[Paper Review] Causality Constraint on Circuit Complexity from $COSMOEFT$

Sayantan Choudhury, Arghya Mukherjee+2 more|arXiv (Cornell University)|Nov 22, 2021

Quantum Mechanics and Applications4 citations

TL;DR

This paper investigates the impact of causality—enforced via the effective speed of sound $ c_s \leq 1 $—on quantum circuit complexity (QCC) in cosmological effective field theory (COSMOEFT). Using two-mode squeezed states in a quasi-de Sitter background, it computes QCC and entanglement entropy via Nielsen’s and covariance matrix approaches, revealing a non-universal relationship between complexity and entropy within $ 0.024 \leq c_s \leq 1 $, with implications for thermalization and quantum gravity.

ABSTRACT

In this article, we investigate the physical implications of the causality constraint via effective sound speed $c_s(\leq 1)$ on Quantum Circuit Complexity(QCC) in the framework of Cosmological Effective Field Theory (COSMOEFT) using the two-mode squeezed quantum states. This COSMOEFT setup is constructed using the St$\ddot{ ext{u}}$ckelberg trick with the help of the lowest dimensional operators, which are broken under time diffeomorphism. In this setup, we consider only the contribution from two derivative terms in the background quasi de Sitter metric. Next, we compute the relevant measures of circuit complexity and their cosmological evolution for different $c_s$ by following two different approaches, Nielsen's and Covariance matrix method. Using this setup, we also compute the Von-Neumann and Rényi entropy, which finally establishes an underlying connecting relationship between the entanglement entropy and circuit complexity. Essentially, we study the behaviour of the circuit complexity measures and entanglement entropy with respect to the scale factor and $c_s$ and find various interesting unexplored features within the window, $0.024\leq c_s\leq 1$, which is supported by both causality and cosmological observation. Finally, we also comment on the connection between the circuit complexity, entanglement entropy and equilibrium temperature for different $c_s$ lying within the mentioned window.

Motivation & Objective

• To explore how causality, enforced through $ c_s \leq 1 $, constrains quantum circuit complexity (QCC) in cosmological effective field theory (COSMOEFT).
• To investigate the interplay between QCC, entanglement entropy (Von Neumann and Rényi), and the effective speed of sound $ c_s $ in a quasi-de Sitter spacetime.
• To test whether the $ \frac{dC}{dt} = TS $ relation—valid in black hole physics—extends to inflationary cosmology, and if deviations occur within the $ c_s \in [0.024,1] $ window.
• To establish a quantitative link between circuit complexity and entanglement entropy using two distinct computational frameworks: Nielsen’s geometric approach and the covariance matrix method.

Proposed method

• Constructing a COSMOEFT framework using the Stueckelberg trick and lowest-dimension operators broken under time diffeomorphism, focusing on two-derivative terms in a quasi-de Sitter background.
• Defining the effective speed of sound $ c_s $ via $ c_s = \left(1 - \frac{2M_2^4}{\dot{H}M_p^2}\right)^{-1/2} $, with $ c_s \leq 1 $ enforced by causality and cosmological observations.
• Employing two-mode squeezed vacuum states to model quantum states in de Sitter space, enabling computation of QCC and entanglement entropy.
• Applying Nielsen’s geometric approach to circuit complexity using the Fubini-Study metric on SU(2^K) gates, and the covariance matrix approach via Gaussian state formalism.
• Computing Von Neumann and Rényi entropies from the reduced density matrix to quantify entanglement, and comparing their time evolution with QCC.
• Analyzing the cosmological evolution of QCC and entanglement entropy as functions of scale factor $ a(t) $ and $ c_s $, focusing on the $ 0.024 \leq c_s \leq 1 $ window.

Experimental results

Research questions

• RQ1 How does the causality constraint $ c_s \leq 1 $ affect the growth rate of quantum circuit complexity in inflationary cosmology?
• RQ2 Is the $ \frac{dC}{dt} = TS $ relation—known in black hole physics—valid or modified in the context of cosmological effective field theory?
• RQ3 What is the functional relationship between quantum circuit complexity and entanglement entropy (Von Neumann and Rényi) for different values of $ c_s $ in the range $ 0.024 \leq c_s \leq 1 $?
• RQ4 Do the two computational methods—Nielsen’s geometric approach and the covariance matrix method—yield consistent results for QCC in this setup?
• RQ5 How does the equilibrium temperature $ T $, inferred from $ \frac{dC}{dt} = TS $, vary with $ c_s $, and what does this imply for thermalization in early-universe quantum states?

Key findings

• For $ 0.024 \leq c_s \leq 1 $, both Nielsen’s and covariance matrix approaches yield consistent, non-monotonic growth of quantum circuit complexity, with distinct time evolution depending on $ c_s $.
• The Von Neumann and Rényi entropies grow linearly with time for all $ c_s $ in the allowed window, but their growth rates are modulated by $ c_s $, indicating $ c_s $-dependent entanglement dynamics.
• A non-universal relationship is found between circuit complexity and entanglement entropy: while both grow linearly, their proportionality constant depends on $ c_s $, deviating from the $ C \sim S $ scaling seen in black holes.
• The effective speed of sound $ c_s $ is constrained via $ -867.556 \leq \frac{M_2^4}{\dot{H}M_p^2} \leq 0 $, ensuring $ c_s \in [0.024, 1] $, consistent with observational bounds from CMB and large-scale structure.
• The relation $ \frac{dC}{dt} = TS $ holds approximately for $ c_s \in [0.024, 1] $, with the equilibrium temperature $ T $ scaling inversely with $ c_s $, suggesting faster thermalization for smaller $ c_s $.
• The analysis reveals unexplored features in the $ c_s \in [0.024, 1] $ window, including non-trivial interplay between complexity, entanglement, and the effective field theory parameters, suggesting new avenues in quantum gravity and early-universe cosmology.