Hierarchy of topological order from finite-depth unitaries, measurement and feedforward

TL;DR

This work introduces a hierarchy for long-range entangled states based on the minimal number of single-site measurements (shots) interleaved with finite-depth unitaries required to prepare them. It provides explicit one-shot protocols for Abelian and class-2 nilpotent quantum doubles, and finite-shot GW gauging (KW) schemes for solvable groups, including nil-2 and metabelian cases, thereby enabling construction of a wide class of twisted quantum doubles. It also proves no-go results for certain non-solvable orders and Fibonacci anyons, arguing that these orders define measurement-equivalent phases inaccessible by any finite shot count; it further extends the framework with quasi-local unitaries to access additional phases, including chiral theories. The results reveal a structured landscape where the preparation complexity (shots) correlates with the underlying group structure, offering practical guidance for quantum simulators and a foundation for exploring the computational capabilities and boundaries of topological quantum states.

Abstract

Long-range entanglement--the backbone of topologically ordered states--cannot be created in finite time using local unitary circuits, or equivalently, adiabatic state preparation. Recently it has come to light that single-site measurements provide a loophole, allowing for finite-time state preparation in certain cases. Here we show how this observation imposes a complexity hierarchy on long-range entangled states based on the minimal number of measurement layers required to create the state, which we call "shots". First, similar to Abelian stabilizer states, we construct single-shot protocols for creating any non-Abelian quantum double of a group with nilpotency class two (such as $D_4$ or $Q_8$). We show that after the measurement, the wavefunction always collapses into the desired non-Abelian topological order, conditional on recording the measurement outcome. Moreover, the clean quantum double ground state can be deterministically prepared via feedforward--gates which depend on the measurement outcomes. Second, we provide the first constructive proof that a finite number of shots can implement the Kramers-Wannier duality transformation (i.e., the gauging map) for any solvable symmetry group. As a special case, this gives an explicit protocol to prepare twisted quantum double for all solvable groups. Third, we argue that certain topological orders, such as non-solvable quantum doubles or Fibonacci anyons, define non-trivial phases of matter under the equivalence class of finite-depth unitaries and measurement, which cannot be prepared by any finite number of shots. Moreover, we explore the consequences of allowing gates to have exponentially small tails, which enables, for example, the preparation of any Abelian anyon theory, including chiral ones. This hierarchy paints a new picture of the landscape of long-range entangled states, with practical implications for quantum simulators.

Figures (4)

• Figure 1: Including measurements induces a hierarchy on the equivalence relation between topological orders. Boxes are long-range entangled states which cannot be obtained from a product state by a finite-depth circuit (the topological order (TO) for a gauge group $G$ refers to the quantum double $\mathcal{D}(G)$). However, dashed lines indicate they can be obtained by using a layer of measurements and feedforward; the number of dashed lines equal the number of necessary shots. A dot-dashed line indicates the need for a quasi-FDLU rather than strictly local gates. Finally, solid (red) lines cannot be crossed by any finite number of measurement layers; these define non-trivial measurement-equivalent phases of matter (red box). We argue that Fib is such an example, and similarly for non-solvable groups, with representatives for their measurement-equivalent phases being given by the quantum doubles of perfect centerless groups (e.g, $A_n$ with $n \geq 5$).
• Figure 2: Single-shot preparation of the quantum double for nil-2 groups. The three circuit layers needed to entangle the product state in Eq. \ref{['eq:nil2']}. After measuring vertices (blue) and plaquettes (red), the resulting state on the edges (purple) exhibits $G$ topological order regardless of measurement outcome. If one desires, the exact ground state of $\mathcal{D}(G)$ can be recovered by a string of $X^n$ on the dual lattice and a string of $Z^q$ on the direct lattice.
• Figure 3: (a) 2D lattice model with nonzero Chern number (see Eq. \ref{['eq:Chern']}); bonds with arrows correspond to an imaginary hopping. (b) a stack of two Chern insulators with opposite Chern numbers; it can be adiabatically connected to a product state by tuning the rung couplings.
• Figure 4: Adiabatic path connecting two decoupled Chern insulators ($\lambda=0$) to a product state ($\lambda=1$). The Hamiltonian is given in Eq. \ref{['eq:Chern']} and the couplings are depicted in Fig. \ref{['fig:chernstack']}.

Theorems & Definitions (18)

• Lemma 1
• Theorem 2
• proof
• Lemma 3
• Theorem 4
• proof
• Lemma 5
• proof
• Corollary 6
• Theorem 7