The Signal Horizon: Local Blindness and the Contraction of Pauli-Weight Spectra in Noisy Quantum Encodings

TL;DR

The paper addresses how much class information remains operable when measurements are locality-constrained and noise is present in quantum encodings. It introduces the Signal Horizon concept and the LO(k) norm to quantify locally accessible information, along with the computable Pauli-accessible amplitude $A_k(p)$ that lower-bounds achievable accuracy under $k$-local Pauli measurements. A weight-contraction mechanism under independent depolarizing noise is derived via Pauli weights, yielding an operational bound ${\text{Acc}}_k(p) \ge \tfrac{1}{2}+\tfrac{1}{4}A_k(p)$ and a practical estimator for $A_k(p)$ tested on 4-qubit encodings; this reveals a threshold $p^*$ where locality-limited classifiers become indistinguishable from random guessing despite persistent global distinguishability. The results highlight a fundamental separation between globally present information and what is measurably extractable in NISQ devices, informing design principles that separately optimize encoding geometry, noise handling, and measurement locality for robust quantum learning.

Abstract

The performance of quantum classifiers is typically analyzed through global state distinguishability or the trainability of variational models. This study investigates how much class information remains accessible under locality-constrained measurements in the presence of noise. The authors formulate binary quantum classification as constrained quantum state discrimination and introduce a locality-restricted distinguishability measure quantifying the maximum bias achievable by observables acting on at most $k$ subsystems. For $n$-qubit systems subject to independent depolarizing noise, the locally accessible signal is governed by a Pauli-weight-dependent contraction mechanism. This motivates a computable predictor, the $k$-local Pauli-accessible amplitude $A_{k}(p)$, which lower bounds the optimal $k$-local classification advantage. Numerical experiments on four-qubit encodings demonstrate quantitative agreement between empirical accuracy and the prediction across noise levels. The research identifies an operational breakdown threshold where $k$-local classifiers become indistinguishable from random guessing despite persistent global distinguishability.

Figures (2)

• Figure 1: Product encoding ($n=4$). (Top row) Global trace norm $\|\mathcal{N}_p(\Delta\rho)\|_1$ and local amplitude $A_k(p)$ for $k=1,2,3$. (Middle row) Accessible fraction $A_k / \|\Delta\rho\|_1$ and gap $\|\Delta\rho\|_1 - A_k$. (Bottom row) Empirical classification accuracy and theoretical prediction $\tfrac{1}{2}+\tfrac{1}{4} A_k(p)$, together with optimal Pauli weight $w(P^\star)$.
• Figure 2: Entangling encoding ($n=4$). (Top row) Global trace norm and local amplitude $A_k(p)$. (Middle row) Accessible fraction and global--local gap. (Bottom row) Empirical versus predicted accuracy and optimal Pauli weight flow under noise.

Theorems & Definitions (11)

• Theorem 1: Helstrom optimum for unconstrained measurements
• proof
• Definition 1: $\mathrm{LO}(k)$ seminorm
• Lemma 1: Optimal $k$-local accuracy as constrained testing
• proof
• Definition 2: $k$-local Pauli-accessible amplitude
• Remark 1: Normalization
• Lemma 2: Pauli restriction lower-bounds $\mathrm{LO}(k)$ distinguishability
• proof
• Definition 3: Operational breakdown threshold