[Paper Review] Exploiting non-quantum entanglement to widen applicability of limited-entanglement classical simulations of quantum systems

N. Ratanje, S. Virmani|arXiv (Cornell University)|Jan 3, 2012

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper demonstrates that using a generalized notion of entanglement—beyond quantum separability—can significantly extend the range of classical simulations for stabilizer-based quantum computers. By redefining separability with respect to restricted measurement sets, the Harrow-Nielsen simulation algorithm achieves efficient classical simulation of magic-state quantum computers with up to 56% joint depolarizing noise on CNOT gates, improving upon the 66% threshold when using standard quantum entanglement criteria.

ABSTRACT

It is known that if the quantum gates in a proposed quantum computer are so noisy that they are incapable of generating entanglement, then the device can be efficiently simulated classically. If the measurements and single particle operations are restricted, then the same statement can be true for generalised non-quantum notions of entanglement. Here we show that this can improve the applicability of limited-entanglement simulation algorithms. In particular, we show that by using a non-quantum notion of separability a classical simulation algorithm of Harrow & Nielsen can efficiently simulate magic state quantum computers with noisy CNOTs (but otherwise ideal) for joint depolarising strengths of 272/489 ~ 56%, in comparison to noise levels of 2/3 ~ 66% required if the algorithm uses quantum notions of separability. This suggests that quantum entanglement may not be the most appropriate notion of entanglement to use when discussing the power of stabilizer based quantum computers.

Motivation & Objective

• To investigate whether alternative notions of entanglement beyond quantum separability can improve the range of classical simulation algorithms.
• To assess the applicability of the Harrow-Nielsen simulation algorithm under generalized entanglement criteria in stabilizer-based quantum computation.
• To determine whether non-stabilizer resources (e.g., magic states) can be treated as classical when combined with non-quantum entanglement measures.
• To identify the minimal noise level on CNOT gates that still allows efficient classical simulation under generalized separability.
• To explore whether truncated cubes in the Bloch sphere provide optimal sets for achieving maximal noise tolerance in such simulations.

Proposed method

• Adapts the Harrow-Nielsen classical simulation algorithm by replacing quantum separability with C-separability, where C is a convex set of single-particle states closed under single-qubit operations.
• Defines C-separability using the dual of the measurement set M, ensuring that measurement probabilities remain non-negative during simulation.
• Applies the generalized separability framework to magic-state quantum computing, where single-qubit operations are Clifford gates and measurements are restricted to Pauli bases.
• Analyzes the action of a noisy controlled-Z (CZ) gate under joint depolarizing noise, modeling the output state as a mixture of product states from a truncated cube set C.
• Derives necessary inequalities for separability using Pauli measurement outcomes, particularly focusing on X and Y basis probabilities.
• Uses analytical and numerical methods to compute the minimal noise parameter λ for which the output state remains C-separable, with r parameterizing the truncated cube.

Experimental results

Research questions

• RQ1 Can non-quantum notions of entanglement extend the range of classical simulation beyond what is possible with standard quantum entanglement?
• RQ2 What is the maximal noise level on CNOT gates that still allows efficient classical simulation of magic-state quantum computers under generalized separability?
• RQ3 Does the use of generalized entanglement criteria improve the simulation threshold compared to the standard quantum separability condition?
• RQ4 Are truncated cubes in the Bloch sphere optimal sets C for maximizing noise tolerance in such simulations?
• RQ5 Can analytical bounds on the required noise level be derived and shown to be tight for specific parameter regimes?

Key findings

• The Harrow-Nielsen simulation algorithm can be extended to work under generalized C-separability, enabling classical simulation of stabilizer-based quantum circuits with non-quantum entanglement.
• For magic-state quantum computers, the algorithm achieves efficient classical simulation up to 56% joint depolarizing noise on CNOT gates, improving from the 66% threshold under standard quantum separability.
• At r = 1/2, the minimal noise threshold λ = 272/489 ≈ 0.5562 is analytically achievable, and this value is nearly tight, with a necessary lower bound of 5/9 ≈ 0.5556.
• For r = 1/2, Cases 2, 3, and 4 of the analysis all match their respective analytical lower bounds, indicating that the derived noise threshold is optimal for these cases.
• Case 1 remains analytically unresolved, but numerical evidence suggests the same threshold is achievable in its vicinity.
• The results suggest that quantum entanglement may not be the most appropriate measure of non-classicality in fault-tolerant quantum computation, with generalized entanglement offering better predictive power for classical simulability.