20 problems

Clifford bound conjecture. The cardinality of any unitary 2-design is at least

Let ${\cal C}_m$ be the Clifford group acting on the sphere associated with the code and invariants discussed above. For $m=2$, there are real points on $S^3$ at which the harmonic…

Let the configuration space of finite-dimensional quantum mechanics be a finite abelian group $A$ of even order, so that $|A|$ is even. Consider the associated projective Clifford…

Clifford extension splitting conjecture. The Clifford extension splits if and only if $4\nmid |A|$.

Direct-product distillation conjecture. The state

Mana-maximization conjecture. Based on numerical results, stabilized subspaces of finite subgroups of the Clifford group whose Wigner functions are everywhere non-vanishing maximiz…

Let $W_2$ denote the two-anyon-pair state, and let $\Sigma_2$ be a genus-$2$ handlebody. The Clifford group orbit of $W_2$ is represented by removing two solid tori from the interi…

Let $d$ be prime, let $\boldsymbol{1}_{2t}d0 (1,1,\dots,1)^\top$ be the all-ones vector in $\mathbb{F}_d^{2t}$, and let $\Sigma_{t,t}(d)$ denote the set of stochastic Lagrangian su…

Let $d$ be a positive integer and let $\ket{f}$ be a fiducial vector for a Weyl–Heisenberg SIC. A unitary $A$ is a canonical order-three unitary when its Clifford trace satisfies…

Normal-subgroup conjecture. When $n\geq 3$, the $n$-qubit Pauli group $\mathcal{P}_n$ is the only non-trivial proper normal subgroup of $\mathcal{C}_n$.

Faithfulness conjecture. The matrix representation $\mathcal{M}_{2^n}$ for $\mathcal{C}_n$ is faithful.

Integrality and self-duality conjecture. The entries in the character table of $\mathcal{C}_n$ are integers, and hence all irreducible characters of $\mathcal{C}_n$ are self-dual.

Stabilizerness conjecture. Some measure of stabilizerness, similar to stabilizer rank or mana, can be defined using the order of the state's stabilizer subgroup in $\mathcal{C}_n$.

Fix $t$ and let $n$ be sufficiently large. Let $C_{\mathbf{1}_t}$ be the code space appearing in the construction, let $T'$ be the associated symplectic space, let…

The real Clifford group and a symplectic group act on a given maximal-rank subspace, yielding the duality under consideration. Real Clifford–symplectic correspondence. In analogy w…

Let an orbit of the complex Clifford group be a finite set of pure quantum states, and call it a projective $t$-design when its moments up to degree $t$ agree with those of the uni…

Let $d$ be an odd prime power with $d\equiv3\pmod 4$, let $|\psi\rangle$ be the MUB-balanced state associated with the MUB-cycler $U_{G_0}$, and let the extended Clifford group act…

Let $U_{\mathcal{Z}}$ be the Zauner unitary associated with the Zauner symplectic matrix … A Weyl–Heisenberg SIC fiducial is a fiducial vector generating a symmetric informationall…

Let $d$ be the prime-power dimension under consideration, let $U_{G_0}$ be the MUB-cycler defined in the paper, and let $|\psi\rangle$ be the corresponding MUB-balanced state. The…

Let a MUB-balanced state be a quantum state for which the $d+1$ probability vectors obtained by projection onto the $d+1$ mutually unbiased bases are identical up to permutations o…