Topological order in symmetric blockade structures

TL;DR

The work tackles designing prescribed quantum phases from simple two-level blockade systems by introducing blockade-graph automorphisms as a general symmetry tool. It develops fully-symmetric blockade structures and proves that uniform quantum fluctuations stabilize a unique ground state that is an equal-weight superposition over the classical logical states, enabling robust topological order. By constructing a tessellated, quasi-two-dimensional blockade structure with edge-encoded qubits and FSU-based XOR/XNOR constraints, the authors realize a ${\mathbb{Z}_2}$ loop condensate in the same phase as the toric code, without relying on numerics. They further demonstrate how local automorphisms give rise to a gapped, topologically ordered ground state, and discuss the remaining challenges, including a rigorous bulk-gap proof and planar embeddings. These results expand the bottom-up toolbox for engineered quantum matter using purely two-body blockade interactions and uniform fluctuations.

Abstract

The bottom-up design of strongly interacting quantum materials with prescribed ground state properties is a highly nontrivial task, especially if only simple constituents with realistic two-body interactions are available on the microscopic level. Here we study two- and three-dimensional structures of two-level systems that interact via a simple blockade potential in the presence of a coherent coupling between the two states. For such strongly interacting quantum many-body systems, we introduce the concept of blockade graph automorphisms to construct symmetric blockade structures with strong quantum fluctuations that lead to equal-weight superpositions of tailored states. Drawing from these results, we design a quasi-two-dimensional periodic quantum system that - as we show rigorously - features a topological $\mathbb{Z}_2$ spin liquid as its ground state. Our construction is based on the implementation of a local symmetry on the microscopic level in a system with only two-body interactions.

Figures (17)

• Figure 1: Rationale. The "inverse problem" of condensed matter physics studies the construction of prescribed, robust quantum phases from a given set of simple, microscopic constituents. Here we consider a toolbox motivated by (but not specific to) the Rydberg platform: two-level systems can be placed freely in two and three dimensions and interact via a simple blockade mechanism. Our contribution is an extension of the theoretical foundations of this toolbox to construct interesting quantum phases.
• Figure 2: Setting & Objective. (a) A two-dimensional arrangement $\mathcal{C}=(\boldsymbol{r}_i,\Delta_i)_{i\in V}$ of atoms $i\in V$ with position $\boldsymbol{r}_i$ and detuning $\Delta_i$ is governed by the Hamiltonian $H_\mathcal{C}$ that describes a blockade interaction with radius ${r_{\text{\tinyB}}}$. The detunings are encoded by colors. (b) Interactions and detunings are conveniently represented by a vertex-weighted blockade graph $G_\mathcal{C}$. (c) For $\Omega=0$, the Hamiltonian gives rise to a degenerate low-energy eigenspace $\mathcal{H}_\mathcal{C}<\mathcal{H}$, separated from the excited states by a gap $\Delta E$. The ground states correspond to maximum-weight independent sets of the blockade graph. (d) Our objective is to design blockade structures in which quantum fluctuations $\Omega\neq 0$ stabilize a ground state that is an equal-weight superposition of the states in $\mathcal{H}_\mathcal{C}$.
• Figure 3: Review of blockade structures. (a) The simplest blockade structure $\mathcal{C}_\texttt{NOT}$ consists of two equally detuned atoms in blockade: $\Delta_Q=\Delta_A=1\Delta$ (blue squares) (i). Its degenerate ground state manifold $\mathcal{H}_{\mathcal{C}_\texttt{NOT}}$ is spanned by the two states $\mathinner{|{01}\rangle}$ and $\mathinner{|{10}\rangle}$ (ii). It therefore realizes the language $L_\texttt{NOT}=\{01,10\}$, which contains bit strings that correspond to the rows of the truth table of the NOT-gate (iii). (b) The five-atom structure $\mathcal{C}_\texttt{NOR}$ consists of three ports (squares) and two ancillas (disks) (i). Two ports and both ancillas are detuned by $1\Delta$ (blue) while the central port is detuned by $\Delta_Q=2\Delta$ (green). As a consequence, the structure has a four-fold degenerate ground state manifold (ii). If one ignores the ancillas and labels the ground states by the excitation patterns of the ports, one finds the language $L_\texttt{NOR}=\{001,010,100,110\}$ which corresponds to the truth table of the universal NOR-gate (Not-OR) (iii). (c) The NOT-structure $\mathcal{C}_\texttt{NOT}$ can be "glued" to the output port of the NOR-structure $\mathcal{C}_\texttt{NOR}$ by identifying this port atom with an input port atom of $\mathcal{C}_\texttt{NOT}$, thereby adding up their detunings to $3\Delta$ (red) (i). This procedure is called amalgamation and results in an OR-structure, as can bee seen from the ground state patterns (ii) that realize the truth table (iii) of an OR-gate. The existence of the NOR-structure and the possibility of amalgamation make the toolbox of blockade structures functionally complete.
• Figure 4: Motivation: Logic gates. (a) (i) The NOR-gate introduced in Ref. Stastny2023a. It is related to a triangle-based family of primitives that exhaust all fundamental logic gates. (ii) The structure has four degenerate ground states that realize the truth table of a NOR-gate (iii). The numbers at arrows in (ii) denote Hamming distances between excitation patterns. The gray boxes indicate the orbits (for a definition see main text) in which the excitation patterns transform under the symmetry $S$ highlighted in (i). (iv-v) With quantum fluctuations, the ground state of $\mathcal{C}_\texttt{NOR}$ is not equal-weight, but favors one logic configuration above all others. (b) By contrast, the (inverted) crossing $\mathcal{C}_\texttt{ICRS}$ has an equal-weight ground state for $\Omega\neq 0$ (iv-v). It is also highly symmetric (i-ii): The symmetries $S_1$ and $S_2$ generate graph automorphisms that transform all ground state patterns into each other [gray box in (ii)].
• Figure 5: Fully-symmetric universal (FSU) gate. (a) The FSU-gate $\mathcal{C}_\texttt{FSU}$ is constructed from the inverted crossing by embedding it in 3D and attaching two additional atoms. The resulting structure has the full tetrahedral symmetry as automorphism group. Here we show the Klein four-group $V=\{1,C_{2,x},C_{2,y},C_{2,z}\}$, where $C_{2,\alpha}$ denotes the $\pi$-rotation about one of the three symmetry axes $\alpha=x,y,z$. (b) The four degenerate ground states form a single orbit under the action of $V$, thereby ensuring their equal-weight superposition in the ground state for $\Omega\neq 0$. Furthermore, the Hamming distances (numbers on arrows) of all transitions are equal [cf. \ref{['fig:motivation']} (b-ii)]. (c) The gate implements most Boolean primitive gates by choosing different atoms as ports. The six atoms on the "wings" form logically inverted antipodal pairs. We highlight the gate choice for the XOR-gate which becomes important below. $\texttt{INH}_{xy}=\neg x\wedge y$ denotes the "inhibition gate" where $x$ inhibits $y$. (d) Weight of the logical states for $\Omega\neq 0$ and logical overlap in the ground state (upper panel) and low-energy spectrum (lower panel).

Theorems & Definitions (29)

• Definition 1: Graph automorphism
• Definition 2: Fully-symmetric blockade structure
• Proposition 1
• Proposition 2
• Proposition 3
• proof
• Lemma 1
• Theorem 1: Perron-Frobenius
• proof
• proof