Dynamically generated concatenated codes and their phase diagrams

TL;DR

The paper studies dynamically generated concatenated quantum codes built from unitary circuits on expanding tree geometries, showing that certain gate classes yield zero-rate codes with code distance growing exponentially with tree depth. It develops a tensor-network framework and complete vector/coset weight enumerators to perform optimal decoding, interpreting bulk noise as an effective temperature and surface noise as the decoding initialization, which maps to a spin-glass-like phase structure on trees. For heralded noise, the recursion becomes exactly solvable, enabling analytic phase diagrams and the notion of conditional code distance, while unheralded noise reveals rich spin-glass landscapes with fixed-point distributions and phase transitions. The work identifies two high-performing tree codes—the optimal-distance self-concatenated code with $d(t)\sim 1.521^t$ and the Bell tree with $d(t)=2^{\lfloor t/2\rfloor}$ (CSS, permitting separate X and Z decoding)—and connects these quantum codes to classical broadcasting on trees and LDPC-type classical codes on locally tree-like graphs. These results illuminate how simple, depth-growing encodings can achieve large distances and robust thresholds, guiding fault-tolerant code design and offering exact analytic control in the heralded-noise limit.

Abstract

We formulate code concatenation as the action of a unitary quantum circuit on an expanding tree geometry and find that for certain classes of gates, applied identically at each node, a binary tree circuit encodes a single logical qubit with code distance that grows exponentially in the depth of the tree. When there is noise in the bulk or at the end of this encoding circuit, the system undergoes a phase transition between a coding phase, where an optimal decoder can successfully recover logical information, and a non-coding phase. Leveraging the tree structure, we combine the formalism of "tensor enumerators" from quantum coding theory with standard recursive techniques for classical spin models on the Bethe lattice to explore these phases. In the presence of bulk errors, the coding phase is a type of spin glass, characterized by a distribution of failure probabilities. When the errors are heralded, the recursion relation is exactly solvable, giving us an analytic handle on the phase diagram.

Figures (1)

• Figure 1: Binary tree encoding circuit. This tree has depth 4 and produces a $[[16,1,d(4)]]$ stabilizer code. Blue and yellow "starbursts" indicate single-qubit errors on the branches and leaves, respectively, of the tree; the rates of these errors parameterize the phase diagrams studied in this work.