Quantifying the Expressive Capacity of Quantum Systems: Fundamental Limits and Eigentasks

TL;DR

The paper tackles how quantum sampling noise limits the expressive capacity of quantum systems used for learning. It introduces EC as the maximum information extractable via linear readouts from finitely-sampled quantum measurements, and derives a closed-form bound $C_T(\boldsymbol{\theta}) = \mathrm{Tr}((\mathbf{G} + \frac{1}{S}\mathbf{V})^{-1}\mathbf{G}) = \sum_{k=0}^{K-1} \frac{1}{1 + \beta_k^2(\boldsymbol{\theta})/S}$, where $\beta_k^2$ are NSR eigenvalues from $\mathbf{V}\mathbf{r}^{(k)} = \beta_k^2 \mathbf{G}\mathbf{r}^{(k)}$ and $y^{(k)}(u) = \sum_j r_j^{(k)} x_j(u)$ are eigentasks. The authors analytically solve the EC for quantum 2-designs, revealing a sharp drop in finite-S capacity for large systems, and demonstrate experimentally on IBMQ devices that increasing quantum correlations boosts EC and improves learning robustness. They further propose an eigentask-based learning strategy that truncates to $K_c(S) = \max_k\{\beta_k^2 < S\}$, mitigating overfitting and leveraging the noise structure to optimize performance. Overall, the work provides a practical, computable metric (EC) and a noise-aware basis (eigentasks) to guide circuit design and learning in noisy intermediate-scale quantum devices, with implications for QML and quantum sensing.

Abstract

The expressive capacity of quantum systems for machine learning is limited by quantum sampling noise incurred during measurement. Although it is generally believed that noise limits the resolvable capacity of quantum systems, the precise impact of noise on learning is not yet fully understood. We present a mathematical framework for evaluating the available expressive capacity of general quantum systems from a finite number of measurements, and provide a methodology for extracting the extrema of this capacity, its eigentasks. Eigentasks are a native set of functions that a given quantum system can approximate with minimal error. We show that extracting low-noise eigentasks leads to improved performance for machine learning tasks such as classification, displaying robustness to overfitting. We obtain a tight bound on the expressive capacity, and present analyses suggesting that correlations in the measured quantum system enhance learning capacity by reducing noise in eigentasks. These results are supported by experiments on superconducting quantum processors. Our findings have broad implications for quantum machine learning and sensing applications.

Figures (16)

• Figure 1: (a) Representation of the learning framework considered in this work: inputs $\bm{u}$ are transformed to a set of outputs via a parameterized feature generator, here implemented using a finitely-sampled quantum system as shown in (b). Inputs are encoded in the state of a quantum system via a general quantum channel $\mathcal{U}$, and information is then extracted through a positive operator-valued measure. This framework describes a wide range of practical quantum systems, from quantum circuits used in supervised or generative quantum machine learning, to quantum annealers exhibiting continuous evolution, and beyond, all defined by a general quantum channel with parameters $\bm{\theta}$. Extracted information takes the form of $K$ stochastic features $\bar{\bm{X}}$ obtained under finite shots $S$. The geometric structure of distributions of these measured features is fundamentally determined by quantum sampling noise, which depends on the quantum state $\hat{\rho}(\bm{u};\bm{\theta})$, and hence on the nature of the mapping from input $\bm{u}$ to this quantum state. We show four obtained distributions differing only in the values of inputs $\bm{u}$ to highlight this dependence. As shown in (a), learned estimates for desired functions are then constructed via a linear combination ${\bm{w}}$ of $\bar{\bm{X}}$, with a resolution limited by $S$. Capacity $C[f]$ then quantifies the error in the approximation of a target function $f$ via this scheme.
• Figure 2: (a) A representation of the EC analysis, featuring the IBMQ Perth device and a schematic of the quantum circuit considered in this section. On the right, the specific feature plotted is $\bar{X}_1 (u)$ ($\bm{b}_1=000001$) with $S=2^{14}$ shots. (b) Left panel: Device noise-to-signal spectrum $\beta^2_k$ for a specific encoding as a correlated system (CS), $J=\pi/2$ (blue crosses) and product system (PS), $J=0$ (brown diamonds). Ideal (solid) and device noise (dashed) simulations are also shown. Note the agreement between device and simulation, along with distortion from more direct exponential growth in $\beta^2_k$ with $k$ in the ideal case, due to device errors. Right panel: $C_T$ vs. $S$ calculated from the left panel. At a given $S$, the $C_T$ can be approximated by performing the indicated sum over all $\beta_k^2 < S$. (c) Expressive capacity $C_T$ (top panel) and expected total correlation $\bar{\mathcal{T}}$ (lower panel) for the chosen encoding under $S=2^{14}$ from the IBM device, and device noise simulations (dashed peach). Average metrics over 8 random encodings for device noise (solid peach) and ideal (solid gray) simulations are also shown. The $S\to\infty$ expressive capacity of these encodings always attains the ${\rm max}\{C_T\}=64$, indicated in dashed red.
• Figure 3: (a) Device eigentasks for correlated system (CS, left) and product system (PS, right), constructed from noisy features at $S=2^{10}$ and $S=2^{14}$. (b) Classification demonstration on IBMQ Perth. Binary distributions to be classified over the input domain are shown. (c) The classification task can be cast as learning the likelihood function separating the two distributions; this target function is shown in the upper panel. Lower panels show the learned estimate of this target based on the $N_{\rm train}=150$ points shown in (b), using only $K_c(S)$ eigentasks for $S = 2^{14}$; this cutoff is indicated by the dashed red lines. For the correlated system $K_c(S) = 40$, while for the product system $K_c(S) = 29$.
• Figure 4: (a) Training (light) and testing (dark) accuracy for the device encodings of Fig. \ref{['fig:Genc3']}(a), as a function of the number of eigentasks used to approximate the target function. Markers indicate performance on the dataset shown in Fig. \ref{['fig:Genc3']}(b), and solid lines are the average over $10$ random selections of training and test sets. The shaded region denotes the maximum and minimum test accuracy observed. The optimal test set performance is found near the noise-to-signal cutoff $K_c(S=2^{14})$ (dash-dotted lines) informed by the quantum system's noise-to-signal spectra. (b) Testing set classification accuracy as a function of $J$ for our optimal learning method. In all cases, the average performance over the $10$ task permutations is reported, using $K_c(S=2^{14})$. Markers indicate device results for the chosen encoding, and the corresponding simulation is shown in solid peach. Dashed peach shows the average of these results over the $8$ device noise simulation encodings, and dashed grey the ideal simulation performance in the $S\to\infty$ limit, where all $K=64$ features are used. The horizontal line denotes the performance of a software neural network with $K_{\rm L}=64$ nodes (and $1153 \gg K_c$ trained parameters) for comparison.
• Figure 5: Schematic of a simple $L=2$ qubit circuit, comprised of a CNOT gate sanwiched by input-dependent local $x$-rotation gates $\{R_i(\bm{u})\}$. Different $2$D inputs shown on the left are mapped to the finite-$S$ feature space on the right via this circuit. Specifically, a $2$D slice ($\bar{X}_{00}$ and $\bar{X}_{11}$) of the $4$D feature space is shown. Each point represents an individual sample or experiment, i.e. an output constructed with $S<\infty$ shots via Eq. \ref{['eq:Xsum']}. Distinct values of $S=10^2,10^3,10^4$ are shown in different colors (blue, red, green). For each input $\bm{u}$ and shots $S$, the simulation is conducted for $100$ repetition.