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Circuit Complexity across a Topological Phase Transition
Fangli Liu, Seth Whitsitt, Jonathan B. Curtis, Rex Lundgren, Paraj Titum, Zhi-Cheng Yang, James R. Garrison, Alexey V. Gorshkov
TL;DR
This paper asks how Nielsen’s circuit complexity behaves across topological phase transitions in the Kitaev chain. It applies a geometric circuit-complexity approach and finds nonanalytic behavior in ground states and nonequilibrium steady states at critical points, while optimal-circuit locality distinguishes same-phase from cross-phase ground states.
Problem
The paper examines how Nielsen’s circuit complexity behaves across topological phase transitions in the Kitaev chain.
Source: p. 1Method
The authors use Nielsen’s geometric approach to quantify circuit complexity in Kitaev-chain ground states and nonequilibrium steady states.
Source: pp. 1, 4Results
Circuit complexity is nonanalytic at critical points for both ground states and nonequilibrium steady states, and the optimal Hamiltonian is nonlocal in real space when connecting states across a critical point.
Source: pp. 1, 4Takeaways & Limitations
Circuit complexity can detect equilibrium and dynamical topological phase transitions, while optimal-circuit locality provides a way to distinguish states in the same phase from states in different phases.
Source: pp. 1, 3, 4
Abstract
from arXiv · showhide
We use Nielsen's geometric approach to quantify the circuit complexity in a one-dimensional Kitaev chain across a topological phase transition. We find that the circuit complexities of both the ground states and non-equilibrium steady states of the Kitaev model exhibit non-analytical behaviors at the critical points, and thus can be used to detect both {\it equilibrium} and {\it dynamical} topological phase transitions. Moreover, we show that the locality property of the real-space optimal Hamiltonian connecting two different ground states depends crucially on whether the two states belong to the same or different phases. This provides a concrete example of classifying different gapped phases using Nielsen's circuit complexity. We further generalize our results to a Kitaev chain with long-range pairing, and discuss generalizations to higher dimensions. Our result opens up a new avenue for using circuit complexity as a novel tool to understand quantum many-body systems.
Supplemental Material
The Supplemental Material supplies derivations and analyses supporting the paper’s main results, spanning fermion-pair complexity, nonanalyticities, optimal-circuit locality, quenches, and higher dimensions.
Supplemental Material
- The supplement derives circuit complexity for a fermion pair and analyzes ground-state nonanalyticities and the real-space structure of the optimal circuit.It also presents evidence for nonanalytic steady-state complexity after a quantum quench and analyzes a p + ip topological superconductor.Source: Supplemental Material · p. 7
- The supplement also analyzes nonanalytic steady-state complexity after a quantum quench and circuit complexity for a p + ip topological superconductor.Source: Supplemental Material · p. 7
- The supplement provides detailed derivations of ground-state complexity nonanalyticities and discusses the real-space structure of the optimal circuit.Source: Supplemental Material · p. 7
I. DERIVATION OF CIRCUIT COMPLEXITY FOR A PAIR OF FERMIONS
This section derives the circuit complexity for a pair of fermionic modes by parameterizing a unitary path, expressing its generators, and minimizing the associated cost.
I. DERIVATION OF CIRCUIT COMPLEXITY FOR A PAIR OF FERMIONS
- The derivation parameterizes the momentum-sector unitary U(s) and expresses it using generators of U(2).The generators’ strengths are then used to write the cost functional.Source: §I · p. 7
- The path begins at the identity, and its endpoint is constrained by a boundary condition.Source: §I · p. 7
- Choosing β = φ1 = 0 and φ2 = π/2 removes three non-negative cost terms, while a linear ω(s) = s∆θ minimizes the remaining term.Source: §I · p. 7
II. ANALYTICAL DERIVATION OF DIVERGENT DERIVATIVES IN GROUND STATES
The section derives how ground-state complexity derivatives become nonanalytic at phase boundaries, using contour integrals whose singularities arise when branch points cross the unit circle.
II. ANALYTICAL DERIVATION OF DIVERGENT DERIVATIVES IN GROUND STATES
- The divergence of the first derivative maps the phase boundary for a reference state within a phase and targets approaching arbitrary points in the phase diagram.The derivation is stated for the thermodynamic limit.Source: §II · p. 7
- The derivation expresses the Bogoliubov-angle difference and its target-parameter derivatives as integrals, then rewrites them as closed contour integrals using z = e^ik.The integrals are taken counter-clockwise over |z| = 1.Source: §II · p. 8
- Branch points cross the unit-circle contour at phase transitions, causing the contour integral and complexity derivatives to diverge.The phase diagram can be characterized by which branch points lie inside or outside the contour.Source: §II · p. 8, Figure S1 · p. 8
- For µT > 1, the contour is deformed around branch cuts; summing the four contour contributions cancels their log ϵ divergences.Source: §II · pp. 8–9
- For µT > 1, the ∂µT C/L divergence is log(µT − 1)/8∆T when ∆T ≠ 0, but the derivative is analytic on approach to ∆T = 0.The reference-state-dependent term remains analytic as the transition is approached.Source: §II · p. 9
- The target-state derivative ∂µT C/L is analytic away from µT = ±1 and diverges logarithmically near those transitions when ∆T ≠ 0.At ∆T = 0, the derivative does not diverge.Source: §II · p. 9
- The ∂∆T C/L derivative is analytic whenever ∆T ≠ 0, while its divergence at ∆T = 0 depends on the approach to the phase boundary.Approaching from topological phases gives log |∆T|/4 for |µT| < 1, crossing over to log |∆T|/2 near multicritical points.Source: §II · p. 9
III. REAL-SPACE BEHAVIOR OF THE OPTIMAL CIRCUITS
This section connects momentum-space optimal circuits to real-space pairing terms and shows that their locality depends on whether the reference and target states occupy the same topological phase.
III. REAL-SPACE BEHAVIOR OF THE OPTIMAL CIRCUITS
- The single-sector optimal circuit is generated by a time-independent Hamiltonian, and the full ground-state circuit combines contributions from momentum sectors.The Hamiltonian is written in operator form to analyze its real-space behavior.Source: §III · p. 9
- For states in the same phase, ∆θ(0) = ∆θ(π) = 0 allows uniform convergence and finite-order truncation to arbitrary accuracy, yielding a short-range circuit in the infinite-system limit.Source: §III · p. 10
- A Fourier harmonic sin(lk) in the pairing channel corresponds to real-space fermion operators separated by l sites.Source: §III · p. 10
- For states in different phases, ∆θ(π) = π/2 while ∆θ(0) = 0 obstructs uniform convergence, so a fixed-error finite Fourier truncation is impossible.Source: §III · p. 10
- With fixed evolution time, the cross-phase optimal Hamiltonian is long-range because its Fourier series cannot be accurately truncated.Source: §III · p. 10
IV. NUMERICAL EVIDENCE FOR NONANALYTICITY OF QUENCH DYNAMICS
The numerical analysis attributes the steady-state complexity’s kink to saturation of the time-averaged angular contribution after a quench across the critical point.
IV. NUMERICAL EVIDENCE FOR NONANALYTICITY OF QUENCH DYNAMICS
- Because φkn(t) is too complex for the authors to derive an analytic time-averaged complexity, they use numerical plots to show its nonanalyticity.The plots examine the maximum and time-averaged angular contribution across the critical point.Source: §IV · p. 10
- For µf < 1, max[φkn(t)] is smooth in kn, whereas above 1 it develops a kink and reaches π/2 when 1 + µf cos kn = 0.The kink occurs at the momentum where the envelope reaches its maximum.Source: §IV · p. 11
- At the special momentum sector, the time-averaged φkn(t) is π/4, bounding its post-transition maximum independently of the pre-quench parameters.The special sector satisfies 1 + µf cos kn = 0.Source: §IV · p. 11
- At µf = 1, the complexity integral becomes constant after the transition for the parameters shown, producing a nonanalytic kink.Below the transition, the integral increases smoothly; after the angular contribution saturates its bound, the dependence changes form.Source: §IV · p. 11
V. CIRCUIT COMPLEXITY FOR TWO-DIMENSIONAL p + ip TOPOLOGICAL SUPERCONDUCTORS
The section generalizes the circuit-complexity calculation to a two-dimensional p + ip topological superconductor, formulating its ground state and complexity in momentum space.
V. CIRCUIT COMPLEXITY FOR TWO-DIMENSIONAL p + ip TOPOLOGICAL SUPERCONDUCTORS
- The note generalizes its 1D Kitaev-chain results to 2D by considering a p + ip topological superconductor whose momentum-space Hamiltonian is summed over the 2D Brillouin zone.Source: §V · p. 11
- The ground-state angle obeys tan(2θk) = |∆k|/εk, and the full-wavefunction complexity follows the same construction used for the 1D chain.Source: §V · p. 11
- In the continuum limit, the momentum sum becomes an integral, with ε(k) ≈ k^2/(2m) − µ and ∆(k) ≈ i∆(kx + iky).Source: §V · p. 11
- For the calculation, the reference is the trivial no-particle vacuum at µR → −∞, and tuning µ gives a transition at µ = 0.The passage states that the authors expect the nonanalyticity not to depend on the reference-state choice.Source: §V · p. 11