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Learning T-conjugated stabilizers: The multiple-squares dihedral StateHSP

Gideon Lee, Jonathan A. Gross, Masaya Fukami, Zhang Jiang

TL;DR

The paper tackles the non-abelian StateHSP by solving the Multiple-Squares StateHSP for $\mathcal{D}_4^N$ using $|\\Psi\rangle$-state inputs and a representation-based approach. The core strategy combines quotient-based abelianization via parity- and Bell-based Fourier sampling, a maximal-rotation step to extract a usable $w^{\max}$, and a T-conjugation (S correction) to reduce the problem to an abelian StateHSP solvable by Learning Pauli Stabilizers. The main result shows the problem can be solved with high probability using $O(\varepsilon^{-2}(N^2+\log(N/\delta)))$ samples, with constant-depth quantum circuits, and it generalizes to arbitrary representations with similar guarantees. The work connects StateHSP to practical tasks in stabilizer learning and Hamiltonian spectroscopy, and it provides a blueprint for tackling other non-abelian StateHSPs by exploiting representation structure and abelian quotients. Overall, this presents a concrete, resource-efficient route to non-abelian StateHSPs and highlights the pivotal role of representation theory in quantum-sampling-based learning.

Abstract

The state hidden subgroup problem (StateHSP) is a recent generalization of the hidden subgroup problem. We present an algorithm that solves the non-abelian StateHSP over $N$ copies of the dihedral group of order $8$ (the symmetries of a square). This algorithm is of interest for learning non-Pauli stabilizers, as well as related symmetries relevant for the problem of Hamiltonian spectroscopy. Our algorithm is polynomial in the number of samples and computational time, and requires only constant depth circuits. This result extends previous work on the abelian StateHSP and, as a special case, provides a solution for the ordinary hidden subgroup problem on this specific non-abelian group.

Learning T-conjugated stabilizers: The multiple-squares dihedral StateHSP

TL;DR

The paper tackles the non-abelian StateHSP by solving the Multiple-Squares StateHSP for using -state inputs and a representation-based approach. The core strategy combines quotient-based abelianization via parity- and Bell-based Fourier sampling, a maximal-rotation step to extract a usable , and a T-conjugation (S correction) to reduce the problem to an abelian StateHSP solvable by Learning Pauli Stabilizers. The main result shows the problem can be solved with high probability using samples, with constant-depth quantum circuits, and it generalizes to arbitrary representations with similar guarantees. The work connects StateHSP to practical tasks in stabilizer learning and Hamiltonian spectroscopy, and it provides a blueprint for tackling other non-abelian StateHSPs by exploiting representation structure and abelian quotients. Overall, this presents a concrete, resource-efficient route to non-abelian StateHSPs and highlights the pivotal role of representation theory in quantum-sampling-based learning.

Abstract

The state hidden subgroup problem (StateHSP) is a recent generalization of the hidden subgroup problem. We present an algorithm that solves the non-abelian StateHSP over copies of the dihedral group of order (the symmetries of a square). This algorithm is of interest for learning non-Pauli stabilizers, as well as related symmetries relevant for the problem of Hamiltonian spectroscopy. Our algorithm is polynomial in the number of samples and computational time, and requires only constant depth circuits. This result extends previous work on the abelian StateHSP and, as a special case, provides a solution for the ordinary hidden subgroup problem on this specific non-abelian group.

Paper Structure

This paper contains 29 sections, 29 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup
  3. Problem and motivation
  4. HSP vs. StateHSP: The non-abelian case
  5. Connection to previous solutions for the regular HSP
  6. Dihedral preliminaries
  7. Algorithm
  8. The Multiple-Squares StateHSP algorithm
  9. Subroutines
  10. Main theorem and supporting results
  11. Proofs
  12. Bell-resolvable sets are efficiently obtainable
  13. Bell-resolvable sets are useful
  14. Good nullspaces exist, and they are efficiently obtainable
  15. Preliminaries: Annihilator subgroup formalism
  16. ...and 14 more sections

Key Result

Corollary 4.1

Let $| \Psi \rangle$ be a state that is stabilized by a doubled Pauli $P \otimes P$, i.e. $P \otimes P | \Psi \rangle = \pm | \Psi \rangle$, and for any $P' \neq P$, satisfies $| \langle \Psi | P' \otimes P' | \Psi \rangle | \leq 1 - \varepsilon$. Then, there exists an algorithm to determine $P$ wit

Figures (1)

  • Figure 1: (a) Schematic of the parity sampling subroutine. A tranversal $Z \otimes Z$ measurement is performed on pairs of qubits, yielding an outcome $\pi$, and a post-measurement state $| \Psi(\pi) \rangle$. (b) Schematic of a projected state $| \Psi(\pi) \rangle$, where the labels on each site are given by $\pi_1, ... \pi_4$. (c) Schematic of a Bell-resolvable set, comprising three projected states with $\pi^0 = (1, 1, 0, 0), \pi^1 = (0, 1, 1, 0), \pi^2 = (1, 0, 1, 0)$. Bell measurements are performed on pairs of sites with $\pi_j = 1$, and $X$ measurements on the remaining sites. These measurement outcomes are processed to give the irrep labels $q_j, p_j$.

Theorems & Definitions (60)

  • Definition 1.1: State Hidden Subgroup Problem (StateHSP) Bouland_arXiv_2024_stateHSP
  • Definition 2.1: Multiple squares StateHSP I
  • Corollary 4.1: Learning Pauli Stabilizers
  • Definition 4.1: Parity Sampling Subroutine
  • Definition 4.2: Bell-resolvable set
  • Definition 4.3: Bell-resolution subroutine
  • Definition 4.4: Maximal rotation subroutine
  • Theorem 1: Multiple-Squares StateHSP
  • Proposition 5.1: Sample complexity for obtaining a basis for parity
  • Proposition 5.2: Abelianization of Multiple-Squares
  • ...and 50 more