Draft:3D toric code

• Comment: Missing inline citations for entire sections. Aydoh8^{[what have I done now?]} 14:26, 21 July 2025 (UTC)

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The 3D Toric Code is a quantum error-correcting code and a model of a topologically ordered phase of matter in three spatial dimensions. It is a generalization of the well-known two-dimensional Toric Code introduced by Alexei Kitaev. Its primary significance lies in being the simplest model of a self-correcting quantum memory, a system that can passively protect its encoded quantum information against thermal noise without the need for active error correction, provided the temperature is below a critical threshold.

The model provides a concrete example of a $Z_{2}$ lattice gauge theory in 3+1 dimensions and exhibits a rich interplay between its point-like and loop-like elementary excitations.

Model Definition

The 3D Toric Code is defined on a three-dimensional cubic lattice with periodic boundary conditions (i.e., a 3-torus, T³). The degrees of freedom are qubits placed on the edges of this lattice. The model's behavior is governed by a stabilizer Hamiltonian, which is a sum of commuting local operators:

$H=-J_{e}\sum _{v}A_{v}-J_{m}\sum _{p}B_{p}$

where $J_{e}$ and $J_{m}$ are positive coupling constants. The two types of stabilizer operators are:

• Vertex term ($A_{v}$): For each vertex v of the lattice, the operator $A_{v}$ is the product of the Pauli-X operators on the 6 edges connected to that vertex.

$A_{v}=\prod _{j\in {\text{edges}}(v)}X_{j}$

• Plaquette term ($B_{p}$): For each elementary square face p (a plaquette) of the lattice, the operator $B_{p}$ is the product of the Pauli-Z operators on the 4 edges forming the boundary of that plaquette.

$B_{p}=\prod _{j\in {\text{edges}}(p)}Z_{j}$

All $A_{v}$ and $B_{p}$ operators commute with each other. This can be seen because any pair of operators either acts on disjoint sets of qubits or shares exactly two qubits, on which the Pauli operators (X and Z) commute. Because the Hamiltonian is a sum of commuting terms, it is exactly solvable.

Ground State and Excitations

The ground state |GS⟩ of the system is the unique state (on a simply connected space) that is a +1 eigenstate of all stabilizer operators:

$A_{v}|GS\rangle =|GS\rangle \quad \forall v$

$B_{p}|GS\rangle =|GS\rangle \quad \forall p$

Excitations above the ground state correspond to violations of these stabilizer conditions.

• Point-like Excitations (e): If the stabilizer condition is violated at a vertex v such that $A_{v}=-1$, this corresponds to a point-like quasiparticle excitation located at that vertex. These are often called "electric charges" in the language of $Z_{2}$ gauge theory. Such an excitation can be created by applying a string of Pauli-Z operators along a path ending at the vertex.

• Loop-like Excitations (m): If the stabilizer condition is violated at a plaquette p such that B_p = -1, this corresponds to an excitation on the loop forming the boundary of that plaquette. These are often called "magnetic fluxes." Unlike the point-like e charges, these excitations are fundamentally loop-like. Crucially, a single loop excitation is immobile under local operators. To move the loop, one must apply a "membrane" of Pauli-X operators, which changes the location of the loop but creates a line of high-energy e charge violations at its boundary. This energetic cost of moving an m-loop is a key feature of the model.

The immobility of the m loop excitations is the key difference from the 2D Toric Code (where both e and m are mobile point-like anyons) and is the source of the model's self-correcting properties. This type of emergent $Z_{2}$ gauge structure with deconfined point-like charges and immobile flux loops finds a remarkable physical analogue in spin ice materials, where violations of the local "ice rules" behave as mobile magnetic monopoles (analogous to e charges).

Topological Degeneracy and Logical Operators

When defined on a 3-torus T³, the 3D Toric Code has a topologically protected ground state degeneracy. This degeneracy arises because of the existence of non-local logical operators that commute with the Hamiltonian but are not products of the local stabilizers. These operators transform one ground state into another.

For a T³, there are three independent non-contractible directions. The number of logical qubits the system can store is given by the first Betti number of the manifold, $b_{1}(T^{3})=3$. Therefore, the 3D Toric Code on a T³ can store 3 logical qubits, leading to a ground state degeneracy of 2³ = 8.

The logical operators for these three qubits are:

• Logical Z operators ($Z_{L,i}$): A logical Z operator is a membrane of Pauli-Z operators wrapping a non-contractible 2D plane of the torus. For example, $Z_{L,1}$ would be the product of all Z operators on edges crossing the yz-plane at a fixed x coordinate. This operator commutes with all $A_{v}$ terms (as Z and X commute) and all $B_{p}$ terms (as it intersects any plaquette an even number of times).

• Logical X operators ($X_{L,i}$): A logical X operator is a string of Pauli-X operators wrapping a non-contractible 1D loop of the torus. For example, $X_{L,1}$ would be the product of all X operators on edges along a loop in the x-direction. This operator commutes with all $B_{p}$ terms and all $A_{v}$ terms (as it touches any vertex an even number of times).

The logical operators for the different qubits satisfy the correct algebra. For instance, the $Z_{L,1}$ membrane (on the yz-plane) and the $X_{L,1}$ string (along the x-direction) must intersect at exactly one point. At this point, the Pauli Z and X operators anti-commute, causing the logical operators as a whole to anti-commute: $Z_{L,1}X_{L,1}=-X_{L,1}Z_{L,1}$. However, logical operators from different pairs (e.g., $Z_{L,1}$ and $X_{L,2}$) commute because they act on disjoint sets of qubits.

Self-Correcting Quantum Memory

The most significant property of the 3D Toric Code is its capacity to function as a self-correcting quantum memory. This means the system can passively protect its encoded logical quantum state from thermal errors without requiring active measurement and feedback, provided its temperature T is below a certain critical temperature $T_{c}$.

This property arises from the energetic cost of creating a logical error. A logical error corresponds to applying a non-trivial logical operator (or a deformation thereof).

• A logical Z error requires creating an entire membrane of Z operator applications.
• A logical X error requires creating a non-contractible string of X operator applications.

Consider a logical X error. The string of X operators violates the $B_{p}$ stabilizers on all plaquettes along its boundary, creating a large tube of m loop excitations. To create such a logical error via thermal fluctuations, the system must overcome an energy barrier that grows with the length of the shortest non-contractible loop of the system (L). For a large system, this energy barrier is substantial. At a sufficiently low temperature, the probability of such a macroscopic error occurring is exponentially suppressed. The immobility of the magnetic loop excitations prevents them from diffusing and spreading across the lattice to cause a logical error, a failure mode that plagues the 2D Toric Code at any non-zero temperature.

Bosonic vs. Fermionic Toric Codes

The standard model described above is a bosonic toric code. Its elementary excitations (the point-like e charge and the loop-like m flux) are bosons. However, it is possible to construct fermionic toric codes that describe topological phases of fermions. These models are crucial for understanding fermionic topological phases and require a different construction.

One common approach, developed by Kitaev and Bravyi , involves placing Majorana fermion modes on the elements of the lattice (e.g., vertices and edges) and defining a Hamiltonian of four-fermion interaction terms. The resulting model can also have a $Z_{2}$ gauge structure, but its properties are distinct:

• Fermionic Excitations: The point-like (e) excitation can be a fermion. This means creating two such excitations and exchanging their positions results in a phase factor of -1.
• Ground State Degeneracy: The ground state degeneracy on a given manifold can depend on the spin structure chosen for that manifold.
• Different Logical Operators: The structure of the logical operators must be adapted to the fermionic nature of the underlying degrees of freedom.

While the bosonic 3D Toric Code serves as a model for a robust quantum memory, the fermionic versions are of fundamental importance for classifying phases of matter in condensed matter physics, particularly in the study of fermionic topological insulators and superconductors.