Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spread and circuit complexity as a measure of particle content and phase space fluctuations

Satyaki Chowdhury

TL;DR

This work connects two notions of quantum complexity—Nielsen-style circuit complexity and Krylov/spread complexity—to physically meaningful quantities in a time-dependent harmonic oscillator, providing a bridge to quantum field theory in curved spacetimes. By solving the Gaussian dynamics with a time-dependent mass m(t) and frequency omega(t) and introducing the excitation parameter z(t), the authors derive exact relations: the circuit complexity C(t) can be written in terms of the particle content <n(t)> and its rate, or in terms of the phase-space variances <q^2> and <p^2>, and the energy E(t); the spread complexity C_S(t) is computed via su(1,1) decoupling and relates to a Lanczos-like parameter Gamma_+(t), which in turn connects to z. A central result is the universal relation C_S(t) = sinh^2(C(t)), linking Krylov and geometric complexity in this non-holographic setting, and the framework provides a physically transparent interpretation of complexity in terms of particle production and phase-space fluctuations. The findings offer a concrete methodology for studying complexity in quantum fields on curved backgrounds and may inform how complexity encodes dynamical features such as particle creation in time-dependent spacetimes.

Abstract

In this work, we investigate the relation between different notions of quantum complexity, namely, circuit and spread complexity and physically meaningful quantities such as the particle content of the quantum state and the variances of position and momentum operators. Using a harmonic oscillator with time-dependent mass and frequency as a toy model, we show that both circuit and spread complexity at any instant is determined by the mean number of quanta and its rate of change. Furthermore, both complexity and its growth are directly linked to the variances of the position and momentum operators, providing a clear physical interpretation of complexity in terms of the state's excitation and phase-space fluctuation. Although the analysis is carried out for a single time-dependent oscillator, the results have direct relevance for quantum field theory in curved backgrounds, where individual field modes effectively behave as time-dependent oscillators. This offers new insights into how quantum complexity encodes particle production and phase space fluctuations in non-holographic systems. Finally, we establish a precise and potentially universal relation between spread and circuit complexity for the time evolved state suggesting deeper connections between different complexity measures in the context of field theories on curved backgrounds.

Spread and circuit complexity as a measure of particle content and phase space fluctuations

TL;DR

This work connects two notions of quantum complexity—Nielsen-style circuit complexity and Krylov/spread complexity—to physically meaningful quantities in a time-dependent harmonic oscillator, providing a bridge to quantum field theory in curved spacetimes. By solving the Gaussian dynamics with a time-dependent mass m(t) and frequency omega(t) and introducing the excitation parameter z(t), the authors derive exact relations: the circuit complexity C(t) can be written in terms of the particle content <n(t)> and its rate, or in terms of the phase-space variances <q^2> and <p^2>, and the energy E(t); the spread complexity C_S(t) is computed via su(1,1) decoupling and relates to a Lanczos-like parameter Gamma_+(t), which in turn connects to z. A central result is the universal relation C_S(t) = sinh^2(C(t)), linking Krylov and geometric complexity in this non-holographic setting, and the framework provides a physically transparent interpretation of complexity in terms of particle production and phase-space fluctuations. The findings offer a concrete methodology for studying complexity in quantum fields on curved backgrounds and may inform how complexity encodes dynamical features such as particle creation in time-dependent spacetimes.

Abstract

In this work, we investigate the relation between different notions of quantum complexity, namely, circuit and spread complexity and physically meaningful quantities such as the particle content of the quantum state and the variances of position and momentum operators. Using a harmonic oscillator with time-dependent mass and frequency as a toy model, we show that both circuit and spread complexity at any instant is determined by the mean number of quanta and its rate of change. Furthermore, both complexity and its growth are directly linked to the variances of the position and momentum operators, providing a clear physical interpretation of complexity in terms of the state's excitation and phase-space fluctuation. Although the analysis is carried out for a single time-dependent oscillator, the results have direct relevance for quantum field theory in curved backgrounds, where individual field modes effectively behave as time-dependent oscillators. This offers new insights into how quantum complexity encodes particle production and phase space fluctuations in non-holographic systems. Finally, we establish a precise and potentially universal relation between spread and circuit complexity for the time evolved state suggesting deeper connections between different complexity measures in the context of field theories on curved backgrounds.

Paper Structure

This paper contains 9 sections, 145 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General formalism of a time-dependent oscillator
  3. Circuit complexity of the time evolved state
  4. Spread complexity of the time evolved state
  5. Discussion and Summary
  6. A brief review of the Lie algebra decoupling theorem
  7. Relation between Gamma3(t) and GammaPlus(t)
  8. Derivation of the relation between Gamma+ and the excitation parameter z
  9. Application: Oscillator with exponentially varying mass

Figures (3)

  • Figure 1: Variation of the mean particle number and the modulus of the excitation parameter with time for different damping scenarios. $\omega$ is fixed to 1.
  • Figure 2: Time evolution of circuit and spread complexity for the different damping scenarios. $\omega$ is fixed to 1.
  • Figure 3: Time evolution of rate of change of circuit and spread complexity for different damping situations. $\omega$ fixed at 1.