We analyse the role of the Extended Clifford group in classifying the spectra of phase point operators within the framework laid out by Gibbons et al for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of ${\rm ESL}(2, \mathbb{F}_N)$.
Farinholt gives a characterization of Clifford operators for qudits; d both odd and even. In this comment it is shown that the necessary gates for the construction of Clifford operators; N both odd and even, are obtained directly from operations that appear in SU(N)1. A witness for W3 states in SU(2)1 is discussed. See e.g. [1-4].
The Clifford group is the quotient of the normalizer of the Weyl-Heisenberg group in dimension $d$ by its centre. We prove that when $d$ is not prime the Clifford group is not a group unitary $2$-design. Furthermore, we prove that the multipartite Clifford group is not a group unitary 2-design except for the known cases wherein the local Hilbert space dimensions are a constant prime number. We also clarify the structure of projective group unitary $2$-designs. We show that the adjoint action induced by a group unitary $2$-design decomposes into exactly two irreducible components; moreover, a g…
We propose a normal form for single-qudit gates composed of Clifford and $T$-gates for qudits of odd prime dimension $p\geq 5$. We prove that any single-qudit Clifford+$T$ operator can be re-expressed in this normal form in polynomial time. We also provide strong numerical evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an algorithm for exact synthesis of any single-qudit Clifford+$T$ operator with minimal $T$-count.
We introduce a family of ZX-calculi which axiomatise the stabiliser fragment of quantum theory in odd prime dimensions. These calculi recover many of the nice features of the qubit ZX-calculus which were lost in previous proposals for higher-dimensional systems. We then prove that these calculi are complete, i.e. provide a set of rewrite rules which can be used to prove any equality of stabiliser quantum operations. Adding a discard construction, we obtain a calculus complete for mixed state stabiliser quantum mechanics in odd prime dimensions, and this furthermore gives a complete axiomatisat…
How Clifford algebra helps understand second quantized quarks and leptons and corresponding vector and scalar boson fields, {\it opening a new step beyond the standard model}
We study locality preserving automorphisms of operator algebras on $D$-dimensional uniform lattices of prime $p$-dimensional qudits (QCA), specializing in those that are translation invariant (TI) and map every prime $p$-dimensional Pauli matrix to a tensor product of Pauli matrices (Clifford). We associate antihermitian forms of unit determinant over Laurent polynomial rings to TI Clifford QCA with lattice boundaries, and prove that the form determines the QCA up to Clifford circuits and shifts (trivial). It follows that every 2D TI Clifford QCA is trivial since the antihermitian form in this…
Entropy cones for SU(N)1 Chern-Simons theory are discussed. It is shown that stabilizer states can be constructed from topological operators in SU(N)1 for N odd prime, but not for SU(N)K; K >= 2. This implies that the topological entropy cone is properly contained in the stabilizer entropy cone for SU(N)K; K >= 2.
Graph states and hypergraph states can be constructed from products of basic operations that appear in SU(N)1. The level-rank dual of a theorem of Salton, Swingle, and Walter implies that these operations can be prepared topologically in the n-torus Hilbert space of Chern-Simons theory for N neq 5 mod 4. For SU(N)1, N = 5 mod 4, only stabilizer states can be prepared on the n-torus Hilbert space, which restricts the construction to graph states.
Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary $t$-designs with $t\geq3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension~4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, t…
