Coupled-Layer Construction of Quantum Product Codes

Abstract

Product codes are a class of quantum error correcting codes built from two or more constituent codes. They have recently gained prominence for a breakthrough yielding quantum low-density parity-check (qLDPC) codes with favorable scaling of both code distance and encoding rate. However, despite its powerful algebraic formulation, the physical mechanism for assembling a general product code from its constituents remains unclear. In this letter, we show that the tensor and balanced product codes admit an intuitive coupled-layer construction by taking a stack of one code and condensing a set of excitations in the pattern given by the checks of the other code. Our framework accommodates both classical or quantum CSS input codes, unifies known physical mechanisms for constructing higher dimensional topological phases via anyon condensation, and naturally extends to non-topological codes.

Figures (24)

• Figure 1: The tensor product complex
• Figure 1: Left: The stacking configuration of 2D TC $\otimes$ 2D TC. The ancillas are not displayed for clarity. Right: condensing anyons on a 2D layout. The blue box indicates $\mathsf{m}$ anyons are condensed around the vertex, and red box shows $\mathsf{e}$ anyons are condensed around the plaquette. After condensing, both $\mathsf{m}$ and $\mathsf{e}$ form loops, giving the 4D loop-only TC.
• Figure 2: The stabilizers of the 2D toric code $\otimes$ [[4,2,2]].
• Figure 2: Left: a unit cell of the RBH cluster state. The gray layers are three stacks of 2D TC labeled by $2i-1$, $2i$ and $2i+1$. The black qubits are the TC qubits. The blue qubits are $X$-ancillas, and red qubits are $Z$-ancillas. Right: the first term condenses $\mathsf{m}$ anyons in three TC centered on an odd layer. The second term condenses three $\mathsf{e}$ anyons in three TC centered on an even layer. The qubit $q$ is chosen to be on a vertical edge in each term.
• Figure 3: The coupled layer construction from the chain complex perspective. Starting with stacks of CSS$_1$ given by the complex in the grey band along with $X/Z$ ancillas, Measuring $X/Z$-stabilizers given by the red/blue arrow ($\alpha(q_1,a_2)/ \beta(q_1,b_2)$) induces the blue/red dashed arrow, completing $\zeta(b_1,q_2)/\xi(a_1,q_2)$.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Definition 5
• Definition 6
• Remark 2
• Remark 3
• Remark 4