17 problems

Kong–Li–Liu's iSWAP optimality conjecture. For a general fixed connected graph, Kong, Li, and Liu conjectured that $\mathrm{iSWAP}$ maximises the Hermitian second-moment spectral g…

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

Kong–Li–Liu's ironed-gadget degeneracy conjecture. Every ironed gadget with KAK parameter $a=5/9$ has the same second-moment spectral gap as $\mathrm{iSWAP}$.

Linear-graph connection optimality conjecture. No other graph on $n$ qudits requires more connections to form an $\epsilon$-approximate $t$-design than the linear graph, which requ…

Consider random circuits formed by placing random gates on the edges of a connected interaction graph, and let $n$ denote the number of qudits. A $k$-design is an ensemble reproduc…

The 1D brickwork channel is defined by applying random two-qudit unitaries in alternating layers: $\Phi_{BR1}$ acts on $(1,2),(3,4),\dots$, $\Phi_{BR2}$ acts on $(2,3),(4,5),\dots$…

Let $d$ be the dimension of the unitary group and let $t$ denote the design order in a relation between $epsilon$-nets and $delta$-approximate $t$-designs on the space of unitary c…

Let $n$ be the number of qubits in the brickwork model, let $t$ denote the circuit depth or number of layers, and let…

Let $K_n$ be the complete graph on $n$ vertices, and consider a gadget associated with a family of 2-local gates having parameter $a=\frac{5}{9}$ in the notation of the paper. Let…

Let $n\geq 3$, let $\mathcal{G}$ be a fixed connected graph, and let $\mathcal{E}$ be a 2-local unitary circuit ensemble. Write…

Let $G_{\star}$ be a star graph with $n_{\star}$ vertices, local dimension $q$, and moment $k$, and let $\Delta(G_{\star},n_{\star},k)$ denote its spectral gap. Higher-moment star-…

For a star graph $G_{\star}$ with $n_{\star}$ vertices, let $\Delta(G_{\star},n_{\star},k)$ denote the spectral gap of its Hamiltonian at moment $k$. Low-moment star graph conjectu…

Consider the one-dimensional CQA ensemble on $n$ qudits with SU$(d)$ symmetry, and let $\epsilon>0$ denote the target approximation precision. CQA convergence conjecture. Based on…

Let $n$ qubits be equipped with a 2-local one-dimensional CQA random circuit, and let $\theta$ denote the circuit depth. For $\epsilon>0$, an ensemble is an $\epsilon$-approximate…

Optimal-scaling conjecture. With some work, it should be possible to obtain

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…

Linear-depth conjecture. For the general random quantum circuit model, a depth of order $O(nt)$ suffices to approximate $t$-designs.