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Exponential speedup in measurement property learning with post-measurement states

Zhenhuan Liu, Qi Ye, Zhenyu Cai, Jens Eisert

TL;DR

This work identifies a measurement learning task for which access limited to classical measurement outcomes leads to an exponential lower bound on the query complexity, established via a distinguishing task between a genuine quantum projective measurement and a purely classical random number generator.

Abstract

Learning properties of quantum states and channels is known to benefit from resources such as entangled operations, auxiliary qubits, and adaptivity, whereas the resource structure of measurement learning, namely, learning properties of quantum measurement operators, remains poorly understood. In this work, we identify a measurement learning task for which access limited to classical measurement outcomes leads to an exponential lower bound on the query complexity, established via a distinguishing task between a genuine quantum projective measurement and a purely classical random number generator. Remarkably, this hardness persists even when arbitrary entangled operations, auxiliary systems, and fully adaptive strategies are allowed, indicating that conventional resources for state and channel learning are ineffective in this task. In contrast, when access to the post-measurement quantum state is available, the same task can be solved with constant query complexity using a simple measuring-twice protocol, without requiring resources that are useful for state and channel learning. Our results reveal post-measurement states as a qualitatively new and decisive resource for measurement learning, suggesting potential implications for the design of practical quantum certification protocols.

Exponential speedup in measurement property learning with post-measurement states

TL;DR

This work identifies a measurement learning task for which access limited to classical measurement outcomes leads to an exponential lower bound on the query complexity, established via a distinguishing task between a genuine quantum projective measurement and a purely classical random number generator.

Abstract

Learning properties of quantum states and channels is known to benefit from resources such as entangled operations, auxiliary qubits, and adaptivity, whereas the resource structure of measurement learning, namely, learning properties of quantum measurement operators, remains poorly understood. In this work, we identify a measurement learning task for which access limited to classical measurement outcomes leads to an exponential lower bound on the query complexity, established via a distinguishing task between a genuine quantum projective measurement and a purely classical random number generator. Remarkably, this hardness persists even when arbitrary entangled operations, auxiliary systems, and fully adaptive strategies are allowed, indicating that conventional resources for state and channel learning are ineffective in this task. In contrast, when access to the post-measurement quantum state is available, the same task can be solved with constant query complexity using a simple measuring-twice protocol, without requiring resources that are useful for state and channel learning. Our results reveal post-measurement states as a qualitatively new and decisive resource for measurement learning, suggesting potential implications for the design of practical quantum certification protocols.
Paper Structure (8 sections, 5 theorems, 25 equations, 4 figures)

This paper contains 8 sections, 5 theorems, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Measurement learning
  3. Exponential lower bound with only classical outputs
  4. Constant query complexity with having access to post-measurement states
  5. Quantumness certification
  6. Discussion
  7. Distinguishing between uniform and Porter-Thomas distributions
  8. General query complexity lower bound with only classical output

Key Result

Theorem 1

For any quantum protocol, the query complexity for solving the task introduced in Definition def:certification with only classical output with constant success probability is $\Omega(\sqrt{d})$.

Figures (4)

  • Figure 1: A quantum information processing task consists of state initialization, quantum evolution, and measurement. Although various quantum resources have been identified that yield exponential speedups for state and channel learning, the resource structure of measurement learning remains largely unexplored.
  • Figure 2: The black box distinguishing task.
  • Figure 3: (a) A distinguishing protocol by inputting a fixed state and comparing the output distribution. (b) The most general distinguishing protocol utilizing adaptivity, auxiliary qubits, entangled operation, and controllable measurements. All the measurement outcomes, including the black box measurements and the controllable measurements, represented by blue boxes, will be utilized for the distinguishing task.
  • Figure 4: (a) Protocol for estimating sharpness with the post-measurement state. (b) The adversarial robust protocol with one bit of quantum randomness.

Theorems & Definitions (8)

  • Definition 1: Black-box distinguishing task
  • Theorem 1: Quantum birthday paradox
  • Corollary 1: Query complexity bound
  • Theorem 2: Estimating sharpness
  • Theorem 3: Sample complexity of discriminating the Porter-Thomas from the uniform distribution
  • proof
  • Theorem 4: Query complexity bound without access to post-measurement states
  • proof