Complexity phase transition for continuous-variable cluster state

TL;DR

This paper investigates how the level of squeezing in continuous-variable cluster states influences the classical computational complexity of measurement-based linear optics, revealing a phase transition between tractable and intractable regimes.

Contribution

It introduces an explicit framework for analyzing the classical complexity of MBLO with CV cluster states and identifies squeezing thresholds that determine computational feasibility.

Findings

Identifies squeezing thresholds for classical intractability.

Reveals a complexity phase transition driven by squeezing levels.

Highlights the importance of scaling squeezing or error correction for quantum computation.

Abstract

Continuous-variable (CV) cluster states offer a promising platform for large-scale measurement-based quantum computations (MBQC). However, finite squeezing inevitably introduces Gaussian noise during MBQC. While fault-tolerant MBQC schemes exist in principle, they require the scalable incorporation of non-Gaussian resources, such as GKP states, which remain experimentally challenging. Consequently, a central question at this stage is how finite squeezing fundamentally constrains the intrinsic computational power of CV cluster states themselves. In this work, we address this question by analyzing the classical complexity of measurement-based linear optics (MBLO) implemented with such states, motivated by its near-term feasibility and recent experimental progress. We develop an explicit MBLO framework and examine how the squeezing level governs the complexity of the classical simulation of the resulting output states. Specifically, we identify squeezing-level thresholds that delineate classically tractable and intractable regimes, thereby revealing a squeezing-driven complexity phase transition. These findings advance our understanding of the squeezing resources necessary for meaningful quantum computation in current experimental regimes. Furthermore, they underscore the critical need to either scale the squeezing level or integrate error-correction schemes to achieve reliable, large-scale quantum computation with CV cluster states.