Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays

TL;DR

A fault tolerant construction for error correction and computation using two punctured quantum Reed-Muller (PQRM) codes and it is shown that the entire logical Clifford group can be achieved using only permutations, transversal gates, and fold-transversal gates.

Abstract

We give a fault tolerant construction for error correction and computation using two punctured quantum Reed-Muller (PQRM) codes. In particular, we consider the $[[127,1,15]]$ self-dual doubly-even code that has transversal Clifford gates (CNOT, H, S) and the triply-even $[[127,1,7]]$ code that has transversal T and CNOT gates. We show that code switching between these codes can be accomplished using Steane error correction. For fault-tolerant ancilla preparation we utilize the low-depth hypercube encoding circuit along with different code automorphism permutations in different ancilla blocks, while decoding is handled by the high-performance classical successive cancellation list decoder. In this way, every logical operation in this universal gate set is amenable to extended rectangle analysis. The CNOT exRec has a failure rate approaching $10^{-9}$ at $10^{-3}$ circuit-level depolarizing noise. Furthermore, we map the PQRM codes to a 2D layout suitable for implementation in arrays of trapped atoms and try to reduce the circuit depth of parallel atom movements in state preparation. The resulting protocol is strictly fault-tolerant for the $[[127,1,7]]$ code and practically fault-tolerant for the $[[127,1,15]]$ code. Moreover, each patch requires a permutation consisting of $7$ sub-hypercube swaps only. These are swaps of rectangular grids in our 2D hypercube layout and can be naturally created with acousto-optic deflectors (AODs). Lastly, we show for the family of $[[2^{2r},{2r\choose r},2^r]]$ QRM codes that the entire logical Clifford group can be achieved using only permutations, transversal gates, and fold-transversal gates.

Figures (10)

• Figure 1: Two equivalent ways of understanding the classical Reed-Muller code. (a) via the hypercube encoding circuit. (b) polynomial evaluation. The information position (marked with $0$/$1$) on the left corresponds to which rows (evaluation vectors) are chosen from the encoding matrix.
• Figure 2: The QRM encoding circuit is the same as the one for classical RM codes, just the input are quantum states. (a) Showing the $X$ and $Z$ stabilizers of quantum RM codes obtained by propagating the stabilizers from the input (acting on a single qubit) to the output. To create the logical zero state $|0\rangle_L$ of punctured QRM code, remove the bottom-most wire and all CNOT gates involving it. (b) To create the logical plus state $|+\rangle_L$ with the correct chirality, one reverses the direction of all the CNOT gates, as well as bit-reverse and the input states. It is still the bottom-most wire that is removed.
• Figure 3: Code switching via Steane error correction (a) from $\text{PQRM}(3,3,7)$ to $\text{PQRM}(2,4,7)$, (b) the reverse direction. These two circuits hold for all PQRM codes. (c) Stabilizer containment relationship. $C_X$/$C_Z$ being the $X$/$Z$-type stabilizers of $\text{PQRM}(3,3,7)$ and $C'_X/C'_Z$ are those of $\text{PQRM}(2,4,7)$. The shown distance, e.g., $d'_X=31$ is the minimum distance between $C'_X=\overline{\text{RM}}(2,7)$ and $\mathbf{1}+C'_X$.
• Figure 4: Data qubit noise decoding on BSC($p$). List size $8$ for all simulations. The decoder needs to distinguish between whether the added noise is closer to $\overline{\text{RM}}(r,m)$ or $\mathbf{1}+\overline{\text{RM}}(r,m)$ for $r=4$ (purple), $3$ (green), $2$ (blue).
• Figure 5: State preparation circuit to create verified $|0\rangle_L$ and $|+\rangle_L$ state. Only transversal CNOT and measurements are used in these verification circuits. Logical states are still encoded using the hypercube encoding circuit, but each of them undergoes a carefully chosen automorphism permutation before verification circuits to make the preparation fault-tolerant. The output patch is accepted if all transversal measurement results turn out to be stabilizers of the desired state.

Theorems & Definitions (13)

• Theorem 1
• Definition 2
• Lemma 3
• Definition 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Theorem 10