Toric-code Hamiltonians and local indistinguishability

Here a closed surface $S$ means a nonempty, compact, connected, boundaryless topological $2$-manifold in the convention of the earlier definition Topological manifolds with and without boundary: $S$ is Hausdorff (distinct points have disjoint open neighborhoods), has an at-most-countable basis of open sets (every open set is a union of basis members), and for every $x\in S$ there are an open neighborhood $U$ and a homeomorphism $h:U\to V$ onto an open subset $V\subseteq {\mathbb{R}}_{+}^2$, where ${\mathbb{R}}_{+}^2=\{(t_1,t_2)\in {\mathbb{R}}^2:t_2\ge 0\}$. The model boundary is $t_2=0$, and the manifold boundary subset defined by that chart convention is empty; equivalently, every point has a neighborhood homeomorphic to an open subset of ${\mathbb{R}}^2$. It is orientable when it has an $R$-orientation for $R=\mathbb{Z}$ under the earlier definition R-orientation of a topological manifold: a continuous section of the local homology system with fibers $H_2(S,S\setminus \{x\};\mathbb{Z})$ that generates each fiber.

A finite cellulation $K$ of $S$ is a finite $2$-dimensional CW decomposition with vertex, edge, and face sets $V,E,F$ and no higher-dimensional cells: each edge has a characteristic map from $[0,1]$ whose restriction to $(0,1)$ maps homeomorphically onto that edge and whose endpoints map to vertices; each face has a characteristic map from the closed disk whose interior maps homeomorphically onto that face and whose boundary follows a finite closed edge path in the resulting graph. The quotient is $S$, and loops, multiple edges, and repeated face-edge incidences are allowed. The primal graph has vertices $V$ and edges $E$. The dual graph has one vertex for each face and one dual edge crossing each primal edge, joining its two incident face sides; loops and multiple edges are allowed. Assume both graphs are connected. Let ${\mathbb{F}}_2=\{0,1\}$ with addition modulo two, and for each finite cell set $W$ use the standard dot product $⟨r,s⟩={\sum }_{w\in W}{r}_w{s}_w$ on ${\mathbb{F}}_2^W$; transposes are taken in the cell bases. Incidence multiplicities are reduced modulo two: ${\mathrm{\partial }}_1:{\mathbb{F}}_2^E\to {\mathbb{F}}_2^V$ sends each edge to the sum of its endpoint incidences, and ${\mathrm{\partial }}_2:{\mathbb{F}}_2^F\to {\mathbb{F}}_2^E$ sends each face to its boundary-edge incidence vector. Since each face boundary is a closed edge path, ${\mathrm{\partial }}_1{\mathrm{\partial }}_2=0$. Since $S$ has no boundary, each edge has two incident face sides, counted with multiplicity, so ${\mathrm{\partial }}_2({\sum }_{f\in F}f)=0$. No orientation of the surface or of its cells is chosen or needed for these mod-two constructions.

Identify $E$ with a finite region $\mathrm{\Lambda }\subset \mathbb{Z}$ and use the finite local spin matrix algebra on $\mathrm{\Lambda }$ with spin dimension $q=2$. The edge Hilbert space is ${\mathcal{H}}_K=({\mathbb{C}}^2{)}^{\otimes E}$, so $\dim {\mathcal{H}}_K=2^{\mathrm{\mid }E\mathrm{\mid }}$. Let $X=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and $Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$. For $a,b\in {\mathbb{F}}_2^E$, define $X(a)={⨂}_{e\in E}{X}^{a_e}$, $Z(b)={⨂}_{e\in E}{Z}^{b_e}$, and $P(a,b)=X(a)Z(b)$. For a vertex $v$ and face $f$, put $A_v=X({\mathrm{\partial }}_1^{T}v)$ and $B_f=Z({\mathrm{\partial }}_{2}f)$, where $v$ and $f$ also denote the corresponding basis vectors. Thus $A_v$ is the product of $X$ over incident edges with odd incidence, and $B_f$ is the product of $Z$ over boundary edges with odd incidence. The identity ${\mathrm{\partial }}_1{\mathrm{\partial }}_2=0$ makes every $A_v$ commute with every $B_f$; each is a self-adjoint involution.

For positive real coefficients $J_v,K_f$, define the toric-code Hamiltonian

$H_K={\sum }_{v\in V}{J}_v\frac{I-A_v}{2}+{\sum }_{f\in F}{K}_f\frac{I-B_f}{2}\mathrm{.}$

The Pauli operators are dimensionless. In a physical interpretation, $J_v,K_f$ have energy units (joules in SI); here they are positive mathematical coefficients and no measured values are asserted. A simultaneous $+1$ eigenvector of all $A_v,B_f$ is called a ground vector. The ground space is their common $+1$ eigenspace, equivalently $\ker H_K$.

For the square $L\times L$ torus, $L\ge 3$, use vertices ${\mathbb{Z}}_L^2$, horizontal and vertical nearest-neighbor edges, and the $L^2$ square faces, with periodic boundary conditions. A coordinate cycle is a closed edge path winding once around one coordinate direction. Let $p_1,p_2$ be the horizontal and vertical coordinate cycles in the primal graph, and let $q_1,q_2$ be the corresponding vertical and horizontal coordinate cycles in the dual graph, chosen so that $I(p_i,q_j)={\delta }_{ij}$. The mod-two intersection number is the parity of primal edges crossed by a dual cycle. A topological disk is a subset homeomorphic to the closed unit disk. A finite edge support is disk-local with full complement cycles if it is contained in a topological disk whose complement contains representatives of these two primal and two dual coordinate cycles. The phrase “both independent homology cycles” below refers to these two primal cycles and their dual crossing representatives.

An edge path in a graph with vertex set $W$ and edge set $D$ is a finite sequence $w_0,d_1,w_1,\dots ,d_n,w_n$ in which each $d_i$ joins $w_{i-1}$ to $w_i$. Its mod-two edge chain counts each edge occurrence modulo two; the boundary of this chain is $w_0+w_n$. An open primal path is such a path in the vertex-edge graph with distinct endpoint vertices $w_0\mathrm{\ne }{w}_n$; its $Z$ string is $Z(b)$ for its edge chain $b$. An open dual path is such a path in the face-adjacency graph with distinct endpoint faces $w_0\mathrm{\ne }{w}_n$; its $X$ string is $X(a)$ for the primal edges crossed by its dual edges. The same formulas define strings along closed paths, whose endpoints coincide ($w_0=w_n$). A closed string has zero boundary in the corresponding graph. The support of an operator is a finite edge set $S$ such that the operator is a matrix on $({\mathbb{C}}^2{)}^{\otimes S}$ tensored with the identity on $E\setminus S$.

Topological manifolds with and without boundary

Local spin matrix algebras and their factor inclusions

R-orientation of a topological manifold

A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” §1

John Preskill, Lecture Notes for Physics 219: Quantum Computation, §9.7

Assume the Axiom of Choice. Let $K$ be a finite cellulation of a closed, connected, orientable surface of genus $g\ge 0$, with the incidence and Hamiltonian conventions of the toric-code definition. For positive couplings $J_v,K_f$, the Hamiltonian is positive and frustration free, meaning its positive penalty terms have a common nonzero vector in their kernels. Its ground-space dimension is $2^{2g}$. Define its spectral gap as the least positive eigenvalue of $H_K$, or $+\mathrm{\infty }$ if there is no positive eigenvalue. This gap is at least $2m$, where $m=\min (\{J_v:v\in V\}\cup \{K_f:f\in F\})>0$, and in particular is at least $m$. On a uniform $L\times L$ square torus with $L\ge 3$ and couplings $J_v=J>0$, $K_f=K>0$, the exact gap is $2\min (J,K)$.

A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” §1

John Preskill, Lecture Notes for Physics 219: Quantum Computation, §§9.6–9.7

The gap statement concerns the exact finite commuting-projector Hamiltonian. No perturbation-stability, thermodynamic-limit, or empirical claim is asserted.

Let $K_L$ be the square $L\times L$ torus cellulation with $L\ge 3$. For $a,b\in {\mathbb{F}}_2^E$, let $P(a,b)=X(a)Z(b)$, where $X(a)={⨂}_e{X}^{a_e}$ and $Z(b)={⨂}_e{Z}^{b_e}$. Suppose its edge support is contained in a disk whose complement contains primal cycles $p_1,p_2$ and dual cycles $q_1,q_2$ with $I(p_i,q_j)={\delta }_{ij}$. Write $A_v=X({\mathrm{\partial }}_1^{T}v)$ and $B_f=Z({\mathrm{\partial }}_{2}f)$. If $P(a,b)$ commutes with every $A_v$ and $B_f$, then there are vertex and face chains $u\in {\mathbb{F}}_2^V$ and $w\in {\mathbb{F}}_2^F$ such that $a={\mathrm{\partial }}_1^{T}u$ and $b={\mathrm{\partial }}_{2}w$. Consequently $P(a,b)$ is a product of the vertex and face stabilizers.

A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” §1

John Preskill, Lecture Notes for Physics 219: Quantum Computation, §§9.6–9.7

The disk and complement condition is essential to this argument: it makes local cycles pair trivially with both complementary dual generators and local cocycles vanish on both complementary primal generators. The lemma concerns the exact unperturbed stabilizer algebra only.

Assume the Axiom of Choice and use the square $L\times L$ torus cellulation, $L\ge 3$. A closed primal $Z$ string and a closed dual $X$ string commute or anticommute according to the parity of their edge crossings: their commutator sign is $(-1{)}^I$, where $I$ is the mod-two intersection number. Let $P_0$ be the orthogonal projection onto the common $+1$ eigenspace of all vertex and face stabilizers. For any operator $O$ supported on an edge set contained in a disk whose complement contains the coordinate cycles $p_1,p_2,q_1,q_2$ specified in the definition, there is a scalar $\lambda (O)\in \mathbb{C}$ such that $P_{0}O{P}_0=\lambda (O)P_0$. This is exact local indistinguishability for the unperturbed finite Hamiltonian.

A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” §1

John Preskill, Lecture Notes for Physics 219: Quantum Computation, §§9.6–9.7

The proof is independent of the diagram. The picture shows one representative pair of fundamental cycles on the square fundamental domain; opposite sides are identified, the blue solid path is primal, and the red dashed path is dual.

At the displayed crossing the primal and dual strings meet on one edge, so their intersection parity is one and their Pauli strings anticommute. A diagram illustrates the convention; it supplies no proof step. No perturbation-stability claim follows from this exact compression identity.