Adaptivity can help exponentially for shadow tomography

In recent years there has been significant interest in understanding the statistical complexity of learning from quantum data under the constraint that one can only make unentangled measurements. While a key challenge in establishing tight lower bounds in this setting is to deal with the fact that the measurements can be chosen in an adaptive fashion, a recurring theme has been that adaptivity offers little advantage over more straightforward, nonadaptive protocols. In this note, we offer a counterpoint to this. We show that for the basic task of shadow tomography, protocols that use adaptively chosen two-copy measurements can be exponentially more sample-efficient than any protocol that uses nonadaptive two-copy measurements.

Gradient dynamics for low-rank fine-tuning beyond kernels

LoRA has emerged as one of the de facto methods for fine-tuning foundation models with low computational cost and memory footprint. The idea is to only train a low-rank perturbation to the weights of a pre-trained model, given supervised data for a downstream task. Despite its empirical sucess, from a mathematical perspective it remains poorly understood what learning mechanisms ensure that gradient descent converges to useful low-rank perturbations. In this work we study low-rank fine-tuning in a student-teacher setting. We are given the weights of a two-layer base model $f$, as well as i.i.d. samples $(x,f^*(x))$ where $x$ is Gaussian and $f^*$ is the teacher model given by perturbing the weights of $f$ by a rank-1 matrix. This generalizes the setting of generalized linear model (GLM) regression where the weights of $f$ are zero. When the rank-1 perturbation is comparable in norm to the weight matrix of $f$, the training dynamics are nonlinear. Nevertheless, in this regime we prove under mild assumptions that a student model which is initialized at the base model and trained with online gradient descent will converge to the teacher in $dk^{O(1)}$ iterations, where $k$ is the number of neurons in $f$. Importantly, unlike in the GLM setting, the complexity does not depend on fine-grained properties of the activation's Hermite expansion. We also prove that in our setting, learning the teacher model "from scratch'' can require significantly more iterations.

Predicting quantum channels over general product distributions

We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an $n$-qubit channel $E$ and an observable $O$, we aim to learn the mapping \begin{equation*} \rho \mapsto \mathrm{Tr}(O E[\rho]) \end{equation*} to within a small error for most $\rho$ sampled from a distribution $D$. Previously, Huang, Chen, and Preskill proved a surprising result that even if $E$ is arbitrary, this task can be solved in time roughly $n^{O(\log(1/\epsilon))}$, where $\epsilon$ is the target prediction error. However, their guarantee applied only to input distributions $D$ invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states $\rho$. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution $D$, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.

Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation

We study the task of agnostic tomography: given copies of an unknown $n$-qubit state $\rho$ which has fidelity $\tau$ with some state in a given class $C$, find a state which has fidelity $\ge \tau - \epsilon$ with $\rho$. We give a new framework, stabilizer bootstrapping, for designing computationally efficient protocols for this task, and use this to get new agnostic tomography protocols for the following classes: Stabilizer states: We give a protocol that runs in time $\mathrm{poly}(n,1/\epsilon)\cdot (1/\tau)^{O(\log(1/\tau))}$, answering an open question posed by Grewal, Iyer, Kretschmer, Liang [40] and Anshu and Arunachalam [6]. Previous protocols ran in time $\mathrm{exp}(\Theta(n))$ or required $\tau>\cos^2(\pi/8)$. States with stabilizer dimension $n - t$: We give a protocol that runs in time $n^3\cdot(2^t/\tau)^{O(\log(1/\epsilon))}$, extending recent work on learning quantum states prepared by circuits with few non-Clifford gates, which only applied in the realizable setting where $\tau = 1$ [30, 37, 46, 61]. Discrete product states: If $C = K^{\otimes n}$ for some $\mu$-separated discrete set $K$ of single-qubit states, we give a protocol that runs in time $(n/\mu)^{O((1 + \log (1/\tau))/\mu)}/\epsilon^2$. This strictly generalizes a prior guarantee which applied to stabilizer product states [39]. For stabilizer product states, we give a further improved protocol that runs in time $(n^2/\epsilon^2)\cdot (1/\tau)^{O(\log(1/\tau))}$. As a corollary, we give the first protocol for estimating stabilizer fidelity, a standard measure of magic for quantum states, to error $\epsilon$ in $n^3 \mathrm{quasipoly}(1/\epsilon)$ time.

Optimal high-precision shadow estimation

We give the first tight sample complexity bounds for shadow tomography and classical shadows in the regime where the target error is below some sufficiently small inverse polynomial in the dimension of the Hilbert space. Formally we give a protocol that, given any $m\in\mathbb{N}$ and $\epsilon \le O(d^{-12})$, measures $O(\log(m)/\epsilon^2)$ copies of an unknown mixed state $\rho\in\mathbb{C}^{d\times d}$ and outputs a classical description of $\rho$ which can then be used to estimate any collection of $m$ observables to within additive accuracy $\epsilon$. Previously, even for the simpler task of shadow tomography -- where the $m$ observables are known in advance -- the best known rates either scaled benignly but suboptimally in all of $m, d, \epsilon$, or scaled optimally in $\epsilon, m$ but had additional polynomial factors in $d$ for general observables. Intriguingly, we also show via dimensionality reduction, that we can rescale $\epsilon$ and $d$ to reduce to the regime where $\epsilon \le O(d^{-1/2})$. Our algorithm draws upon representation-theoretic tools recently developed in the context of full state tomography.

Faster Diffusion-based Sampling with Randomized Midpoints: Sequential and Parallel

In recent years, there has been a surge of interest in proving discretization bounds for diffusion models. These works show that for essentially any data distribution, one can approximately sample in polynomial time given a sufficiently accurate estimate of its score functions at different noise levels. In this work, we propose a new discretization scheme for diffusion models inspired by Shen and Lee's randomized midpoint method for log-concave sampling~\cite{ShenL19}. We prove that this approach achieves the best known dimension dependence for sampling from arbitrary smooth distributions in total variation distance ($\widetilde O(d^{5/12})$ compared to $\widetilde O(\sqrt{d})$ from prior work). We also show that our algorithm can be parallelized to run in only $\widetilde O(\log^2 d)$ parallel rounds, constituting the first provable guarantees for parallel sampling with diffusion models. As a byproduct of our methods, for the well-studied problem of log-concave sampling in total variation distance, we give an algorithm and simple analysis achieving dimension dependence $\widetilde O(d^{5/12})$ compared to $\widetilde O(\sqrt{d})$ from prior work.

Learning general Gaussian mixtures with efficient score matching

We study the problem of learning mixtures of $k$ Gaussians in $d$ dimensions. We make no separation assumptions on the underlying mixture components: we only require that the covariance matrices have bounded condition number and that the means and covariances lie in a ball of bounded radius. We give an algorithm that draws $d^{\mathrm{poly}(k/\varepsilon)}$ samples from the target mixture, runs in sample-polynomial time, and constructs a sampler whose output distribution is $\varepsilon$-far from the unknown mixture in total variation. Prior works for this problem either (i) required exponential runtime in the dimension $d$, (ii) placed strong assumptions on the instance (e.g., spherical covariances or clusterability), or (iii) had doubly exponential dependence on the number of components $k$. Our approach departs from commonly used techniques for this problem like the method of moments. Instead, we leverage a recently developed reduction, based on diffusion models, from distribution learning to a supervised learning task called score matching. We give an algorithm for the latter by proving a structural result showing that the score function of a Gaussian mixture can be approximated by a piecewise-polynomial function, and there is an efficient algorithm for finding it. To our knowledge, this is the first example of diffusion models achieving a state-of-the-art theoretical guarantee for an unsupervised learning task.

Critical windows: non-asymptotic theory for feature emergence in diffusion models

We develop theory to understand an intriguing property of diffusion models for image generation that we term critical windows. Empirically, it has been observed that there are narrow time intervals in sampling during which particular features of the final image emerge, e.g. the image class or background color (Ho et al., 2020b; Georgiev et al., 2023; Raya & Ambrogioni, 2023; Sclocchi et al., 2024; Biroli et al., 2024). While this is advantageous for interpretability as it implies one can localize properties of the generation to a small segment of the trajectory, it seems at odds with the continuous nature of the diffusion. We propose a formal framework for studying these windows and show that for data coming from a mixture of strongly log-concave densities, these windows can be provably bounded in terms of certain measures of inter- and intra-group separation. We also instantiate these bounds for concrete examples like well-conditioned Gaussian mixtures. Finally, we use our bounds to give a rigorous interpretation of diffusion models as hierarchical samplers that progressively "decide" output features over a discrete sequence of times. We validate our bounds with synthetic experiments. Additionally, preliminary experiments on Stable Diffusion suggest critical windows may serve as a useful tool for diagnosing fairness and privacy violations in real-world diffusion models.

An optimal tradeoff between entanglement and copy complexity for state tomography

There has been significant interest in understanding how practical constraints on contemporary quantum devices impact the complexity of quantum learning. For the classic question of tomography, recent work tightly characterized the copy complexity for any protocol that can only measure one copy of the unknown state at a time, showing it is polynomially worse than if one can make fully-entangled measurements. While we now have a fairly complete picture of the rates for such tasks in the near-term and fault-tolerant regimes, it remains poorly understood what the landscape in between looks like. In this work, we study tomography in the natural setting where one can make measurements of $t$ copies at a time. For sufficiently small $\epsilon$, we show that for any $t \le d^2$, $\widetilde{\Theta}(\frac{d^3}{\sqrt{t}\epsilon^2})$ copies are necessary and sufficient to learn an unknown $d$-dimensional state $\rho$ to trace distance $\epsilon$. This gives a smooth and optimal interpolation between the known rates for single-copy and fully-entangled measurements. To our knowledge, this is the first smooth entanglement-copy tradeoff known for any quantum learning task, and for tomography, no intermediate point on this curve was known, even at $t = 2$. An important obstacle is that unlike the optimal single-copy protocol, the optimal fully-entangled protocol is inherently biased and thus precludes naive batching approaches. Instead, we devise a novel two-stage procedure that uses Keyl's algorithm to refine a crude estimate for $\rho$ based on single-copy measurements. A key insight is to use Schur-Weyl sampling not to estimate the spectrum of $\rho$, but to estimate the deviation of $\rho$ from the maximally mixed state. When $\rho$ is far from the maximally mixed state, we devise a novel quantum splitting procedure that reduces to the case where $\rho$ is close to maximally mixed.

Provably learning a multi-head attention layer

The multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. Given a sequence length $k$, attention matrices $\mathbf{\Theta}_1,\ldots,\mathbf{\Theta}_m\in\mathbb{R}^{d\times d}$, and projection matrices $\mathbf{W}_1,\ldots,\mathbf{W}_m\in\mathbb{R}^{d\times d}$, the corresponding multi-head attention layer $F: \mathbb{R}^{k\times d}\to \mathbb{R}^{k\times d}$ transforms length-$k$ sequences of $d$-dimensional tokens $\mathbf{X}\in\mathbb{R}^{k\times d}$ via $F(\mathbf{X}) \triangleq \sum^m_{i=1} \mathrm{softmax}(\mathbf{X}\mathbf{\Theta}_i\mathbf{X}^\top)\mathbf{X}\mathbf{W}_i$. In this work, we initiate the study of provably learning a multi-head attention layer from random examples and give the first nontrivial upper and lower bounds for this problem: - Provided $\{\mathbf{W}_i, \mathbf{\Theta}_i\}$ satisfy certain non-degeneracy conditions, we give a $(dk)^{O(m^3)}$-time algorithm that learns $F$ to small error given random labeled examples drawn uniformly from $\{\pm 1\}^{k\times d}$. - We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable. We focus on Boolean $\mathbf{X}$ to mimic the discrete nature of tokens in large language models, though our techniques naturally extend to standard continuous settings, e.g. Gaussian. Our algorithm, which is centered around using examples to sculpt a convex body containing the unknown parameters, is a significant departure from existing provable algorithms for learning feedforward networks, which predominantly exploit algebraic and rotation invariance properties of the Gaussian distribution. In contrast, our analysis is more flexible as it primarily relies on various upper and lower tail bounds for the input distribution and "slices" thereof.