SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Hao-kai Zhang $^{1,3}$ , Chenghong Zhu $^{2,3}$ , Mingrui Jing $^{2,3}$ , Xin Wang $^{2,3*}$ $^{1}$  Institute for Advanced Study, Tsinghua University, Beijing 100084, China  
 $^{2}$  Thrust of Artificial Intelligence, Information Hub, Hong Kong University of Science and Technology (Guangzhou), China  
 $^{3}$  Institute for Quantum Computing, Baidu Research, Beijing, China
Quantum neural networks (QNNs) have been a promising framework in pursuing near-term quantum advantage in various fields, where many applications can be viewed as learning a quantum state that encodes useful data. As a quantum analog of probability distribution learning, quantum state learning is theoretically and practically essential in quantum machine learning. In this paper, we develop a no-go theorem for learning an unknown quantum state with QNNs even starting from a high-fidelity initial state. We prove that when the loss value is lower than a critical threshold, the probability of avoiding local minima vanishes exponentially with the qubit count, while only grows polynomially with the circuit depth. The curvature of local minima is concentrated to the quantum Fisher information times a loss-dependent constant, which characterizes the sensibility of the output state with respect to parameters in QNNs. These results hold for any circuit structures, initialization strategies, and work for both fixed ansatzes and adaptive methods. Extensive numerical simulations are performed to validate our theoretical results. Our findings place generic limits on good initial guesses and adaptive methods for improving the learnability and scalability of QNNs, and deepen the understanding of prior information's role in QNNs.
Recent experimental progress towards realizing quantum information processors [1-3] has fostered the thriving development of the emerging field of quantum machine learning (QML) [4-18], pursuing quantum advantages in artificial intelligence. Different QML algorithms have been proposed for various topics, e.g., quantum simulations [19-23], chemistry [24-28], quantum data compression [29, 30], generative learning [31, 32] and reinforcement learning [33], where quantum neural networks (QNNs) become a leading framework due to the hardware restriction from noisy intermediate scale quantum (NISQ) [34] devices. As quantum analogs of artificial neural networks, QNNs typically refer to parameterized quantum circuits which are trainable based on quantum measurement results.
However, QNNs face a severe scalability barrier which might prevent the realization of potential quantum advantages. A notorious example is the barren plateau phenomenon [35] which shows that the gradient of the loss function vanishes exponentially in the system size with a high probability for randomly initialized deep QNNs, giving rise to an exponential training cost. To address this issue, a variety of training strategies has been proposed, such as local loss functions [36], correlated parameters [37], structured architectures [38-40], good initial guesses [41, 42], initialization heuristics near the identity [43-46], adaptive methods [27] and layerwise training [47], etc. Nevertheless, there
*felixxinwang@hkust-gz.edu.cn
(c)
Figure 1: Sketches of our work characterizing the statistical performance of QNNs on quantum state learning tasks. (a) indicates the existence of a critical loss value  $\mathcal{L}_c = 1 - 2^{-N}$  below which the local minima start to become severe to trap the training process. (b) depicts the setup of quantum state learning tasks where a QNN is used to learn an unknown target state encoding practical data. All target states with a constant distance to the output state  $|\psi^{*}\rangle$  form our ensemble  $\mathbb{T}$ , depicted by the contour on the Bloch sphere. (c) shows typical loss curves for different qubit counts of  $N = 2,6,10$  and circuit depths of  $D = 1,3,5$ . The intensity of the background colors represents the magnitude of our theoretical bound on the probability of encountering a local minimum, which hence signifies the hardness of optimization. One can find that the theoretical bound appropriately reflects the hardness encountered by the practical loss curves.
is a lack of scalability analyses for these strategies to guarantee their effectiveness. Especially, since the identity initialization and adaptive or layerwise training methods do not require uniformly random initialization of deep QNNs, they are out of the scope of barren plateaus and urgently need a comprehensive theoretical analysis to ascertain their performance under general conditions.
In this work, we analyze the learnability of QNNs from a statistical perspective considering the information of loss values. Specifically, given a certain loss value during the training process, we investigate the statistical properties of surrounding training landscapes. Here we mainly focus on quantum state learning tasks [48, 49], which can be seen as a quantum analog of probability distribution learning and play a central role in QML. To summarize, our contributions include:
- We prove a no-go theorem stating that during the process of learning an unknown quantum state with QNNs, the probability of avoiding local minima is of order  $\mathcal{O}(N^2 2^{-N}D^2 /\epsilon^2)$  as long as the loss value is lower than a critical threshold (cf. Fig. 1). The bound vanishes exponentially in the qubit count  $N$  while only increases polynomially with the circuit depth  $D$ . The curvature of local minima is concentrated to the quantum Fisher information times a loss-dependent constant. The proof is mainly based on the technique of "subspace Haar integration" we developed in Appendix A.1. A generalized version for the local loss function is provided in Appendix C.
- We conduct extensive numerical experiments to verify our theoretical findings. We first compare our bound with practical loss curves to show the prediction ability on the statistical behavior of the actual training process. Then we sample landscape profiles to visualize the existence of asymptotic local minima. Finally, we compute the gradients and diagonalize the Hessian matrices to directly verify the correctness of our bound.
- Our results place general limits on the learnability of QNNs, especially for the training strategies beyond the scope of barren plateaus, including high-fidelity initial guesses, initialization heuristics near the identity, and adaptive and layerwise training methods. Moreover, our results provide a theoretical basis for the necessity of introducing prior information into QNN designs and hence draw a guideline for future QNN developments.
The barren plateau phenomenon was first discovered by [35], which proves that the variance of the gradient vanishes exponentially with the system size if the randomly initialized QNN forms a unitary 2-design. Thereafter, [36] finds the dependence of barren plateaus on the circuit depth for loss functions with local observables. [50] proves that training QNNs is in general NP-hard. [51] introduces barren plateaus from uncertainty which precludes learning scramblers. [51, 52] establish connections among the expressibility, generalizability and trainability of QNNs. [53] and [54] show that apart from barren plateaus, QNNs also suffer from local minima in certain cases.
On the other hand, many training strategies have been proposed to address barren plateaus. Here we only list a small part relevant to our work. [43-46] suggest that initializing the QNN near the identity could reduce the randomness and hence escape from barren plateaus. [27] and [47] propose adaptive and layerwise training methods which avoid using randomly initialized QNNs in order to avoid barren plateaus, whereas [55] finds counterexamples where the circuit training terminates close to the identity and remains near to the identity for subsequently added layers without effective progress.
2 Quantum computing basics and notations
We use  $\| \cdot \| _p$  to denote the  $l_{p}$ -norm for vectors and the Schatten-  $p$  norm for matrices.  $A^{\dagger}$  is the conjugate transpose of matrix  $A$ . tr  $A$  represent the trace of  $A$ . The  $\mu$ -th component of the vector  $\theta$  is denoted as  $\theta_{\mu}$  and the derivative with respect to  $\theta_{\mu}$  is simply denoted as  $\partial_{\mu} = \frac{\partial}{\partial\theta_{\mu}}$ . We employ  $\mathcal{O}$  as the asymptotic notation of upper bounds.
In quantum computing, the basic unit of quantum information is a quantum bit or qubit. A single-qubit pure state is described by a unit vector in the Hilbert space  $\mathbb{C}^2$ , which is commonly written in Dirac notation  $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ , with  $|0\rangle = (1,0)^T$ ,  $|1\rangle = (0,1)^T$ ,  $\alpha, \beta \in \mathbb{C}$  subject to  $|\alpha|^2 + |\beta|^2 = 1$ . The complex conjugate of  $|\psi\rangle$  is denoted as  $\langle \psi | = |\psi\rangle^\dagger$ . The Hilbert space of  $N$  qubits is formed by the tensor product “ $\otimes$ ” of  $N$  single-qubit spaces with dimension  $d = 2^N$ . We denote the inner product of two states  $|\phi\rangle$  and  $|\psi\rangle$  as  $\langle \phi|\psi\rangle$  and the overlap is defined as  $|\langle \phi|\psi\rangle|$ . General mixed quantum states are represented by the density matrix, which is a positive semidefinite matrix  $\rho \in \mathbb{C}^{d\times d}$  subject to  $\operatorname{tr}\rho = 1$ . Quantum gates are unitary matrices, which transform quantum states via the matrix-vector multiplication. Common single-qubit rotation gates include  $R_x(\theta) = e^{-i\theta X/2}$ ,  $R_y(\theta) = e^{-i\theta Y/2}$ ,  $R_z(\theta) = e^{-i\theta Z/2}$ , which are in the matrix exponential form of Pauli matrices
$$
X = \left( \begin{array}{c c} 0 & 1 \\ 1 & 0 \end{array} \right), \qquad Y = \left( \begin{array}{c c} 0 & - i \\ i & 0 \end{array} \right), \qquad Z = \left( \begin{array}{c c} 1 & 0 \\ 0 & - 1 \end{array} \right). \tag {1}
$$
Common two-qubit gates include controlled-X gate CNOT  $= I \oplus X$  ( $\oplus$  is the direct sum) and controlled-Z gate CZ  $= I \oplus Z$ , which can generate quantum entanglement among qubits.
2.1 Framework of Quantum Neural Networks
Quantum neural networks (QNNs) typically refer to parameterized quantum circuits  $\mathbf{U}(\pmb{\theta})$  where the parameters  $\pmb{\theta}$  are trainable based on the feedback from quantum measurement results using a classical optimizer. By assigning some loss function  $\mathcal{L}(\pmb{\theta})$ , QNNs can be used to accomplish various tasks just like artificial neural networks. A general form of QNNs reads  $\mathbf{U}(\pmb{\theta}) = \prod_{\mu=1}^{M} U_{\mu}(\theta_{\mu}) W_{\mu}$ , where  $U_{\mu}(\theta_{\mu}) = e^{-i\Omega_{\mu}\theta_{\mu}}$  is a parameterized gate such as single-qubit rotations with  $\Omega_{\mu}$  being a Hermitian generator.  $W_{\mu}$  is a non-parameterized gate such as CNOT and CZ. The product  $\prod_{\mu=1}^{M}$  is by default in the increasing order from the right to the left.  $M$  denotes the number of trainable parameters. Note that QNNs with intermediate classical controls [56] can also be included in this general form theoretically. Commonly used templates of QNNs include the hardware efficient ansatz [26], the alternating-layered ansatz (ALT) [57] and the tensor-network-based ansatz [36, 58], which are usually composed of repeated layers. The number of repeated layers is called the depth of the QNN, denoted as  $D$ . Fig. 2 depicts an example of the ALT circuit. The gradients of loss functions of certain QNNs are evaluated by the parameter-shift rule [59-61] on real quantum devices. Hence we can train QNNs efficiently with gradient-based optimizers [62].
Here we focus on quantum state learning tasks, the objective of which is to learn a given target state  $|\phi \rangle$  encoding practical data via minimizing the distance between the target state  $|\phi \rangle$  and the output
Figure 2: The quantum circuit of the alternating-layered ansatz on 4 qubits. The circuit starts with a  $R_y$  layer and a  $R_x$  layer, followed by  $D$  repeated layers, where each layer contains alternating 2-qubit unit blocks of a CZ gate, two  $R_y$  gates and two  $R_z$  gates.
state from the QNN  $|\psi(\pmb{\theta}) \rangle = U(\pmb{\theta}) |0\rangle^{\otimes N}$ . The corresponding loss function is usually chosen as the fidelity distance
$$
\mathcal {L} (\boldsymbol {\theta}) = 1 - \left| \langle \phi | \psi (\boldsymbol {\theta}) \rangle \right| ^ {2}. \tag {2}
$$
which can be efficiently calculated on quantum computers using the swap test [63]. An important quantity we used below characterizing the sensitivity of the QNN output state  $|\psi (\pmb {\theta})\rangle$  regarding its parameters  $\pmb{\theta}$ , the quantum Fisher information (QFI) matrix  $\mathcal{F}_{\mu \nu}$  [64], which is defined as the Riemannian metric induced from the fidelity distance (cf. Appendix A.4)
$$
\mathcal {F} _ {\mu \nu} (\boldsymbol {\theta}) = 2 \operatorname {R e} \left[ \left\langle \partial_ {\mu} \psi \mid \partial_ {\nu} \psi \right\rangle - \left\langle \partial_ {\mu} \psi \mid \psi \right\rangle \left\langle \psi \mid \partial_ {\nu} \psi \right\rangle \right]. \tag {3}
$$
If the QFI is not full-rank, we say  $|\psi(\pmb{\theta})\rangle$  is over-parameterized [65, 66] meaning that there are redundant degrees of freedom in the parameters over the manifold dimension of the state.
3 Statistical characterization of quantum state learning in QNNs
In this section, we develop a no-go theorem characterizing the limitation of quantum neural networks in state learning tasks from a statistical perspective. In short, we prove that the probability of avoiding local minima during the process of learning an unknown quantum state with a QNN is of order  $\mathcal{O}(2^{-N}M^2/\epsilon^2)$ , where  $N$  is the number of qubits,  $M$  is the number of trainable parameters and  $\epsilon$  represents the typical precision of measurements. The detailed upper bound also depends on the overlap between the value of the loss function and the QFI of the QNN. Our bounds significantly improve existing results of the trainability analysis of QNNs, which mainly focus on the randomness from the initialization and neglect the information of the loss function value. We will first introduce our ensemble setting in Section 3.1, then present our main theorem on local minima in Section 3.2 and finally show some results beyond local minima in Section 3.4.
3.1 Ensemble of the unknown target state
We first introduce the probability measure used in this work. The randomness studied by most of the previous work on the trainability analyzes of QNNs originates from the random initialization of trainable parameters [35], which usually depends on the circuit depth, specific choices of the QNN architecture and initialization strategies [44]. Meanwhile, the randomness can also come from the lack of prior information, such as learning an unknown quantum state or an unknown scrambler like a black hole [67]. We focus on the latter in the present work.
The usage of adaptive methods is usually not covered by common trainability analyses, however, the training process often tends to stagnate. With the aim of investigating the trainability at a specific loss function value, the ensemble is constructed by a uniform measure of overall pure states that have the same overlap with the current output state of the QNN. Specifically, suppose  $\pmb{\theta}^{*}$  is the current value of the trainable parameters. The overlap, or fidelity, between the output state  $|\psi^{*}\rangle = |\psi (\pmb{\theta}^{*})\rangle$  and the target state  $|\phi \rangle$  equals to  $|\langle \phi |\psi^{*}\rangle | = p$ . Thus, the target state can be decomposed as
$$
| \phi \rangle = p | \psi^ {*} \rangle + \sqrt {1 - p ^ {2}} | \psi^ {\perp} \rangle , \tag {4}
$$
where  $|\psi^{\perp}\rangle$  represents the unknown component in the target state  $|\phi \rangle$  orthogonal to the learnt component  $|\psi^{*}\rangle$ . If no more prior information is known about the target state  $|\phi \rangle$  except for the overlap
$p$ , in the spirit of Bayesian statistics,  $|\psi^{\perp}\rangle$  is supposed to be a random state uniformly distributed in the orthogonal complement of  $|\psi^{*}\rangle$ , denoted as  $\mathcal{H}^{\perp}$ . Such a Haar-random state can induce an ensemble of the unknown target state via Eq. (4), which we denote as  $\mathbb{T} = \{| \phi \rangle | |\psi^{\perp}\rangle$  is Haar-random in  $\mathcal{H}^{\perp}\}$ . Graphically,  $\mathbb{T}$  can be understood as a contour on the Bloch sphere of an  $N$ -qubit system with a constant distance to  $|\psi^{*}\rangle$  as shown in Fig. 1(b). We remark that  $\theta^{*}$  can be interpreted as either an initial guess or an intermediate value during the training process so that our following results can be applied to the entire process of learning a quantum state. See Appendix A.1 for more details on the ensemble setting.
3.2 Exponentially likely local minima
We now investigate the statistical properties of the gradient  $\nabla \mathcal{L}$  and the Hessian matrix  $H_{\mathcal{L}}$  of the loss function  $\mathcal{L}(\pmb{\theta}^{*})$  at the parameter point  $\pmb{\theta} = \pmb{\theta}^{*}$  regarding the ensemble  $\mathbb{T}$ , and hence derive an upper bound of the probability that  $\pmb{\theta}^{*}$  is not a local minimum. For simplicity of notation, we represent the value of a certain function at  $\pmb{\theta} = \pmb{\theta}^{*}$  by appending the superscript "*, e.g.,  $\nabla \mathcal{L}|_{\pmb{\theta} = \pmb{\theta}^{*}}$  as  $\nabla \mathcal{L}^*$  and  $H_{\mathcal{L}}|_{\pmb{\theta} = \pmb{\theta}^{*}}$  as  $H_{\mathcal{L}}^*$ . We define that  $\pmb{\theta}^{*}$  is a local minimum up to a fixed precision  $\epsilon = (\epsilon_1, \epsilon_2)$  if and only if each of the gradient components is not larger than  $\epsilon_1$  and the minimal eigenvalue of the Hessian matrix is not smaller than  $-\epsilon_2$ , i.e.,
$$
\operatorname {L o c a l M i n} \left(\boldsymbol {\theta} ^ {*}, \epsilon\right) = \bigcap_ {\mu = 1} ^ {M} \left\{\left| \partial_ {\mu} \mathcal {L} ^ {*} \right| \leq \epsilon_ {1} \right\} \cap \left\{H _ {\mathcal {L}} ^ {*} \succ - \epsilon_ {2} I \right\}. \tag {5}
$$
If  $\epsilon_{1}$  and  $\epsilon_{2}$  both take zero, Eq. (5) is reduced back to the common exact definition of the local minimum. However, noises and uncertainties from measurements on real quantum devices give rise to a non-zero  $\epsilon$ , where the estimation cost scales as  $\mathcal{O}(1 / \epsilon^{\alpha})$  for some power  $\alpha$  [68]. Specially, if  $|\psi (\pmb{\theta}^{*})\rangle$  approaches the true target state  $|\phi \rangle$  such that  $\mathcal{L}^{*}\rightarrow 0$ , we say  $\pmb{\theta}^{*}$  is a global minimum. That is to say, here "local minima" are claimed with respect to the entire Hilbert space instead of training landscapes created by different ansatzes. The expectation and variance of the first and second-order derivatives of the loss function are calculated and summarized in Lemma 1, with the detailed proof in Appendix B utilizing the technique we dubbed "subspace Haar integration" in Appendix A.1.
Lemma 1 The expectation and variance of the gradient  $\nabla \mathcal{L}$  and Hessian matrix  $H_{\mathcal{L}}$  of the fidelity loss function  $\mathcal{L}(\pmb {\theta}) = 1 - |\langle \phi |\psi (\pmb {\theta})\rangle |^2$  at  $\pmb {\theta} = \pmb{\theta}^{*}$  with respect to the target state ensemble  $\mathbb{T}$  satisfy
$$
\mathbb {E} _ {\mathbb {T}} \left[ \nabla \mathcal {L} ^ {*} \right] = 0, \quad \operatorname {V a r} _ {\mathbb {T}} \left[ \partial_ {\mu} \mathcal {L} ^ {*} \right] = f _ {1} (p, d) \mathcal {F} _ {\mu \mu} ^ {*}. \tag {6}
$$
$$
\mathbb {E} _ {\mathbb {T}} \left[ H _ {\mathcal {L}} ^ {*} \right] = \frac {d p ^ {2} - 1}{d - 1} \mathcal {F} ^ {*}, \quad \operatorname {V a r} _ {\mathbb {T}} \left[ \partial_ {\mu} \partial_ {\nu} \mathcal {L} ^ {*} \right] \leq f _ {2} (p, d) \| \Omega_ {\mu} \| _ {\infty} ^ {2} \| \Omega_ {\nu} \| _ {\infty} ^ {2}. \tag {7}
$$
where  $\mathcal{F}$  denote the QFI matrix in Eq. (3) and  $\Omega_{\mu}$  is the generator of the gate  $U_{\mu}(\theta_{\mu})$ .  $f_{1}$  and  $f_{2}$  are functions of the overlap  $p$  and the Hilbert space dimension  $d$ , i.e.,
$$
f _ {1} (p, d) = \frac {p ^ {2} \left(1 - p ^ {2}\right)}{d - 1}, \quad f _ {2} (p, d) = \frac {3 2 \left(1 - p ^ {2}\right)}{d - 1} \left[ p ^ {2} + \frac {2 \left(1 - p ^ {2}\right)}{d} \right]. \tag {8}
$$
The exponentially vanishing variances in Lemma 1 imply that the gradient and Hessian matrix concentrate to their expectations exponentially in the number of qubits  $N$  due to  $d = 2^{N}$  for a  $N$ -qubit system. Thus the gradient concentrates to zero and the Hessian matrix concentrates to the QFI  $\mathcal{F}^*$  times a non-vanishing coefficient proportional to  $(p^2 - 1/d)$ . Since the QFI is always positive semidefinite, the expectation of the Hessian matrix is either positive semidefinite  $\mathcal{L}^* = 1 - p^2 < 1 - 1/d$ , or negative semidefinite if  $\mathcal{L}^* > 1 - 1/d$ , as illustrated in Fig. 1(a). The critical point  $\mathcal{L}_c = 1 - 1/d$  coincides with the average fidelity distance of two Haar-random pure states, which means that as long as  $|\psi^*\rangle$  has a higher fidelity than the average level of all states, the expectation of the Hessian matrix would be positive semidefinite.
Using Lemma 1, we establish an exponentially small upper bound on the probability that  $\theta^{*}$  is not a local minimum in the following Theorem 2, where the generator norm vector  $\omega$  is defined as  $\omega = (\| \Omega_1\|_{\infty},\| \Omega_2\|_{\infty},\dots ,\| \Omega_M\|_{\infty})$
Theorem 2 If the fidelity loss function satisfies  $\mathcal{L}(\pmb{\theta}^{*}) < 1 - 1 / d$ , the probability that  $\pmb{\theta}^{*}$  is not a local minimum of  $\mathcal{L}$  up to a fixed precision  $\epsilon = (\epsilon_{1},\epsilon_{2})$  with respect to the target state ensemble  $\mathbb{T}$  is upper bounded by
$$
\Pr_ {\mathbb {T}} [ \neg \operatorname {L o c a l M i n} (\boldsymbol {\theta} ^ {*}, \epsilon) ] \leq \frac {2 f _ {1} (p , d) \| \boldsymbol {\omega} \| _ {2} ^ {2}}{\epsilon_ {1} ^ {2}} + \frac {f _ {2} (p , d) \| \boldsymbol {\omega} \| _ {2} ^ {4}}{\left(\frac {d p ^ {2} - 1}{d - 1} e ^ {*} + \epsilon_ {2}\right) ^ {2}}, \tag {9}
$$
where  $e^{*}$  denotes the minimal eigenvalue of the QFI matrix at  $\pmb{\theta} = \pmb{\theta}^{*}$ .  $f_{1}$  and  $f_{2}$  are defined in Eq. (8) which vanish at least of order  $1 / d$ .
A sketch version of the proof is as follows, with the details in Appendix B. By definition in Eq. (5), the left-hand side of Eq. (9) can be upper bounded by the sum of two terms: the probability that one gradient component is larger than  $\epsilon_{1}$ , and the probability that the Hessian matrix is not positive definite up to  $\epsilon_{2}$ . The first term can be bounded by Lemma 1 and Chebyshev's inequality, i.e.,
$$
\Pr_ {\mathbb {T}} \left[ \bigcup_ {\mu = 1} ^ {M} \left\{\left| \partial_ {\mu} \mathcal {L} ^ {*} \right| > \epsilon_ {1} \right\} \right] \leq \sum_ {\mu = 1} ^ {M} \frac {\operatorname {V a r} _ {\mathbb {T}} \left[ \partial_ {\mu} \mathcal {L} ^ {*} \right]}{\epsilon_ {1} ^ {2}} = \frac {f _ {1} (p , d)}{\epsilon_ {1} ^ {2}} \operatorname {t r} \mathcal {F} ^ {*}, \tag {10}
$$
where the QFI diagonal element is bounded as  $\mathcal{F}_{\mu \mu} \leq 2\|\Omega_{\mu}\|_{\infty}^{2}$  by definition and thus  $\mathrm{tr}\mathcal{F}^* \leq 2\|\boldsymbol{\omega}\|_2^2$ . After assuming  $p^2 > 1 / d$ , the second term can be upper bounded by perturbing  $\mathbb{E}_{\mathbb{T}}[H_{\mathcal{L}}^*]$  to obtain a sufficient condition of positive definiteness (see Appendix A.2), and then utilizing Lemma 1 with the generalized Chebyshev's inequality for matrices (see Appendix A.3), i.e.,