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Circuit Complexity in Fermionic Field Theory

Rifath Khan, Chethan Krishnan, Sanchita Sharma

TL;DR

The paper develops a comprehensive framework for circuit complexity in free fermionic quantum field theories across 1+1 and 3+1 dimensions using Nielsen’s geodesic approach. It employs both lattice BV transformations and continuum squeezing to connect simple reference states to ground states, and it reveals SU(2) and SU(2) x SU(2) structures that organize the complexity calculations. It also examines fermionic cMERA as an alternative circuit path, highlighting cut-off dependencies and their holographic interpretation. The results illuminate how complexity scales with cut-off and mass, generalize to Majorana and higher-dimensional cases, and point toward future work in gauge theories and perturbative string worldsheets.

Abstract

We define and calculate versions of complexity for free fermionic quantum field theories in 1+1 and 3+1 dimensions, adopting Nielsen's geodesic perspective in the space of circuits. We do this both by discretizing and identifying appropriate classes of Bogoliubov-Valatin transformations, and also directly in the continuum by defining squeezing operators and their generalizations. As a closely related problem, we consider cMERA tensor networks for fermions: viewing them as paths in circuit space, we compute their path lengths. Certain ambiguities that arise in some of these results because of cut-off dependence are discussed.

Circuit Complexity in Fermionic Field Theory

TL;DR

The paper develops a comprehensive framework for circuit complexity in free fermionic quantum field theories across 1+1 and 3+1 dimensions using Nielsen’s geodesic approach. It employs both lattice BV transformations and continuum squeezing to connect simple reference states to ground states, and it reveals SU(2) and SU(2) x SU(2) structures that organize the complexity calculations. It also examines fermionic cMERA as an alternative circuit path, highlighting cut-off dependencies and their holographic interpretation. The results illuminate how complexity scales with cut-off and mass, generalize to Majorana and higher-dimensional cases, and point toward future work in gauge theories and perturbative string worldsheets.

Abstract

We define and calculate versions of complexity for free fermionic quantum field theories in 1+1 and 3+1 dimensions, adopting Nielsen's geodesic perspective in the space of circuits. We do this both by discretizing and identifying appropriate classes of Bogoliubov-Valatin transformations, and also directly in the continuum by defining squeezing operators and their generalizations. As a closely related problem, we consider cMERA tensor networks for fermions: viewing them as paths in circuit space, we compute their path lengths. Certain ambiguities that arise in some of these results because of cut-off dependence are discussed.

Paper Structure

This paper contains 33 sections, 298 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lattice of Fermionic Oscillators
  3. A Toy Model
  4. Complexity of the Toy Model Ground State
  5. Squeezing Operators as Gates
  6. 1+1 Dimensional Majorana on a Lattice
  7. Ramond Boundary Condition
  8. Neveu-Schwarz Boundary Condition
  9. Continuum Field Theory
  10. Dirac in 1+1 Dimensions
  11. Ground State Complexity
  12. Ground State Complexity and $SU(2)$ Generators
  13. Another Generator
  14. Majorana Fermions in 1+1 Dimensions
  15. Fermions in Higher Dimensions
  16. ...and 18 more sections

Figures (1)

  • Figure 1: The $|\beta_{+}|$ of first plot is for the geodesic path and the second plot is for the cMERA path. In both cases the target state is the cMERA target state. The above figures shows the plots of $|\beta_{+}|$ vs $k$ for $\Lambda=10$ and $m=1$ for the values of $\sigma=0.2,0.4,0.6,0.8,1.0$, increasing peak of the curve signifies increasing $\sigma$.