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Learning stabilizer states by Bell sampling

Ashley Montanaro

TL;DR

The paper shows that Bell-basis measurements across pairs of copies of an unknown stabilizer state enable efficient learning with only O(n) copies. It constructs an affine subspace T of Pauli indices from Bell-sampling outcomes and uses Gaussian elimination to recover a stabilizer basis, followed by Pauli-eigenbasis measurements to complete the state identification. The proposed algorithm achieves exponential-failure-suppression and runs in time polynomial in n, matching information-theoretic lower bounds for stabilizer-state learning. This approach offers a practical, scalable method for stabilizer-state tomography using simple two-copy Bell measurements and relates to prior Clifford-learning results.

Abstract

We show that measuring pairs of qubits in the Bell basis can be used to obtain a simple quantum algorithm for efficiently identifying an unknown stabilizer state of n qubits. The algorithm uses O(n) copies of the input state and fails with exponentially small probability.

Learning stabilizer states by Bell sampling

TL;DR

The paper shows that Bell-basis measurements across pairs of copies of an unknown stabilizer state enable efficient learning with only O(n) copies. It constructs an affine subspace T of Pauli indices from Bell-sampling outcomes and uses Gaussian elimination to recover a stabilizer basis, followed by Pauli-eigenbasis measurements to complete the state identification. The proposed algorithm achieves exponential-failure-suppression and runs in time polynomial in n, matching information-theoretic lower bounds for stabilizer-state learning. This approach offers a practical, scalable method for stabilizer-state tomography using simple two-copy Bell measurements and relates to prior Clifford-learning results.

Abstract

We show that measuring pairs of qubits in the Bell basis can be used to obtain a simple quantum algorithm for efficiently identifying an unknown stabilizer state of n qubits. The algorithm uses O(n) copies of the input state and fails with exponentially small probability.

Paper Structure

This paper contains 2 sections, 2 theorems, 7 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Learning stabilizer states

Key Result

Theorem 1

There is a quantum algorithm which identifies an unknown stabilizer state ${\left\vert{\psi}\right\rangle}$ of $n$ qubits given access to $O(n)$ copies of ${\left\vert{\psi}\right\rangle}$. The algorithm makes collective measurements across at most two copies of ${\left\vert{\psi}\right\rangle}$ at

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 2
  • proof