Learning to Program Quantum Measurements for Machine Learning

TL;DR

The paper tackles the limitation of fixed, problem-agnostic measurements in variational quantum circuits by introducing Learning to Program Quantum Measurements, a framework where a slow neural controller jointly generates VQC parameters and a data-conditioned Hermitian observable $H(x)$ for each input. This enables end-to-end differentiable training and effectively expands the observable's spectral capabilities beyond the traditional $[-1,1]$ bound. Numerical experiments on synthetic benchmarks (e.g., make_moons, make_circles, make_classification) show that dynamic, input-conditioned measurements yield higher accuracy and more stable convergence than fixed-measurement baselines, with the dual-generator configuration often delivering the best performance. The approach holds promise for enhancing expressivity and robustness of hybrid quantum-classical models on near-term devices by adapting measurements to the input data stream.

Abstract

The rapid advancements in quantum computing (QC) and machine learning (ML) have sparked significant interest, driving extensive exploration of quantum machine learning (QML) algorithms to address a wide range of complex challenges. The development of high-performance QML models requires expert-level expertise, presenting a key challenge to the widespread adoption of QML. Critical obstacles include the design of effective data encoding strategies and parameterized quantum circuits, both of which are vital for the performance of QML models. Furthermore, the measurement process is often neglected-most existing QML models employ predefined measurement schemes that may not align with the specific requirements of the targeted problem. We propose an innovative framework that renders the observable of a quantum system-specifically, the Hermitian matrix-trainable. This approach employs an end-to-end differentiable learning framework, enabling simultaneous optimization of the neural network used to program the parameterized observables and the standard quantum circuit parameters. Notably, the quantum observable parameters are dynamically programmed by the neural network, allowing the observables to adapt in real time based on the input data stream. Through numerical simulations, we demonstrate that the proposed method effectively programs observables dynamically within variational quantum circuits, achieving superior results compared to existing approaches. Notably, it delivers enhanced performance metrics, such as higher classification accuracy, thereby significantly improving the overall effectiveness of QML models.

Figures (15)

• Figure 1: Hybrid quantum-classical computing with learnable observable programmed by the neural network.
• Figure 2: Basic structure of a variational quantum circuit (VQC).
• Figure 3: Our Learning to Program Quantum Measurements is described by a vector bundle $(E, \pi, M)$ with $\pi: E \to M$ the bundle projection. On each classical data point $x\in M$, the living Hilbert space of quantum states is given by $E_x =\pi^{-1}(x) \cong \mathbb{C}^N$ along with a Hermitian inner product $h(x)$ on $E_x$ smoothly changing with respect to $x$. A local section $s: W \subseteq M \to E$ encodes (converts) $x\in M$ into a quantum state $\ket{s(x)} \in E_x$. The inner product $h(x)\left( \ket{s(x)}, \ket{s(x)} \right)$ induces a Hermitian operator $H(x)$ via $\bra{s(x)} H(x) \ket{s(x)}$.
• Figure 4: Schematic of the proposed framework integrating fast weight programmers (FWP) with variational quantum circuits (VQCs). An external neural network controller, trained via gradient descent, takes the classical input and generates two sets of parameters: one for the variational layers of the VQC and another for the learnable quantum observable. The input is encoded via $U(\vec{x})$ and processed through trainable quantum layers $W(\vec{\theta}_j)$ with dynamically generated parameters. The final measurement is performed using a data-conditioned observable, enabling fully input-adaptive quantum inference.
• Figure 5: Visualization of make_moons dataset with different noise configurations.

Theorems & Definitions (2)

• Definition 1
• Definition 2