A Unified Theory of Quantum Neural Network Loss Landscapes

We thank the reviewer for their kind remarks on the novelty and contribution of our results. We now address the five questions of the reviewer, which also addresses the weakness of the paper specified by the reviewer. 1. By our Definition 7, one need only measure the asymptotic scaling of the $\operatorname{Tr}\left(\left(\mathbf{O}^\alpha\right)^2\right)$ and the $\operatorname{Tr}\left(\left(\mathbf{\rho}^\alpha\right)^2\right)$ to determine the asymptotic trainability of a class of quantum neural networks. The latter of these can be explicitly measured on a quantum device through the measurement of basis elements of $\mathcal{A}$; the former, by doing the same on what is called a "block encoding" of $\mathbf{O}$. We now state this following Definition 7. We also provide in the new Appendix E explicit error bounds for this procedure. 2. A quantum protocol for calculating $\operatorname{Tr}\left(\left(\mathbf{O}^\alpha\right)^2\right)$ is now stated following Definition 7. We have also added the new Appendix E, which gives a step-by-step, detailed outline for an implementation of the protocol using the standard linear combination of unitaries subroutine for block encoding ([Quantum Info. Comput. 12, 901](https://doi.org/10.26421/QIC12.11-12-1)), and shows that this implementation of the protocol is efficient. 3. [Anschuetz & Kiani (2022)](https://doi.org/10.1038/s41467-022-35364-5) describe local minima in shallow (not deep), local, brickwork circuits. Their argument essentially is a light cone argument: shallow, local ansatzes with local observables explore a subspace of the full Hilbert space, and brickwork circuits are underparameterized with respect to this local Hilbert space dimension. This is also true for sums of local observables, which the authors approximate by many independent light cones. The results of [Anschuetz & Kiani (2022)](https://doi.org/10.1038/s41467-022-35364-5)---and the equivalent results for other classes of local shallow circuits---are therefore implied by our Corollary 6 by taking the Jordan algebra associated with the network to be a direct sum of algebras associated with each light cone. This is now stated in the new Table 1 (added in response to the reviewer's fourth question), clarified in a footnote in Sec. 1.2, and discussed in Sec. 2.1. This amendment also addresses the weakness of the paper specified by the reviewer. 4. We thank the reviewer for this suggestion. This was the initial idea behind Figure 1 in the initial submission, but we have now replaced it with a table (Table 1) which explicitly references some of the papers our framework generalizes. This table also summarizes what limits of our corollaries encompass these previous results. This amendment also addresses the weakness of the paper specified by the reviewer. 5. Jordan algebras arise naturally when considering quantum neural networks as operations acting on quantum states, which are Hermitian matrices. As Hermitian matrices over a field span one of the classical Jordan algebras (not, for instance, one of the classical Lie algebras), quantum networks can be considered as automorphisms acting on Jordan algebras. As a benefit, this unified description allows us to consider what were previously thought of as "special cases" of algebraically constrained quantum neural networks on the same footing. For instance, quantum networks composed of matchgates are not describable in a Lie algebraic formalism (see, e.g., [arXiv:2310.11505](https://doi.org/10.48550/arXiv.2310.11505)), but we show in what is now Appendix H that they are describable in a Jordan algebraic formalism. The new Table 1 now also explicitly states that the Lie algebraic setting is a limit of our Jordan algebraic setting.

We thank the reviewer for their raising of their score. We now answer the reviewer's follow-up questions. ## Response to Question 1 We assume a 1D brickwork geometry with $n$ qubit input $\mathbf{\rho}$, and assume the objective observable $\mathbf{O}$ only acts locally for simplicity (otherwise, one should perform the block decomposition into light cones as mentioned in our initial response to the reviewer). We also only concretely consider the case when $L\leq\operatorname{O}\left(\log\left(n\right)\right)$ (the "shallow setting") as this is the setting the reviewer previously was commenting on, though calculations in the deep setting proceed similarly. We use $\mathcal{H}_L$ to denote the Hilbert space corresponding the to light cone of $\mathbf{O}$. The number of parameters $p_L$ of the network in the light cone of $\mathbf{O}$ is $p_L\sim L$; similarly, the Hilbert space dimension of the light cone is $N\_L\equiv\dim\left(\mathcal{H}\_L\right)\sim 2^L$ (as there are $n_L\sim L$ qubits in the light cone of $\mathbf{O}$). Assuming $\mathbf{O}$ is given as a Pauli decomposition, \begin{equation} r=\frac{\operatorname{Tr}\left(\mathbf{O}\right)^2}{\operatorname{Tr}\left(\mathbf{O}^2\right)} \end{equation} can be efficiently evaluated classically; typically (i.e., outside of fine-tuned $\mathbf{O}$) this will be $r\sim N\_L$. Similarly, \begin{equation} \mathbf{\rho}\_L\equiv\operatorname{Tr}\_{\mathcal{H}\_L^\complement}\left(\mathbf{\rho}\right) \end{equation} is the state in the light cone of $\mathbf{O}$, and $\operatorname{Tr}\left(\mathbf{\rho}\_L^2\right)$ is just the purity of $\mathbf{\rho}\_L$ which can be estimated via the standard swap test. This gives by our Definition 7: \begin{equation} \operatorname{Var}\_{\mathbf{\theta}}\left[\ell\left(\mathbf{\rho};\mathbf{\theta}\right)\right]=\frac{\operatorname{Tr}\left(\mathbf{O}^2\right)\operatorname{Tr}\left(\mathbf{\rho}\_L^2\right)}{\dim_{\mathbb{R}}\left(\operatorname{SU}\left(N_L\right)\right)}\sim\frac{N_L\operatorname{Tr}\left(\mathbf{\rho}_L^2\right)}{N_L^2}\sim\frac{\operatorname{Tr}\left(\mathbf{\rho}_L^2\right)}{\operatorname{\Omega}\left(\operatorname{poly}\left(n\right)\right)} \end{equation} and \begin{equation} p_L\sim L\ll 2^L\sim N_L. \end{equation} Thus, by our Definition 7, we expect---for $\mathbf{\rho}$ an area law state such that $\operatorname{Tr}\left(\mathbf{\rho}_L^2\right)$ does not vanish exponentially with $n$---the model to have no barren plateaus, but to have poor local minima. In this special setting, the former was proved in [Nat. Commun. 12, 1791](https://doi.org/10.1038/s41467-021-21728-w) and [PRX Quantum 2, 040316](https://doi.org/10.1103/PRXQuantum.2.040316), and the latter in [Nat. Commun. 13, 7760](https://doi.org/10.1038/s41467-022-35364-5). ## Response to Question 2 As discussed in our Sec. 6, we do not believe there are any generic, randomly initialized QNNs which achieve this balance during training. Our results indicate that barren plateaus and poor local minima in randomly initialized QNNs can only be simultaneously avoided if $\dim\left(\mathcal{A}\right)\sim\operatorname{poly}\left(n\right)$ (where $\mathcal{A}$ is the Jordan algebra associated with the QNN). However, this suggests the existence of an efficient classical simulation algorithm, as discussed in our reply to Reviewer 3K5s. As discussed in our Sec. 6, though our results are pessimistic, this still leaves settings with a quantum-classical separation in machine learning (indeed, many separations have been proved unconditionally!). These include: 1. QNNs which have no quantum advantage during training but still having an advantage during inference, as is done in, e.g., [PRX Quantum 4, 020338](https://doi.org/10.1103/PRXQuantum.4.020338) and [arXiv:2402.08606](https://doi.org/10.48550/arXiv.2402.08606). 2. QNNs which are warm started, as is done in [arXiv:2404.10044](https://doi.org/10.48550/arXiv.2404.10044). 3. Quantum models trained via specialized algorithms (i.e., not uniformly-initialized gradient descent), as is done in [Nat. Phys., 17, 1013](https://doi.org/10.1038/s41567-021-01287-z) and [STOC 2024, pp. 1343--1351](https://doi.org/10.1145/3618260.3649722).

We address the reviewer's concerns jointly: Yes, 1D, nearest-neighbor Rzz and Rx gates are a subset of "matchgates"; this is isomorphic to $\operatorname{SO}\left(N\right)$ as a Lie group. So while these gates do not scramble over the full space of unitaries (i.e., $\operatorname{SU}\left(N\right)$), they scramble over $\operatorname{SO}\left(N\right)\subset\operatorname{SU}\left(N\right)$ (see, for instance, [arXiv:2310.11505](https://doi.org/10.48550/arXiv.2310.11505)). Our theorems thus apply exactly as in the $\operatorname{SU}\left(N\right)$ case, with the only difference being instead of taking $\beta=2$ one takes $\beta=1$ in our formulas. This is a setting where the Jordan algebaic description of the network is more natural than, say, a Lie algebraic description. For instance, if the objective observable $\mathbf{O}$ is a real symmetric matrix, it does _not_ belong to the algebra $\mathfrak{so}\left(N\right)$---this precludes any Lie algebraic description (indeed, [arXiv:2310.11505](https://doi.org/10.48550/arXiv.2310.11505)---which studies matchgate networks---includes "beyond the dynamical Lie algebra" in the title). However, $\mathbf{O}$ _does_ belong to the Jordan algebra of real symmetric matrices $\mathcal{H}^N\left(\mathbb{R}\right)$, and matchgates are isomorphic to the automorphism group of this space. Our results thus also hold in this setting, where the associated Jordan algebra $\mathcal{A}\cong\mathcal{H}^N\left(\mathbb{R}\right)$. The Jordan algebraic structure of this network is what allows us to calculate results beyond the existing literature; for instance, for the first time, we calculate the local minima distribution of networks composed of these operations in our Corollary 6. This calculation heavily relies on the Jordan algebraic structure of the network, as it leverages the Wishart process description and real Wishart matrices are not elements of the Lie algebra $\mathfrak{so}\left(N\right)$ but _are_ elements of the Jordan algebra $\mathcal{H}^N\left(\mathbb{R}\right)$.

We thank the reviewer for their kind remarks on the contributions of this work. We apologize that many of the technical details of the work are placed in appendices; this is due to the number of results we show, as well as the lengths of the required proofs and formal definitions. We have updated the manuscript in an effort to make the main text more readable and intuitive, primarily by including many mathematical definitions in the main text that were previously only given in the appendices. A nonexhaustive list of expressions which are now explicitly defined in the main text include: the density of the gamma distribution; the convolution; the projection of an algebra element into a subalgebra; and the semisimple decomposition of an algebra. We have also fixed many typos that were present in the initial submission, which accounted for many mistakenly undefined quantities. We now address the reviewer's concerns regarding the manuscript. 1. We thank the reviewer for pointing out these undefined quantities throughout the paper---many of these were typos, mistakenly unremoved from a previous version. These (and others where found) have now been defined in the main text. We have also removed definitions and cumbersome notation which were unused, which we believe clarifies many statements which were confusing in the original submission. This includes those in the statements of Theorems 2 and 3 that the reviewer mentioned. 2. We again apologize that many of the technical details of the work are in the appendices. In the updated manuscript, we are more comprehensive in our introduction of Jordan algebras in Sec. 2.2 in an effort to make Sec. 3 easier to follow without diving into the appendices. We also make an effort in Secs. 4 and 5 to sketch the techniques we use in the proofs of our results, such that the proof strategies can be understood without requiring all of the technical details supplied in the appendices. Finally, we comment on the reviewer's question on the potential of classical limits within this framework; indeed, a corollary of our result is that each trainable quantum neural network (QNN) has an equivalent, efficient, quantum-inspired classical neural network. As discussed in Sec. 3, every quantum neural network architecture has associated with it a Jordan algebra $\mathcal{A}\cong\bigoplus_\alpha\mathcal{A}\_\alpha$ generated by $\mathbf{U}\_{\mathbf{\theta}}^\dagger\mathbf{O}\mathbf{U}\_{\mathbf{\theta}}$ under the anticommutator; this leads to the general algebraic form of a quantum loss function given in Eq. (16). Note that the argument of the trace in this equation involves matrices in the $N_\alpha\times N_\alpha$ defining representations of the $\mathcal{A}\_\alpha$. If the $N_\alpha$ grow only polynomially with the system size, then, this immediately gives a quantum-inspired classical neural network: given a Hermitian, $N_\alpha\times N_\alpha$-dimensional encoding of the input, one conjugates it with a parameterized unitary matrix, and takes the Frobenius inner product with an $N_\alpha\times N_\alpha$ Hermitian matrix to calculate the loss. This general "dequantization" procedure for algebraically-constrained quantum networks has been previously noted in the literature, see, e.g., [Quantum 7, 1189](https://doi.org/10.22331/q-2023-11-28-1189) and [arxiv:2308.01432](https://doi.org/10.48550/arXiv.2308.01432). As efficient trainability on a quantum computer _also_ requires the $N_\alpha$ to grow only polynomially with the system size by our Definition 7, we state in Sec. 6: "...it seems unlikely that there exists any computational quantum advantage during the training of QNNs...due to efficient classical simulation algorithms for quantum systems algebraically constrained to explore a low-dimensional subspace..." We now more explicitly state in Sec. 6 that this is an implication of our Definition 7. We also now make a similar statement toward the end of Sec. 1.1.

We thank the reviewer for their kind remarks on the organization of the paper and the quality of the technical analysis. We acknowledge that the precise asymptotic limit we are taking was confusingly worded in the theorem statements; we are now more clear, and directly state in both the main text and theorem statements that the limit we are taking is as the dimension of the algebra $\mathcal{A}$ associated with the quantum neural network (QNN) goes to $\infty$ (i.e., the "wide" limit of many qubits). We now address the two questions of the reviewer. ## Response to Question 1 Entanglement arises in a subtle way, which we now make more clear. Eqs. (4) and (6) are functions of the _generalized entanglement_ ([Phys. Rev. Lett. **92**, 107902](https://doi.org/10.1103/PhysRevLett.92.107902)) of the input $\mathbf{\rho}$ and objective observable $\mathbf{O}$, respectively, with respect to the algebraic structure of the Jordan algebra $\mathcal{A}=\bigoplus_\alpha \mathcal{A}_\alpha $ associated with the network. To describe the essence of this property, we first recall that "normal" entanglement is defined as the inability for a state $\mathbf{\rho}$ belonging to a tensor product Hilbert space $\mathcal{H}_A\otimes\mathcal{H}_B$ to be decomposed into a tensor product of states itself. This can be probed by, for instance, taking the partial trace of the state over a subsystem $\mathbf{\rho}_A\equiv\operatorname{Tr}_B\left(\mathbf{\rho}\right)$ and considering the resulting purity $\operatorname{Tr}\left(\mathbf{\rho}_A^2\right)$; $\mathbf{\rho}$ is entangled across the cut between Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$ if and only if $\operatorname{Tr}\left(\mathbf{\rho}_A^2\right)<1$, where the smaller $\operatorname{Tr}\left(\mathbf{\rho}_A^2\right)$ is the more entangled the state is. Similarly, a state $\mathbf{\rho}$ belonging to a direct sum of algebras $\bigoplus_\alpha \mathcal{A}\_\alpha$ exhibits _generalized entanglement_ if $\operatorname{Tr}\left(\left(\mathbf{\rho}^\alpha\right)^2\right)<1$, where $\mathbf{\rho}^\alpha$ is the projection of $\mathbf{\rho}$ into the subalgebra $\mathcal{A}_\alpha$ (see Eq. (12) of the submission for a formal definition). This quantity governs the trainability of the network as described in Definition 7, in the sense that data sets composed of highly entangled states yield barren plateaus and poor local minima in the loss landscape of the QNN. In the latest version of the manuscript, we now state this explicitly at the end of Sec. 5.1 and Appendix C.1. ### Entanglement's Impact on Expressivity What has been described is how entanglement influences the trainability of a given quantum neural network architecture, which is the focus of this work; the reviewer's comment also asks about the potential for an expressivity advantage using entanglement. We note that entanglement has been used to rigorously derive settings in which there is a provable advantage in quantum machine learning over classical machine learning models in learning time series. Typically this is studied through the lens of "quantum contextuality," which is a byproduct of quantum entanglement. We cite two of these works in Sec. 6: [PRX Quantum **4**, 020338](https://doi.org/10.1103/PRXQuantum.4.020338) and [arXiv:2402.08606](https://doi.org/10.48550/arXiv.2402.08606). [arXiv:2410.03094](https://doi.org/10.48550/arXiv.2410.03094) is very recent (released after this work was submitted) but also proves a similar result. See also [arXiv:1403.3351](https://doi.org/10.48550/arXiv.1403.3351) for how contextuality can be used to more efficiently model natural language than noncontextual models. ## Response to Question 2 In an unfortunate overloading of terms, in the quantum information literature the "quantum neural tangent kernel (QNTK) limit" refers to the existence of a Gaussian process limit for the loss function---we agree with the reviewer that the neural tangent kernel is often Wishart-distributed in the classical setting. What we show is that, for QNNs initialized uniformly at random, the _loss function itself_ is not Gaussian-distributed but instead is Wishart-distributed. This leads to the different phenomenological behavior between quantum and classical neural networks, as discussed in Sec. 1.1. Indeed, the covariance of the resulting process in the QNN setting is _not_ Wishart-distributed; see the expression we give for the covariance in Eq. (25). We now clarify throughout the manuscript that we show that the loss function---not the associated kernel matrix---of QNNs is what converges to a Wishart process, and have replaced most uses of the term "QNTK limit" with "Gaussian process limit" to avoid confusion.

We thank the reviewer for their kind remarks on the organization, technical quality, and contributions of the work. We agree with the reviewer that it is unfortunate that these results do not point to any new parameter regimes where one might hope for a generic, large, computational advantage during the training of quantum neural networks. Instead, the value of our work is that the nonexistence of such a setting is _proved_ for networks which do not follow the conditions of Definition 7; we specify exactly the parameter regimes where quantum neural networks (QNNs) are untrainable. In the latest version of the manuscript we also provide a protocol for empirically checking whether or not a given QNN architecture is asymptotically trainable, discussed just after Definition 7. We now address the question on $\mathcal{N}$. Here, we are studying when a class of randomly initialized QNNs with loss function $\ell\left(\mathbf{\rho}\right)$ are such that this loss function asymptotically approaches a Gaussian process. In typical settings, the variance of $\ell\left(\mathbf{\rho}\right)$ vanishes asymptotically; we thus consider normalizing by some $\mathcal{N}\geq 1$ such that $\mathcal{N}\ell\left(\mathbf{\rho}\right)$ has variance nonvanishing asymptotically. However, if there exists no $\mathcal{N}$ such that: 1. this variance is nonvanishing asymptotically, and 2. the higher-order cumulants vanish asymptotically, then $\mathcal{N}\ell\left(\mathbf{\rho}\right)$ cannot form a Gaussian process. What Eq. (25) of the original submission described was when this second condition is satisfied. We acknowledge that the associated section (Sec. 5.2) was written in a confusing manner. We have now rewritten this section. We now emphasize that we are demonstrating when there exists an $\mathcal{N}$ satisfying these two conditions, and explicitly state that (what is now) Eq. (24) describes when the second condition is possible.