[Paper Review] Quantifying non-stabilizerness through entanglement spectrum flatness

Emanuele Tirrito, Poetri Sonya Tarabunga+7 more|arXiv (Cornell University)|Apr 3, 2023

Quantum Computing Algorithms and Architecture9 citations

TL;DR

The paper links non-stabilizerness (magic) to entanglement-spectrum flatness, and offers an efficient Clifford-orbit protocol to witness and quantify non-stabilizerness even with noise.

ABSTRACT

Non-stabilizerness - also colloquially referred to as magic - is the a resource for advantage in quantum computing and lies in the access to non-Clifford operations. Developing a comprehensive understanding of how non-stabilizerness can be quantified and how it relates other quantum resources is crucial for studying and characterizing the origin of quantum complexity. In this work, we establish a direct connection between non-stabilizerness and entanglement spectrum flatness for a pure quantum state. We show that this connection can be exploited to efficiently probe non-stabilizerness even in presence of noise. Our results reveal a direct connection between non-stabilizerness and entanglement response, and define a clear experimental protocol to probe non-stabilizerness in cold atom and solid-state platforms.

Motivation & Objective

• Motivate the need to quantify non-stabilizerness (magic) as a resource for quantum advantage.
• Establish a direct relation between Stabilizer Rényi Entropy and entanglement-spectrum flatness.
• Develop a practical protocol to witness non-stabilizerness in experiments using Clifford-orbit averaging.

Proposed method

• Define Stabilizer Rényi Entropy (SRE) and stabilizer linear entropy as measures of non-stabilizerness.
• Show a theorem that the Clifford-orbit average of entanglement-spectrum flatness equals a constant times M_lin (M_lin = Stabilizer Linear Entropy).
• Relate M_2 to M_lin and derive the flatness formula F_A(Γ|ψ⟩) averaged over Clifford orbit.
• Propose an experimental protocol that detects non-stabilizerness by measuring entanglement-spectrum flatness after applying random Clifford gates.
• Analyze the effect of noise by modeling noisy Clifford gates and showing linear growth of flatness with circuit depth under noise.

Experimental results

Research questions

• RQ1 Can non-stabilizerness be quantified via entanglement-spectrum flatness averaged over the Clifford orbit?
• RQ2 How does stabilizer monotone relate to entanglement properties under Clifford dynamics?
• RQ3 Can one witness non-stabilizerness efficiently in experiments by probing a small bipartition’s entanglement spectrum?
• RQ4 How does noise in Clifford operations affect the flatness-based witness of non-stabilizerness?

Key findings

• A direct relation is established: the Clifford-orbit-averaged entanglement-spectrum flatness is proportional to the stabilizer linear entropy (M_lin).
• The proportionality constant c(d,d_A) scales as roughly (d^2−d_A^2)(d_A^2−1)/((d^2−1)(d+2)d_A^2) for large d.
• A practical protocol is proposed to detect non-stabilizerness by exploring the Clifford orbit up to a modest depth and measuring flatness on small partitions.
• The method remains informative in the presence of noise, with flatness increasing approximately linearly with the number of Clifford layers under noise.
• Numerical experiments demonstrate that the ratio ⟨F_A⟩/(c(d,d_A)M_lin) approaches 1 for large Clifford depth, validating the theorem.