Computational Characterization of Symmetry-Protected Topological Phases in Open Quantum Systems

TL;DR

This work addresses identifying symmetry-protected topological order in open quantum systems by linking MBQC performance to topological structure. It develops a gate-fidelity framework for the AKLT state, showing that the identity gate fidelity coincides with nonlocal string-order markers under strong $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry, and derives analytic results for the Z-rotation gate that reveal finite-size corrections. The authors then explore how various symmetric noise channels affect MBQC, revealing a phase-like structure where identity-only protection, partial gate protection, or none can persist, and conclude that universal one-qubit MBQC generally requires a trivial channel within the studied protocol. The results highlight a richer computational-phase landscape than captured by string order alone and suggest MPO-based extensions for open-system SPT analysis with potential experimental relevance.

Abstract

It is a challenging problem to correctly characterize the symmetry-protected topological (SPT) phases in open quantum systems. As the measurement-based quantum computation (MBQC) utilizes non-trivial edge states of the SPT phases as the logical qubit, its computational power is closely tied to the non-trivial topological nature of the phases. In this paper, we propose to use the gate fidelity which is a measure of the computational power of the MBQC to identify the SPT phases in mixed-state settings. Specifically, we investigate the robustness of the Haldane phase by considering the MBQC on the Affleck-Kennedy-Lieb-Tasaki state subject to different types of noises. To illustrate how our criterion works, we analytically and numerically calculated the gate fidelity to find that its behavior depends crucially on whether the noises satisfy a certain symmetry condition with respect to the on-site $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry. In particular, the fidelity for the identity gate, which is given by the sum of the non-local string order parameters, plays an important role. Furthermore, we demonstrate that a stronger symmetry conditions are required to be able to perform other (e.g., the $Z$-rotation gate) gates with high fidelity. By examining which unitary gates can be implemented with the MBQC on the decohered states, we can gain a useful insight into the richer structure of noisy SPT states that cannot be captured solely by the string order parameters.

Figures (8)

• Figure 1: A schematic diagram of the AKLT state \ref{['MPS_AKLT_state_eq']} coupled with two extra one-half spins. Dots and line segments connecting them denote spin-1/2 degrees of freedom and singlet bonds, respectively. Circles encircling two adjacent dots mean the projection onto the spin-1 degree of freedom.
• Figure 2: Diagrammatic representation of MPS \ref{['MPS_AKLT_state_eq']} for the AKLT state. Large (small) squares respectively denote the MPS tensors $P$ and $A$ in Eq. \ref{['MPS_AKLT_state_eq']}. Horizontal lines connecting the adjacent squares denote entanglement bonds, which are introduced every time when the matrices are multiplied in Eq. \ref{['MPS_AKLT_state_eq']}. The vertical lines denote the physical degrees of freedom (spin-$1/2$ and $1$ here) on each site, which are labeled by $\sigma_{\mathrm{in,out}}$ (for edge spin-$1/2$) and $m_i$ (for the bulk spin-$1$). Here, the exceptional spin-1/2 degrees of freedom on edges are described as short vertical lines.
• Figure 3: A typical term that appears in the first line on the right-hand side of Eq. \ref{['XX_term']}. The symbols and denote the application of the projective measurement \ref{['projective_measuremnt_theta']} with generic $\theta$ and $\theta=0$, respectively.
• Figure 4: Discrete time-evolution of the gate fidelity $F_{U}$ for $U = e^{-i\theta Z/2}$ under Noise 1 (called "dephasing" in Ref. deGroot-T-S-22) in Table \ref{['noises_table']}. The asymptotic value for the gate $e^{-i\theta Z/2}$ is given by $(1+\text{cos}^2\theta)/2$.
• Figure 5: Discrete time-evolution of the gate fidelity $F_{U}$ for $U = e^{-i\theta Z/2}$ under Noise 2 in Table \ref{['noises_table']}. The asymptotic values are the same as in Fig. \ref{['noise1']}.

Theorems & Definitions (11)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Lemma 1
• Lemma 2
• proof
• Corollary 1
• Proposition
• proof