Minimal projective-plane cellulations for single-qubit topological quantum codes: optimality of nine edges and classification of the minimal codes

Cellulations of the real projective plane $RP^2$ define single-qubit homological (surface-type) stabilizer codes, since $H_1(RP^2;Z_2)=Z_2$ has a single essential class; the bit-flip and phase distances are the lengths of the shortest essential (Z_2-homologically nontrivial) cycles in the cellulation and in its dual. Freedman and Meyer constructed three pairwise-inequivalent 9-edge cellulations with both distances 3 — one of which is exactly Shor's original [[9,1,3]] code — and asserted that no cellulation with 5 or 7 edges achieves essential-cycle lengths of at least 3 in both the primal and the dual. The problem, which the source leaves open, is to determine whether nine edges is in fact the minimum number of qubits of such a distance-3 projective-plane code (the edge counts 6 and 8 are never addressed in the source, and $n\leq 4$ is excluded by the quantum Singleton bound), and to classify, up to local-unitary-and-permutation equivalence of the resulting codes, all cellulations achieving that minimum — in particular, whether the three known 9-qubit codes exhaust the list.

Background

A stabilizer quantum error-correcting code encodes $k$ logical qubits into $n$ physical qubits as the joint +1 eigenspace of an abelian group of Pauli operators (the stabilizers); its distance $d$ is the minimum weight of a Pauli operator acting nontrivially on the encoded space, so $d=3$ means an arbitrary single-qubit error is correctable, and the code is denoted $[[n,k,d]]$. The quantum Singleton bound $n-k\geq 2(d-1)$ forces $n\geq 5$ for any $[[n,1,3]]$ code; the perfect five-qubit code and the seven-qubit CSS code (CSS = Calderbank-Shor-Steane, a stabilizer construction whose $X$- and $Z$-type parity checks come from two paired classical binary codes) meet $n=5$ and $n=7$, and Shor's original nine-qubit repetition code is the classic $[[9,1,3]]$ example. Kitaev's toric or surface code construction supplies stabilizer codes with local checks from topology: given a cellulation $\mathcal{C}$ of a closed surface (a finite decomposition into vertices, edges, and disk faces), place one qubit on each edge, and take as stabilizers $A_f$ (a tensor of $\sigma_x$ over the edges on the boundary of each face $f$) and $B_v$ (a tensor of $\sigma_z$ over the edges incident at each vertex $v$); these satisfy exactly two global relations, and the resulting code space is the space of complex functions on the first homology group $H_1(\mathcal{C};Z_2)$, so the code encodes $b_1=\dim_{Z_2}H_1$ qubits. A bit-flip error pattern is a $Z_2$ 1-chain detected at its boundary vertices, and correction fails exactly when the residual error contains an essential cycle, i.e., one representing a nonzero class in $H_1(\mathcal{C};Z_2)$; hence the bit-flip distance equals the length of the shortest essential cycle of $\mathcal{C}$, and the phase distance equals the shortest essential cycle of the dual cellulation $\mathcal{C}^*$ (the cellulation obtained by exchanging the roles of faces and vertices, under which the $X$- and $Z$-type stabilizers are exchanged). A genus-$g$ orientable surface gives $2g$ encoded qubits, but the real projective plane $RP^2$ — the sphere with antipodal points identified, the simplest closed non-orientable surface — has $H_1(RP^2;Z_2)=Z_2$, a single essential $Z_2$-class, so every cellulation of $RP^2$ defines a single-qubit code. Freedman and Meyer studied the smallest such codes: a regular 15-edge triangulation has distances (3,5); three inequivalent 9-edge cellulations have both distances 3, one of them being exactly Shor's $[[9,1,3]]$ code; and the three codes were distinguished by the ranks of reduced density matrices of the code projectors (obtained by tracing out pairs of qubits), which are invariants of the action of local unitaries and qubit permutations, $(U(2))^{\otimes n}\rtimes S_n$, the standard equivalence notion for codes in this context. They also noted that the perfect five-qubit and seven-qubit CSS codes cannot arise this way, since (they assert, without proof) there are no cellulations of $RP^2$ with 5 or 7 edges whose essential and dual essential cycles all have length at least 3. What the paper does not determine is whether nine edges is actually the minimum — the counts of 6 and 8 edges are never addressed, while $n\leq 4$ is impossible for any distance-3 single-qubit code by the quantum Singleton bound — and whether its three codes exhaust the minimal list. (The source also observes that a cellulation and its dual yield equivalent codes via the Hadamard transform on every qubit (the single-qubit unitary $H$ with $H\sigma_x H=\sigma_z$), so code equivalence is coarser than isomorphism of cellulations.)

Problem Statement

Let $\mathcal{C}$ be a cellulation (2-cell decomposition: every face a disk) of the real projective plane $RP^2$ with edge set $E$, and associate to it the homological stabilizer code with one qubit per edge, $X$-type stabilizers on face boundaries, and $Z$-type stabilizers on vertex stars; this is an $[[|E|,1]]$ code (in the standard $[[n,k,d]]$ notation: $n$ physical qubits encoding $k$ logical qubits at distance $d$) whose bit-flip distance $d_X(\mathcal{C})$ is the length of the shortest essential (nonzero in $H_1(\mathcal{C};Z_2)$) cycle of $\mathcal{C}$ and whose phase distance $d_Z(\mathcal{C})$ is the length of the shortest essential cycle of the dual cellulation $\mathcal{C}^*$. Freedman and Meyer constructed three pairwise-inequivalent cellulations with $|E|=9$ and $d_X=d_Z=3$ (one equal to Shor's $[[9,1,3]]$ code) and asserted that no cellulation with $|E|=5$ or $7$ has both essential-cycle lengths at least 3.

(1) Optimality: does there exist a cellulation $\mathcal{C}$ of $RP^2$ with $|E|<9$ such that $d_X(\mathcal{C})\geq 3$ and $d_Z(\mathcal{C})\geq 3$? Since $|E|\leq 4$ is excluded for any distance-3 single-qubit code by the quantum Singleton bound $n-k\geq 2(d-1)$ and the source excludes 5 and 7 by assertion, the unresolved cases are $|E|=6$ and $|E|=8$: equivalently, is $[[9,1,3]]$ the smallest projective-plane homological code that corrects an arbitrary single error, or does a $[[6,1,3]]$ or $[[8,1,3]]$ projective-plane code exist? Settling the optimality question also proves (or corrects) the source's unproven exclusion of the 5- and 7-edge cases.

(2) Classification: determine the complete set of cellulations of $RP^2$ with the minimal number of edges satisfying $d_X\geq 3$ and $d_Z\geq 3$, up to combinatorial isomorphism of the cellulation, and classify the resulting stabilizer codes up to equivalence under local unitaries and qubit permutations, $(U(2))^{\otimes n}\rtimes S_n$ (the equivalence notion used in the source, under which the ranks of reduced density matrices of the code projector are invariants, and under which a cellulation and its dual give equivalent codes via the Hadamard transform on every qubit, the single-qubit unitary exchanging $\sigma_x$ and $\sigma_z$): for every pair of codes, either exhibit an explicit equivalence or certify inequivalence by a sound invariant argument. In particular: are the three 9-qubit codes of Freedman and Meyer the complete list at the minimum, or do further pairwise-inequivalent minimal codes exist?

No restrictions on face sizes, vertex valences, or stabilizer weights are imposed beyond the definition of a cellulation; the distance targets are those of the source paper's regime (correcting one arbitrary error).

References

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