Resource complexity of Symmetry Protected Topological phases

Technical Summary: Resource Complexity of Symmetry Protected Topological Phases

Problem Statement
A central challenge in quantum information science is identifying the specific quantum resources that enable quantum supremacy, distinguishing quantum states that offer computational advantages over classical simulations. While entanglement was historically viewed as the primary bridge between many-body physics and quantum information, it is now understood that entanglement alone is insufficient; highly entangled states can be classically simulable, while some quantum advantages arise with minimal entanglement. Consequently, "quantum magic" (or nonstabilizerness) has emerged as a critical resource, quantifying the distance of a quantum state from the stabilizer manifold (states preparable via Clifford circuits).

Topological phases of matter, particularly Symmetry Protected Topological Phases (SPTPs), are often hypothesized to host high quantum complexity due to their non-local encoding and protection against local perturbations. However, this connection is not universal (e.g., the toric code ground state is a stabilizer state with zero magic). In one-dimensional systems, SPTPs are more fragile than inherent topological phases, yet they serve as resources for measurement-based quantum computation (MBQC). The central question addressed in this work is whether the topological order inherent in 1D SPTPs manifests as an intrinsic increase in quantum magic compared to their topologically trivial counterparts.

Methodology
The authors evaluate quantum magic using the rank-2 Stabilizer Rényi Entropy ($M_2$), defined for a pure state $|\psi\rangle$ of $L$ qubits as:
$$M_2(|\psi\rangle) = \log_2 \left( \frac{1}{2^L} \sum_P |\langle \psi| P |\psi\rangle|^4 \right)$$
where the sum runs over all Pauli strings $P$. The study focuses on one-dimensional models that possess an exact duality mapping a topological phase to a trivial phase. This allows for a stringent comparison of magic at dual points.

The analysis employs three specific models:

1. Dimerized XX Chain (SSH model): Exhibits an SPTP for dimerization $\delta \in (0, 1]$ and a trivial phase for $\delta \in [-1, 0)$.
2. Cluster–Ising Chain: Exhibits an SPTP (cluster phase) when the three-body cluster interaction dominates ($\lambda < 1$) and a trivial antiferromagnetic phase when the Ising coupling dominates ($\lambda > 1$).
3. Transverse-Field Ising Chain: A non-topological model with a self-duality between ferromagnetic and paramagnetic phases, used as a control case.

Numerical simulations utilize the Density Matrix Renormalization Group (DMRG) to obtain ground states via Matrix Product States (MPS). The $M_2$ is evaluated using two complementary approaches, with bond dimensions truncated at $\chi \approx 200-450$ for the Pauli basis expansion. The study systematically compares results under Periodic Boundary Conditions (PBCs), which preserve duality, and Open Boundary Conditions (OBCs), which break duality and introduce edge states.

Key Results
The investigation yields several counter-intuitive findings regarding the relationship between topological order and quantum magic:

• Identical Magic at Dual Points (PBCs): Under periodic boundary conditions, where the duality symmetry is exact, the $M_2$ values for the ground states of the topological and trivial phases at dual points are identical. For the dimerized XX chain, $M_2(\delta) = M_2(-\delta)$; for the Cluster–Ising model, $M_2(\lambda) = M_2(1/\lambda)$. This indicates that the topological nature of the phase does not inherently increase the non-Clifford resources required to prepare the state.
• Subextensive Asymmetry under OBCs: Under open boundary conditions, a difference in magic, $\Delta M_2$, arises between dual points. However, this asymmetry is not a topological invariant. Instead, it is a subextensive correction that depends on microscopic Hamiltonian parameters and system size. As the system size $L$ increases, $\Delta M_2$ converges to a finite, non-constant function of the control parameter (e.g., $\delta$ or $\lambda$).
• Non-Topological Origin: The asymmetry observed under OBCs is attributed to the breaking of duality symmetry by edge effects, rather than the presence of topological order. This is confirmed by the Transverse-Field Ising chain, which lacks topological order entirely but exhibits the same phenomenology: a finite, parameter-dependent $\Delta M_2$ under OBCs that vanishes under PBCs.
• Contrast with Topological Frustration: The results stand in contrast to systems with topological frustration, where delocalized excitations induce a logarithmic growth of magic with system size. In the SPTPs studied here, the total magic is largely insensitive to the topological sector.

Significance and Claims
The authors conclude that topological order in 1D SPTPs is not necessarily a genuine computational resource in terms of quantum magic (nonstabilizerness). The ground states of SPTPs can be prepared using the same amount of non-Clifford resources as their topologically trivial duals.

The paper suggests that the perceived computational advantage of SPTPs in MBQC frameworks may not stem from an intrinsic increase in total magic. Instead, the authors propose that assessing computational complexity in these systems may require alternative quantities, such as Long-Range Magic (LRM), which captures nonstabilizerness components that cannot be eliminated by shallow local circuits. The findings challenge the assumption that topological order automatically equates to higher computational complexity within the Clifford resource framework, highlighting that the "magic" of these states is largely determined by boundary conditions and duality violations rather than the topological phase itself.

The study emphasizes that while SPTPs are valuable resources for specific computational tasks (like MBQC), their classification within a Clifford resource framework remains nontrivial, and the observed complexity may be a manifestation of boundary effects or specific measurement protocols rather than an intrinsic property of the topological order.