Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Predicting symmetries of quantum dynamics with optimal samples

Masahito Hayashi, Yu-Ao Chen, Chenghong Zhu, Xin Wang

TL;DR

The paper develops a unified framework for predicting symmetries of unknown quantum dynamics via unitary subgroup hypothesis testing, combining group representation theory with hypothesis-testing techniques. It proves that parallel testing is optimal for minimizing the type-II error and that adaptive or indefinite-causal-order strategies offer no advantage, with the optimal performance characterized by the quantum max-relative entropy: β_PAR^f(ε) = (1-ε) e^{-D_{max}(ρ_{μ0}^f || ρ_{μ}^f)}. A key contribution is providing an operational meaning to D_{max} and deriving exact sample-complexity scalings for identity, Z-symmetry, and T-symmetry testing in qubits: identity scales as O(δ^{-1/3}) and Z/T-symmetry as O(δ^{-1/2}) under zero type-I error and type-II error ≤ δ. The results offer practical, efficient protocols for symmetry-based unitary property testing in quantum information processing and settle questions about the necessity of complex control sequences in such tests.

Abstract

Identifying symmetries in quantum dynamics, such as identity or time-reversal invariance, is a crucial challenge with profound implications for quantum technologies. We introduce a unified framework combining group representation theory and subgroup hypothesis testing to predict these symmetries with optimal efficiency. By exploiting the inherent symmetry of compact groups and their irreducible representations, we derive an exact characterization of the optimal type-II error (failure probability to detect a symmetry), offering an operational interpretation for the quantum max-relative entropy. In particular, we prove that parallel strategies achieve the same performance as adaptive or indefinite-causal-order protocols, resolving debates about the necessity of complex control sequences. Applications to the singleton group, maximal commutative group, and orthogonal group yield explicit results: for predicting the identity property, Z-symmetry, and T-symmetry of unknown qubit unitaries, with zero type-I error and type-II error bounded by $δ$, we establish the explicit optimal sample complexity which scales as $\mathcal{O}(δ^{-1/3})$ for identity testing and $\mathcal{O}(δ^{-1/2})$ for T/Z-symmetry testing. These findings offer theoretical insights and practical guidelines for efficient unitary property testing and symmetry-driven protocols in quantum information processing.

Predicting symmetries of quantum dynamics with optimal samples

TL;DR

The paper develops a unified framework for predicting symmetries of unknown quantum dynamics via unitary subgroup hypothesis testing, combining group representation theory with hypothesis-testing techniques. It proves that parallel testing is optimal for minimizing the type-II error and that adaptive or indefinite-causal-order strategies offer no advantage, with the optimal performance characterized by the quantum max-relative entropy: β_PAR^f(ε) = (1-ε) e^{-D_{max}(ρ_{μ0}^f || ρ_{μ}^f)}. A key contribution is providing an operational meaning to D_{max} and deriving exact sample-complexity scalings for identity, Z-symmetry, and T-symmetry testing in qubits: identity scales as O(δ^{-1/3}) and Z/T-symmetry as O(δ^{-1/2}) under zero type-I error and type-II error ≤ δ. The results offer practical, efficient protocols for symmetry-based unitary property testing in quantum information processing and settle questions about the necessity of complex control sequences in such tests.

Abstract

Identifying symmetries in quantum dynamics, such as identity or time-reversal invariance, is a crucial challenge with profound implications for quantum technologies. We introduce a unified framework combining group representation theory and subgroup hypothesis testing to predict these symmetries with optimal efficiency. By exploiting the inherent symmetry of compact groups and their irreducible representations, we derive an exact characterization of the optimal type-II error (failure probability to detect a symmetry), offering an operational interpretation for the quantum max-relative entropy. In particular, we prove that parallel strategies achieve the same performance as adaptive or indefinite-causal-order protocols, resolving debates about the necessity of complex control sequences. Applications to the singleton group, maximal commutative group, and orthogonal group yield explicit results: for predicting the identity property, Z-symmetry, and T-symmetry of unknown qubit unitaries, with zero type-I error and type-II error bounded by , we establish the explicit optimal sample complexity which scales as for identity testing and for T/Z-symmetry testing. These findings offer theoretical insights and practical guidelines for efficient unitary property testing and symmetry-driven protocols in quantum information processing.

Paper Structure

This paper contains 14 sections, 6 theorems, 83 equations, 2 figures.

Table of Contents

  1. Parallel Protocols Achieve Optimality.
  2. Connection to Max-Relative Entropy.
  3. Equivalence of Average and Worst-Case Analysis.
  4. Error Tolerance.
  5. Notations
  6. Detailed proofs of the main results
  7. General problem formulation and Statement of Theorem \ref{['NMI']}
  8. Proof of \ref{['KH1']}
  9. Calculation of $D_{\max} (\rho_{\mu_0}\|\rho_{\mu})$
  10. Proof of \ref{['KH2']}: Parallel scheme
  11. Applications via case study
  12. Application: Identity Test via studying the case with $G_0=\{I\}$ and $G$=SU($d$)
  13. Application: Z-symmetry Test via studying the case with commutative $G_0$ and $G$=SU($d$)
  14. Application: T-symmetry Test via studying the case with $G_0=O(d)$ and $G$=U($d$)

Key Result

Theorem 1

For unitary subgroup hypothesis testing involving compact group $G$ and its subgroup $K$ with unitary representation $f$ and an error tolerance of $\epsilon$, the following holds, where $\rho^f_{\mu_0}$ and $\rho^f_{\mu}$ are performance operator of a representation $f$ on the Haar measure of group $G_0$ and $G$, respectively (cf. Eq. eq:average performance state).

Figures (2)

  • Figure 1: An illustration of hypothesis testing for quantum dynamics of interest, considering protocols such as parallel, adaptive, and indefinite causal order strategies.
  • Figure 2: The optimal type-II error scaling with respect to the number of queries in identity, Z-symmetry and T-symmetry identification cases.

Theorems & Definitions (7)

  • Theorem 1: Optimal type-II error of unitary subgroup hypothesis testing
  • Theorem 2: General solutions
  • Theorem 3: Optimal type-II error of unitary subgroup hypothesis testing
  • Theorem 4: General solutions
  • proof
  • Theorem 5
  • Theorem 6