51 problems

Gromov's conjecture. $\mathbb{Z}_2$-systolic freedom does not exist.

Quantum Hamming bound conjecture. All quantum codes satisfy the quantum Hamming bound.

A quantum low-density parity-check code is a quantum error-correcting code with sparse parity checks, and an adversarial error is an error chosen without a probabilistic restrictio…

Let $H_1,H_2,H_3$ be parity-check matrices of three classical binary codes, and let the associated XYZ product code be the non-CSS QLDPC code obtained from the three-fold product c…

Khesin–Lu–Shor distance conjecture. For $m\geq 7$, the distance is $d(m)=m$. This would determine the distance of the Hamming-code-derived family in the stated parameter range; the…

Let $C$ be a 3D Bacon-Shor code defined on a cubic lattice, with sheet-like stabilizer generators; its symmetric family on an $L\times L\times L$ lattice has parameters…

Let $D$ be a dimension and consider families of $D$-dimensional toric codes with appropriate twisted boundary conditions. Let $N$ denote the number of physical qubits, and let the…

Let $C$ be a class of quantum codes called three-dimensional product codes, obtained from the corresponding three-dimensional product construction. A code is single shot when one n…

Let $M$ be a translation-invariant qubit three-dimensional TQFT stabilizer model. Two such models are equivalent under a locality-preserving unitary when a locality-preserving unit…

Fix $t$, and let $n$ be large enough that the parameters $m_2,\ldots,m_t$ defined by … where $c\in(-1,1)$ is chosen so that $\sum_i m_i=n+1$. Let $H$ be the Tanner-code parity-chec…

Gauging-booster conjecture. The gauging booster will allow native logical action in arbitrary Clifford hierarchies implemented via twisted (gauged) 2D product codes, including HGP…

Let $X$ be a $t$-dimensional cubical complex and let $\mathcal{F}$ be a sheaf satisfying the requirement in Dinur (2024). For integers $i,j,k,l$ satisfying … let…

Good-distance conjecture. The code described in this section nevertheless has good distance.

Let $\mathbf{1}$ be an all-one vector of an appropriate length. For $3\leq k\leq 2^m-4$, let $\mathrm{HZ}_k(\mathbb{F}_{2^m}^*,\mathbf{1})$ denote the Han–Zhang code. Heng–Wang con…

Refined Singleton-bound conjecture. The minimum distance satisfies

Let $X$) be a weighted projective variety with orbifold singularities, and let $C$ be a quantum stabilizer code constructed from self-orthogonal algebraic geometry codes on $X$. Wr…

Distance bound conjecture. The distance of an $[[n,1,d]]$ $M_3$-code satisfies $d<d_{\max}$, with

The $44$-dimensional $bZ_2^N$ homological code has logical gate … The construction is explicitly given for $N=2,3,4,5$. Extension conjecture. The same construction can be extended…

Bravyi's conjecture. Higher products of CSS codes with good distance should exist.

Zeng–Pryadko's conjecture. Equality holds in this upper bound for every degree $m$ and all chain complexes $C$ and $D$.

Minimality conjecture. CSS codes constructed from extremal type-II self-dual codes are the smallest stabilizer codes with transversal single-qubit Clifford group for their respecti…

Conjecture. With an appropriate choice of local product codes, these complexes yield good locally testable codes (LTCs). Tensor products of projective Reed–Solomon codes are sugges…

Consider the subspace CSS code obtained by applying the paper's stabilizer-to-CSS mapping to the double semion code, with two physical qudits per edge and eight stabilizer generato…

Let $\mathcal{T}(Q_1,[\sigma_1])$ and $\mathcal{T}(Q_2,[\sigma_2])$ be theories described by bulk surfaces $\mathcal{S}(Q_i,[\sigma_i])$. Suppose that … Here…

Consider a family of generic $3$-manifolds whose interaction hypergraphs are sparse, so that each vertex has constant degree independent of the size of the hypergraph. Such sparsit…