Foliated Quantum Error-Correcting Codes

We show how to construct a large class of quantum error-correcting codes, known as Calderbank-Steane-Shor codes, from highly entangled cluster states. This becomes a primitive in a protocol that foliates a series of such cluster states into a much larger cluster state, implementing foliated quantum error correction. We exemplify this construction with several familiar quantum error-correction codes and propose a generic method for decoding foliated codes. We numerically evaluate the error-correction performance of a family of finite-rate Calderbank-Steane-Shor codes known as turbo codes, finding that they perform well over moderate depth foliations. Foliated codes have applications for quantum repeaters and fault-tolerant measurement-based quantum computation.

Figure 1

Examples of progenitor clusters for clusterized CSS codes. (a) Clusterized Steane code. (b) Clusterized Shor code. (c) Clusterized surface code. Code qubits (the blue circles) are connected by cluster bonds (the black lines) to ancilla qubits (the red squares). An $X$-basis measurement of ancilla $a_k$ projects neighboring code qubits onto an eigenstate of ${\otimes }_{{\mathcal{N}}_{a_k}}Z\in {\mathcal{S}}_Z$.

Figure 2

Examples of foliated, clusterized CSS codes. Code qubits (the blue circles) share cluster bonds (the black lines) with ancilla qubits (the red squares) in the same sheet, and with code qubits in adjacent sheets (the green lines). (a) Foliated Steane code. Being self-dual, primal and dual sheets are identical. The product of cluster stabilizers centered on the numbered qubits generates parity-check operators $C_1{C}_2\cdots C_6=X_1{X}_2\cdots X_6$. (b) Foliated Shor code. This code is not self-dual, so primal and dual sheets are different, and there are two kinds of parity-check operators: $C_a{C}_b\cdots C_h=X_a{X}_b\cdots X_h$, centered on primal sheets, and $C_1{C}_2{C}_3{C}_4=X_1{X}_2{X}_3{X}_4$, centered on dual sheets. (c) Foliated surface code [1], with parity-check operators $C_1\cdots C_6=X_1\cdots X_6$. (d) Foliated self-dual convolutional code with parity-check operator $C_1\cdots C_8=X_1\cdots X_8$. Stabilizers are generated by translations of the kernel (indicated by thick edges) across frames (here, the frame length is 3).

Figure 3

Numerical performance results for a foliated $r=\frac{1}{25}$, $d=25$ turbo code, for different numbers of foliated layers, $L$ (rows). Different colors correspond to different code sizes, $k=nr$; shading indicates $\pm 1\sigma$. A layer consists of a primal code sheet and a dual code sheet [see Fig. 2]. Word error rate (the left column) counts any errors across all $k$ logical qubits. Bit error rate (right column) counts the failure rate per logical qubit.