[Paper Review] Quantum complexity and topological phases of matter

Paweł Caputa, Sinong Liu|arXiv (Cornell University)|May 11, 2022

Quantum many-body systems4 citations

TL;DR

This paper introduces spread complexity as a novel probe for topological phases in quantum many-body systems, demonstrating analytically in the Su-Schrieffer-Heeger (SSH) model that spread complexity becomes constant in the topological phase. The method uses the Krylov basis to compute complexity, revealing distinct dynamical behavior under quantum quen protocols that distinguish topological from trivial phases.

ABSTRACT

In this work, we find that the complexity of quantum many-body states, defined as a spread in the Krylov basis, may serve as a new probe that distinguishes topological phases of matter. We illustrate this analytically in one of the representative examples, the Su-Schrieffer-Heeger model, finding that spread complexity becomes constant in the topological phase. Moreover, in the same setup, we analyze exactly solvable quench protocols where the evolution of the spread complexity shows distinct dynamical features depending on the topological vs non-topological phase of the initial state as well as the quench Hamiltonian.

Motivation & Objective

• To investigate whether quantum complexity, specifically spread complexity, can serve as a universal probe for topological phases of matter.
• To analyze the behavior of spread complexity in the one-dimensional SSH model, a paradigmatic system with topological and trivial phases.
• To test the sensitivity of spread complexity to topological order in both equilibrium ground states and non-equilibrium quench dynamics.
• To compare the performance of spread complexity against traditional probes like entanglement entropy and topological invariants.
• To establish a link between spread complexity and underlying topological order through analytical computation in an exactly solvable model.

Proposed method

• Define spread complexity as the minimum over basis choices of the weighted sum of Krylov basis state overlaps, minimized using the Krylov basis constructed via the Lanczos algorithm.
• Compute the spread complexity of the ground state in the SSH model using the Krylov basis, which diagonalizes the Hamiltonian in a tridiagonal form.
• Use the return amplitude (auto-correlator) S(s) = ⟨Ψ(s)|Ψ₀⟩ to extract Lanczos coefficients aₙ and bₙ that parameterize the Krylov basis.
• Analyze the time evolution of spread complexity under quantum quench protocols, where the system is prepared in an initial state and evolved under a different Hamiltonian.
• Relate the spread complexity to physical observables such as the expectation value of the operator 𝔓ⱼ = i(c†_{Aj}c_{Bj} + h.c.), which acts as a proxy for local entanglement structure.
• Compare the reduced density matrix ρⱼ in both topological and trivial phases to show that spread complexity correlates with nontrivial entanglement in the topological phase.

Experimental results

Research questions

• RQ1 Can spread complexity distinguish between topological and trivial phases in the SSH model?
• RQ2 Does the spread complexity of the ground state remain constant in the topological phase, as observed in the analytical computation?
• RQ3 How does the time evolution of spread complexity differ under quench protocols depending on whether the initial state or quench Hamiltonian is in a topological or trivial phase?
• RQ4 Is there a physical interpretation of spread complexity in terms of entanglement or local correlations in the system?
• RQ5 Can spread complexity serve as a universal probe for topological order in quantum many-body systems, independent of the choice of gates or cost functions as in Nielsen's complexity?

Key findings

• The spread complexity of the ground state in the SSH model becomes constant in the topological phase, indicating a sharp distinction from the non-topological phase.
• In the non-topological phase (t₁=1, t₂=0), the reduced density matrix ρⱼ^{NP} is a pure state, leading to trivial entanglement and a specific expectation value for 𝔓ⱼ.
• In the topological phase (t₁=0, t₂=−1), the reduced density matrix ρⱼ^{TP} contains a mixed sector with nontrivial entanglement, and 𝔓ⱼ captures only part of the information, reflecting the nonlocal nature of topological order.
• The spread complexity density is proportional to the sum of ⟨Ω|𝔓ⱼ|Ω⟩ over all sites j in the large L limit, linking it to the global complexity measure.
• Under quantum quench protocols, the evolution of spread complexity shows distinct dynamical features depending on the topological character of the initial state and quench Hamiltonian.
• The results suggest that spread complexity is a robust, computable, and physically meaningful probe sensitive to topological order, even in integrable and non-chaotic systems.