[Paper Review] Stabilizer Testing and Magic Entropy via Quantum Fourier Analysis

Kaifeng Bu, Weichen Gu+1 more|arXiv (Cornell University)|Jun 15, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces quantum convolution-based protocols for stabilizer testing of quantum states and Clifford gates using swap-tests and circuit implementations over qubits and qudits. It introduces 'magic entropy' as a measurable, experimentally accessible quantifier of non-stabilizer (magic) resource content, with a dimension-independent bound on testing fidelity, enabling robust, scalable verification of quantum resources in near-term devices.

ABSTRACT

Quantum Fourier analysis is an important topic in mathematical physics. We introduce a systematic protocol for testing and measuring ``magic'' in quantum states and gates, using a quantum Fourier approach. Magic, as a quantum resource, is necessary to achieve a quantum advantage in computation. Our protocols are based on quantum convolutions and swap tests, implemented via quantum circuits. We describe this for both qubit and qudit systems. Our quantum Fourier approach offers a unified method to quantify magic, in stabilizer circuits, as well as in matchgate and bosonic Gaussian circuits.

Motivation & Objective

• To develop systematic, experimentally feasible protocols for testing whether a quantum state or gate is a stabilizer state or Clifford unitary.
• To address the limitation of prior stabilizer testing methods that depend on local dimension, by introducing a dimension-independent bound on the probability of acceptance.
• To introduce 'magic entropy' as a measurable, information-theoretic quantifier of non-stabilizer (magic) content in quantum states and gates.
• To unify concepts from classical property testing (e.g., BLR linearity testing) and quantum entanglement entropy with quantum convolution and stabilizer invariance.
• To enable scalable, robust verification of quantum resources in near-term quantum devices using standard quantum circuits and measurements.

Proposed method

• The method employs quantum convolution operations defined via unitary circuits that implement the self-convolution of quantum states and channels, generalizing classical convolution to quantum systems.
• Stabilizer testing is realized via a swap-test-based protocol that measures the purity of the 2-fold quantum convolution of a state, leveraging the fact that stabilizer states are invariant under self-convolution.
• The protocol uses the Fourier transform of the Weyl-Heisenberg operators to define the quantum convolution kernel, enabling efficient circuit implementation on qubit and qudit systems.
• The fidelity of the test is bounded using the trace of the squared quantum convolution, with a dimension-independent upper bound on the acceptance probability for non-stabilizer states.
• Magic entropy is defined as the von Neumann entropy of the quantum convolution of a state with itself, providing a measure of deviation from stabilizer structure.
• Theoretical bounds are derived using properties of the Weyl-Heisenberg group and character sums, showing that the trace of the squared convolution is bounded by 1 − 6ε + O(ε²) for states ε-far from stabilizer.

Experimental results

Research questions

• RQ1 Can stabilizer testing be performed with a dimension-independent bound on the probability of acceptance, independent of the local qudit dimension?
• RQ2 How can the concept of linearity testing in classical property testing be generalized to quantum states and gates using quantum convolution?
• RQ3 Can a physically measurable quantity be defined to quantify the 'magic' or non-stabilizer content of a quantum state or gate?
• RQ4 What is the relationship between the purity of the quantum convolution of a state and its stabilizer structure?
• RQ5 How can magic entropy be experimentally measured using standard quantum circuits and swap tests?

Key findings

• The paper establishes a dimension-independent upper bound on the acceptance probability of the stabilizer test: for a state ε-far from any stabilizer state, the probability of acceptance is at most 1 − 6ε + O(ε²).
• The quantum convolution of a stabilizer state is pure, and this property is used as the basis for the stabilizer testing protocol via the swap-test on the convolved state.
• The magic entropy of a quantum state is defined as the von Neumann entropy of its 2-fold quantum convolution, providing a measurable, information-theoretic quantifier of non-stabilizer content.
• The protocol for stabilizer testing can be implemented using standard quantum circuits involving controlled-SWAP (Fredkin) gates and Weyl-Heisenberg basis operations, enabling experimental realization.
• Theoretical analysis shows that the trace of the squared quantum convolution of a state is bounded by 1 − 6ε + O(ε²) when the state is ε-far from the set of stabilizer states, enabling robust testing.
• The method generalizes the BLR linearity test and separability testing to the quantum stabilizer setting, unifying classical and quantum property testing frameworks via quantum convolution.