Quantum Lego and XP Stabilizer Codes

TL;DR

The paper extends the XP stabilizer formalism to non-Pauli codes by applying the Quantum Lego framework, recasting XP codes as lego blocks built from atomic seeds. It demonstrates that operator matching together with conjoining provides a classically tractable description of XP symmetries for post-trace XP states (notably when the post-trace state remains XP at precision $N=2^t$), constituting a partial extension of the Gottesman–Knill approach. A tensor-network-based maximum-likelihood decoder is developed for XP regular codes via tensor-weight enumerators, and novel XP codes with improved distances are constructed, including a $[[7,1,3]]$ XP code and an $[[8,1,2]]$ code with a fault-tolerant $T$ gate. The work highlights the potential of non-Pauli stabilizer codes and holographic XP code designs, while outlining future routes for systematic XP code discovery, hidden-symmetry identification, and broader classical simulation techniques.

Abstract

We apply the recent graphical framework of "Quantum Lego" to XP stabilizer codes where the stabilizer group is generally non-Abelian. We show that the idea of operator matching continues to hold for such codes and is sufficient for generating all their XP symmetries provided the resulting code is XP. We provide an efficient classical algorithm for tracking these symmetries under tensor contraction or conjoining. This constitutes a partial extension of the algorithm implied by the Gottesman-Knill theorem beyond Pauli stabilizer states and Clifford operations. Because conjoining transformations generate quantum operations that are universal, the XP symmetries obtained from these algorithms do not uniquely identify the resulting tensors in general. Using this extended framework, we provide examples of novel XP stabilizer codes with a higher distance than existing non-trivial XP regular codes and a $[[8,1,2]]$ Pauli stabilizer code with a fault-tolerant $T$ gate. For XP regular codes, we also construct a tensor-network-based maximum likelihood decoder for any independently and identically distributed single qubit error channel using weight enumerators.

Figures (10)

• Figure 1: (a) A tensor $V_{i_1i_2i_3i_4i_5}$ can be expressed graphically where each index is assigned to a dangling leg. The same tensor can be used to construct states or maps but these objects are related to each other via the channel state duality (yellow arrow). The in-going blue arrow marks the input/logical degree of freedom while the out-going blue arrows mark output/physical degrees of freedom. (b) A tensor has a symmetry if it is invariant after contracting with operators. (c) If the local symmetries on the blue and red tensors, marked by $\mathcal{O}$'s and $\mathcal{Q}$'s respectively, are matching, then the operators acting on the connected edges satisfy $\mathcal{Q}_3=\mathcal{O}_3^*$ and $\mathcal{Q}_2=\mathcal{O}_2^*$. It is customary to drop the boxes when writing the symmetries to avoid clutter.
• Figure 2: Pauli stabilizer states (blue box) only cover a fraction of the whole many-body Hilbert space. In contrast, XP states with arbitrary precision (red box) encompass not only Pauli stabilizer states but also the magic state. When combined with the Clifford operations found within PSF, they constitute a universal gate set. Therefore, an XP Lego set is universal.
• Figure 3: Illustration of the two-round decoder.
• Figure 4: Two $[[4,2,2]]$ XP codes are contracted along the same legs to produce a rank-8 tensor. The logical leg is marked by a star.
• Figure 5: A holographic XP code where each heptagon represents the $[[7,1,3]]$ XP Lego that is not LU-equivalent to the Steane code. Red dots mark the logical legs.

Theorems & Definitions (43)

• Proposition 2.1
• Theorem 2.1
• Definition 3.1
• Definition 3.2
• Lemma 3.1
• Theorem 3.1: Operator Matching
• Proposition 3.1
• Theorem 3.2
• Corollary 3.1
• Theorem 3.3