such that
$$
\int_ {\mathcal {U} (d)} d \mu (V) f (V) = \int_ {\mathcal {U} (d)} d \mu (V) f (V U) = \int_ {\mathcal {U} (d)} d \mu (V) f (U V), \tag {S1}
$$
for any integrand  $f$  and group element  $U \in \mathcal{U}(d)$ . If an ensemble  $\mathbb{V}$  of unitaries  $V$  matches the Haar measure up to the  $t$ -degree moment, i.e.,
$$
\mathbb {E} _ {V \in \mathbb {V}} [ p _ {t, t} (V) ] = \int_ {\mathcal {U} (d)} d \mu (V) p _ {t, t} (V), \tag {S2}
$$
then  $\mathbb{V}$  is called a unitary  $t$ -design [78].  $p_{t,t}(V)$  denotes an arbitrary polynomial of degree at most  $t$  in the entries of  $V$  and at most  $t$  in those of  $V^{\dagger}$ .  $\mathbb{E}_{V\in \mathbb{V}}[\cdot ]$  denotes the expectation over the ensemble  $\mathbb{V}$ . The Haar integrals over polynomials can be analytically solved and expressed into closed forms according to the following lemma.
Lemma S1 Let  $\varphi : \mathcal{U}(d) \to \mathrm{GL}(\mathbb{C}^{d'})$  be an arbitrary representation of unitary group  $\mathcal{U}(d)$ . Suppose that the direct sum decomposition of  $\varphi$  to irreducible representations is  $\varphi = \bigoplus_{j,k} \phi_k^{(j)}$ , where  $\phi_k^{(j)}$  denotes the  $k^{\text{th}}$  copy of the irreducible representation  $\phi^{(j)}$ . A set of orthonormal basis in the representation space of  $\phi_k^{(j)}$  is denoted as  $\{|v_{j,k,l}\rangle\}$ . For an arbitrary linear operator  $A: \mathbb{C}^{d'} \to \mathbb{C}^{d'}$ , the following equality holds [79]
$$
\int_ {\mathcal {U} (d)} \varphi (U) A \varphi (U) ^ {\dagger} d \mu (U) = \sum_ {j, k, k ^ {\prime}} \frac {\operatorname {t r} \left(Q _ {j , k , k ^ {\prime}} ^ {\dagger} A\right)}{\operatorname {t r} \left(Q _ {j , k , k ^ {\prime}} ^ {\dagger} Q _ {j , k , k ^ {\prime}}\right)} Q _ {j, k, k ^ {\prime}}, \tag {S3}
$$
where  $Q_{j,k,k'} = \sum_{l} |v_{j,k,l}\rangle \langle v_{j,k',l}|$  is the transfer operator from the representation subspace of  $\phi_{k'}^{(j)}$  to that of  $\phi_k^{(j)}$ . The denominator on the right hand side of (S3) can be simplified as  $\mathrm{tr}(Q_{j,k,k'}^\dagger Q_{j,k,k'}) = \mathrm{tr}(P_{j,k'}) = d_j$ , where  $P_{j,k} = \sum_{l} |v_{j,k,l}\rangle \langle v_{j,k,l}| = Q_{j,k,k}$  is the projector to the representation subspace of  $\phi_k^{(j)}$  and  $d_j$  is the dimension of the representation space of  $\phi^{(j)}$ .
By choosing different representations of the unitary group  $\mathcal{U}(d)$ , some commonly used equalities can be derived, such as
$$
\int_ {\mathcal {U} (d)} V A V ^ {\dagger} d \mu (V) = \frac {\operatorname {t r} (A)}{d} I, \tag {S4}
$$
$$
\int_ {\mathcal {U} (d)} V ^ {\dagger} A V B V ^ {\dagger} C V d \mu (V) = \frac {\operatorname {t r} (A C) \operatorname {t r} B}{d ^ {2}} I + \frac {d \operatorname {t r} A \operatorname {t r} C - \operatorname {t r} (A C)}{d \left(d ^ {2} - 1\right)} \left(B - \frac {\operatorname {t r} B}{d} I\right), \tag {S5}
$$
where  $I$  is the identity operator on the  $d$ -dimensional Hilbert space  $\mathcal{H}$ .  $A, B$  and  $C$  are arbitrary linear operators on  $\mathcal{H}$ . According to the linearity of the integrals, the following equalities can be further derived
$$
\int_ {\mathcal {U} (d)} \operatorname {t r} (V A) \operatorname {t r} \left(V ^ {\dagger} B\right) d \mu (V) = \frac {\operatorname {t r} (A B)}{d}, \tag {S6}
$$
$$
\begin{array}{l} \int_ {\mathcal {U} (d)} \operatorname {t r} \left(V ^ {\dagger} A V B\right) \operatorname {t r} \left(V ^ {\dagger} C V D\right) d \mu (V) = \frac {\operatorname {t r} A \operatorname {t r} B \operatorname {t r} C \operatorname {t r} D + \operatorname {t r} (A C) \operatorname {t r} (B D)}{d ^ {2} - 1} \tag {S7} \\ - \frac {\operatorname {t r} (A C) \operatorname {t r} B \operatorname {t r} D + \operatorname {t r} A \operatorname {t r} C \operatorname {t r} (B D)}{d \left(d ^ {2} - 1\right)}, \\ \end{array}
$$
where  $A, B, C$  and  $D$  are arbitrary linear operators on  $\mathcal{H}$ .
The subspace Haar integration can be regarded as a simple generalization of the formulas above. Suppose that  $\mathcal{H}_{\mathrm{sub}}$  is a subspace with dimension  $d_{\mathrm{sub}}$  of the Hilbert space  $\mathcal{H}$ .  $\mathbb{U}$  is an ensemble whose elements are unitaries in  $\mathcal{H}$  with a block-diagonal structure  $U = \bar{P} + PUP$ .  $P$  is the projector from  $\mathcal{H}$  to  $\mathcal{H}_{\mathrm{sub}}$  and  $PUP$  is a random unitary with the Haar measure on  $\mathcal{H}_{\mathrm{sub}}$ .  $\bar{P} = I - P$  is the projector from  $\mathcal{H}$  to the orthogonal complement of  $\mathcal{H}_{\mathrm{sub}}$ . Integrals with respect to such an ensemble  $\mathbb{U}$  are dubbed as "subspace Haar integrals", which can be reduced back to the common Haar integrals by taking  $\mathcal{H}_{\mathrm{sub}} = \mathcal{H}$ . The corresponding formulas of subspace Haar integrals are developed in the following lemmas, where  $\mathbb{E}_{U \in \mathbb{U}}[\cdot] = \mathbb{E}_{\mathbb{U}}[\cdot]$  denotes the expectation with respect to the ensemble  $\mathbb{U}$ .
Lemma S2 The expectation of a single element  $U \in \mathbb{U}$  with respect to the ensemble  $\mathbb{U}$  equals to the projector to the orthogonal complement, i.e.,
$$
\mathbb {E} _ {\mathbb {U}} [ U ] = I - P. \tag {S8}
$$
Proof The fact that Haar integrals of inhomogeneous polynomials  $p_{t,t'}$  with  $t \neq t'$  over the whole space equals to zero leads to the vanishment of the block in  $\mathcal{H}_{\mathrm{sub}}$ , i.e.,
$$
\mathbb {E} _ {\mathbb {U}} [ U ] = \mathbb {E} _ {\mathbb {U}} [ \bar {P} + P U P ] = \bar {P}, \tag {S9}
$$
which is just the projector to the orthogonal complement  $\mathcal{H}_{\mathrm{sub}}$
Similarly, we know all the subspace Haar integrals involving only  $U$  or  $U^{\dagger}$  will leave a projector after integration. For example, it holds that  $\mathbb{E}_{\mathbb{U}}[UAU] = \bar{P} A\bar{P}$  for an arbitrary linear operator  $A$ .
Lemma S3 For an arbitrary linear operator  $A$  on  $\mathcal{H}$ , the expectation of  $U^{\dagger}AU$  with respect to the random variable  $U\in \mathbb{U}$  is
$$
\mathbb {E} _ {\mathbb {U}} \left[ U ^ {\dagger} A U \right] = \frac {\operatorname {t r} (P A)}{d _ {\text {s u b}}} P + (I - P) A (I - P). \tag {S10}
$$
Proof Eq. (S10) can be seen as a special case of Lemma S1 since  $U$  can be seen as the complete reducible representation of  $\mathcal{U}(d_{\mathrm{sub}})$  composed of  $(d - d_{\mathrm{sub}})$  trivial representations  $\phi_k^{(1)}$  with  $k = 1, \dots, (d - d_{\mathrm{sub}})$  and one natural representation  $\phi^{(2)}$ . This gives rise to