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Learning Quantum Processes with Quantum Statistical Queries

Chirag Wadhwa, Mina Doosti

TL;DR

The paper develops a quantum statistical query framework to learn quantum processes, introducing the QPSQ oracle that yields estimates of $\mathrm{Tr}(O \mathcal{E}(\rho))$ for unknown channels $\mathcal{E}$. It provides an average-case shadow-process tomography algorithm with near-optimal dependence on the number of observables and proves exponential/doubly exponential hardness for learning unitaries in diamond distance, including exact Haar and approximate designs. The authors also demonstrate a cryptanalytic application by attacking CR-QPUF-based authentication protocols, linking learning efficiency to hardware-security vulnerabilities and highlighting practical implications for near-term quantum devices. The work clarifies the trade-offs between query complexity, observables, and state distributions, and outlines future work on extending to broader process classes and experimental validation. Overall, the results deepen our understanding of learnability in quantum-process models and illuminate the intersection of quantum learning with cryptography and hardware security.

Abstract

In this work, we initiate the study of learning quantum processes from quantum statistical queries. We focus on two fundamental learning tasks in this new access model: shadow tomography of quantum processes and process tomography with respect to diamond distance. For the former, we present an efficient average-case algorithm along with a nearly matching lower bound with respect to the number of observables to be predicted. For the latter, we present average-case query complexity lower bounds for learning classes of unitaries. We obtain an exponential lower bound for learning unitary 2-designs and a doubly exponential lower bound for Haar-random unitaries. Finally, we demonstrate the practical relevance of our access model by applying our learning algorithm to attack an authentication protocol using Classical-Readout Quantum Physically Unclonable Functions, partially addressing an important open question in quantum hardware security.

Learning Quantum Processes with Quantum Statistical Queries

TL;DR

The paper develops a quantum statistical query framework to learn quantum processes, introducing the QPSQ oracle that yields estimates of for unknown channels . It provides an average-case shadow-process tomography algorithm with near-optimal dependence on the number of observables and proves exponential/doubly exponential hardness for learning unitaries in diamond distance, including exact Haar and approximate designs. The authors also demonstrate a cryptanalytic application by attacking CR-QPUF-based authentication protocols, linking learning efficiency to hardware-security vulnerabilities and highlighting practical implications for near-term quantum devices. The work clarifies the trade-offs between query complexity, observables, and state distributions, and outlines future work on extending to broader process classes and experimental validation. Overall, the results deepen our understanding of learnability in quantum-process models and illuminate the intersection of quantum learning with cryptography and hardware security.

Abstract

In this work, we initiate the study of learning quantum processes from quantum statistical queries. We focus on two fundamental learning tasks in this new access model: shadow tomography of quantum processes and process tomography with respect to diamond distance. For the former, we present an efficient average-case algorithm along with a nearly matching lower bound with respect to the number of observables to be predicted. For the latter, we present average-case query complexity lower bounds for learning classes of unitaries. We obtain an exponential lower bound for learning unitary 2-designs and a doubly exponential lower bound for Haar-random unitaries. Finally, we demonstrate the practical relevance of our access model by applying our learning algorithm to attack an authentication protocol using Classical-Readout Quantum Physically Unclonable Functions, partially addressing an important open question in quantum hardware security.
Paper Structure (34 sections, 16 theorems, 85 equations, 2 figures, 3 algorithms)

This paper contains 34 sections, 16 theorems, 85 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Quantum Statistical Queries to Quantum Processes (QPSQs):
  4. Predicting properties of quantum processes:
  5. Hardness of diamond-distance learning:
  6. Application to cryptanalysis:
  7. Related work
  8. Quantum Statistical Queries:
  9. Statistical Query Lower Bounds:
  10. Predicting Properties of Quantum Processes:
  11. Concurrent work:
  12. Organization of the paper
  13. Preliminaries
  14. Quantum Information
  15. Unitary Haar Measure and Designs
  16. ...and 19 more sections

Key Result

Lemma 1

We have, for $O \in \mathcal{M}_{N,N}$, For $O \in \mathcal{M}_{N^2,N^2}$, where,

Figures (2)

  • Figure 1: Comparison between simulated errors generated from a normal distribution and those generated using classical shadow tomography to evaluate the EV of the Pauli-Z observable on random single-qubit stabilizer states after evolution under a fixed haar-random unitary. We fix a tolerance value $\tau = 0.2$, and the probability of the deviation lying outside the tolerance, $\delta = 0.0455$
  • Figure 2: Average performance of the learning algorithm on 10 haar-random 6-qubit unitaries, in predicting the outcome of $Z_1$ on three target distributions

Theorems & Definitions (53)

  • Definition 1: Pauli operators
  • Definition 2: Single-qubit stabilizer states
  • Definition 3: Clifford group
  • Definition 4: Quantum process
  • Definition 5: Maximally depolarizing channel
  • Definition 6: Trace Distance and Fidelity
  • Definition 7: Diamond norm and diamond distance
  • Definition 8: POVM
  • Definition 9: Haar measure
  • Definition 10: Exact and Approximate Unitary $t$-Designs
  • ...and 43 more