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Circuit complexity in interacting QFTs and RG flows

Arpan Bhattacharyya, Arvind Shekar, Aninda Sinha

TL;DR

The paper develops a perturbative framework to compute circuit complexity in interacting scalar QFTs, focusing on $\\phi^4$ theory on a lattice and using Nielsen's geometric approach. By solving two- and multi-oscillator systems and extending to continuous targets, it connects complexity to RG flow and the Wilson-Fisher fixed point via epsilon expansion, revealing how complexity grows with dimensionality and can exhibit fractional-volume scaling. The work provides analytic expressions for complexity in low dimensions, renormalized-parameter formulations, and an RG-like flow for complexity, and discusses alternative gate constructions and holographic comparisons. Overall, it suggests circuit complexity as a diagnostic tool for RG structure and fixed-point behavior in quantum field theories, with clear avenues for generalization to more field content and nonperturbative regimes.

Abstract

We consider circuit complexity in certain interacting scalar quantum field theories, mainly focusing on the $φ^4$ theory. We work out the circuit complexity for evolving from a nearly Gaussian unentangled reference state to the entangled ground state of the theory. Our approach uses Nielsen's geometric method, which translates into working out the geodesic equation arising from a certain cost functional. We present a general method, making use of integral transforms, to do the required lattice sums analytically and give explicit expressions for the $d=2,3$ cases. Our method enables a study of circuit complexity in the epsilon expansion for the Wilson-Fisher fixed point. We find that with increasing dimensionality the circuit depth increases in the presence of the $φ^4$ interaction eventually causing the perturbative calculation to breakdown. We discuss how circuit complexity relates with the renormalization group.

Circuit complexity in interacting QFTs and RG flows

TL;DR

The paper develops a perturbative framework to compute circuit complexity in interacting scalar QFTs, focusing on theory on a lattice and using Nielsen's geometric approach. By solving two- and multi-oscillator systems and extending to continuous targets, it connects complexity to RG flow and the Wilson-Fisher fixed point via epsilon expansion, revealing how complexity grows with dimensionality and can exhibit fractional-volume scaling. The work provides analytic expressions for complexity in low dimensions, renormalized-parameter formulations, and an RG-like flow for complexity, and discusses alternative gate constructions and holographic comparisons. Overall, it suggests circuit complexity as a diagnostic tool for RG structure and fixed-point behavior in quantum field theories, with clear avenues for generalization to more field content and nonperturbative regimes.

Abstract

We consider circuit complexity in certain interacting scalar quantum field theories, mainly focusing on the theory. We work out the circuit complexity for evolving from a nearly Gaussian unentangled reference state to the entangled ground state of the theory. Our approach uses Nielsen's geometric method, which translates into working out the geodesic equation arising from a certain cost functional. We present a general method, making use of integral transforms, to do the required lattice sums analytically and give explicit expressions for the cases. Our method enables a study of circuit complexity in the epsilon expansion for the Wilson-Fisher fixed point. We find that with increasing dimensionality the circuit depth increases in the presence of the interaction eventually causing the perturbative calculation to breakdown. We discuss how circuit complexity relates with the renormalization group.

Paper Structure

This paper contains 15 sections, 154 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Circuit complexity with $\phi^4$ interaction--the 2-oscillator case
  3. Generalizing to the $N$-oscillator case
  4. Evaluation of complexity functional and continuous limit
  5. Complexity in terms of renormalized parameters
  6. Comments about $\mathcal{C}^{(2)}_{\kappa}$ and structure of penalty factor
  7. Flow equations
  8. Comparison with Holography
  9. Using circuit complexity for inferring fixed points
  10. An alternative construction
  11. Discussions
  12. Matrix representation of Unitary operators
  13. Counting of dimension of the target matrix and the basis elements for arbitrary $N$:
  14. A general method to do the lattice sums
  15. Structure of the Complexity beyond $\mathcal{O}(\lambda)$

Figures (2)

  • Figure 1: $a_{d+1}$ as a function of $d$.
  • Figure 2: The interacting sum is ${\hat{\lambda}\over 8\,\sqrt{\pi}} \delta^{6-2d} V^{{d-2\over d-1}}$ times the y-axis.