On the Veldkamp Space of GQ(4, 2)

The Veldkamp space, in the sense of Buekenhout and Cohen, of the generalized quadrangle GQ(4, 2) is shown not to be a (partial) linear space by simply giving several examples of Veldkamp lines (V-lines) having two or even three Veldkamp points (V-points) in common. Alongside the ordinary V-lines of size five, one also finds V-lines of cardinality three and two. There, however, exists a subspace of the Veldkamp space isomorphic to PG(3, 4) having 45 perps and 40 plane ovoids as its 85 V-points, with its 357 V-lines being of four distinct types. A V-line of the first type consists of five perps on a common line (altogether 27 of them), the second type features three perps and two ovoids sharing a tricentric triad (240 members), whilst the third and fourth type each comprises a perp and four ovoids in the rosette centered at the (common) center of the perp (90). It is also pointed out that 160 non-plane ovoids (tripods) fall into two distinct orbits -- of sizes 40 and 120 -- with respect to the stabilizer group of a copy of GQ(2, 2); a tripod of the first/second orbit sharing with the GQ(2, 2) a tricentric/unicentric triad, respectively. Finally, three remarkable subconfigurations of V-lines represented by fans of ovoids through a fixed ovoid are examined in some detail.Comment: 6 pages, 7 figures; v2 - slightly polished, subsection on fans of ovoids and three figures adde

Supervised learning with quantum enhanced feature spaces

Machine learning and quantum computing are two technologies each with the potential for altering how computation is performed to address previously untenable problems. Kernel methods for machine learning are ubiquitous for pattern recognition, with support vector machines (SVMs) being the most well-known method for classification problems. However, there are limitations to the successful solution to such problems when the feature space becomes large, and the kernel functions become computationally expensive to estimate. A core element to computational speed-ups afforded by quantum algorithms is the exploitation of an exponentially large quantum state space through controllable entanglement and interference. Here, we propose and experimentally implement two novel methods on a superconducting processor. Both methods represent the feature space of a classification problem by a quantum state, taking advantage of the large dimensionality of quantum Hilbert space to obtain an enhanced solution. One method, the quantum variational classifier builds on [1,2] and operates through using a variational quantum circuit to classify a training set in direct analogy to conventional SVMs. In the second, a quantum kernel estimator, we estimate the kernel function and optimize the classifier directly. The two methods present a new class of tools for exploring the applications of noisy intermediate scale quantum computers [3] to machine learning.Comment: Fixed typos, added figures and discussion about quantum error mitigatio

Mermin's Pentagram as an Ovoid of PG(3,2)

Mermin's pentagram, a specific set of ten three-qubit observables arranged in quadruples of pairwise commuting ones into five edges of a pentagram and used to provide a very simple proof of the Kochen-Specker theorem, is shown to be isomorphic to an ovoid (elliptic quadric) of the three-dimensional projective space of order two, PG(3,2). This demonstration employs properties of the real three-qubit Pauli group embodied in the geometry of the symplectic polar space W(5,2) and rests on the facts that: 1) the four observables/operators on any of the five edges of the pentagram can be viewed as points of an affine plane of order two, 2) all the ten observables lie on a hyperbolic quadric of the five-dimensional projective space of order two, PG(5,2), and 3) that the points of this quadric are in a well-known bijective correspondence with the lines of PG(3,2).Comment: 5 pages, 4 figure

Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits

Given a (2N - 1)-dimensional projective space over GF(2), PG(2N - 1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG(N - 1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG(N - 1, 4). Under such mapping, a non-degenerate quadric surface of the former space has for its image a non-singular Hermitian variety in the latter space, this quadric being {\it hyperbolic} or {\it elliptic} in dependence on N being {\it even} or {\it odd}, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric operators into a new perspective. The N=4 case is taken to illustrate the issue.Comment: 3 pages, no figures/tables; V2 - short introductory paragraph added; V3 - to appear in Int. J. Mod. Phys.

Observation of secondary instability of 2/1 magnetic island in compass high density limit plasmas

Density limit disruptions (DLDs) have been observed in tokamak plasmas when high density regimes are explored. The DLDs are harmless in small size tokamaks like COMPASS,larger tokamaks like JET try to avoid them and they are extremely undesirable in ITER sizetokamaks due to the severe structural damages they can cause. It is very important to understand the dynamics of the DLDs so that better strategies to ameliorate or avoid them can bedeveloped. In this work, following detection in JET [1] of a secondary instability (SI) to thewell-known m/n = 2/1 MHD mode (where m and n are the poloidal and toroidal mode numbers, respectively) in the precursor of DLD, we analyse the evolution of the 2/1 magnetic islandin COMPASS DLD to look for the presence of this SI just close to the onset of energy quenchphase of the disruption. The presence of this SI to the magnetic island was associated with theoccurrence of minor disruptions preceding the major disruption and with the major disruptionitself in [1]. The coherence observed between the perturbations caused by the SI in the magneticpoloidal flux and in the electron temperature was very high (above 0.9), allowing to determinethat the SI perturbations came from the same position as the magnetic island. In the work presented here, only the perturbations in the magnetic poloidal flux are analysed since at the time ofthe experiments in COMPASS, no diagnostics was operational for measuring the time evolutionof the electron temperature with high time rate.Nonlinear MHD numerical simulations have also shown that island deformation during itsrapid growth can lead to the secondary magnetic island formation [2]. A recent review [3] ofthe theory of current sheet formation that leads to magnetic reconnection discusses the role ofplasmoids during magnetic island evolution. Since the validity ranges of the mentioned theoretical works are not directly comparable to the experimental conditions, one cannot claimwith certainty that the SI observed in JET [1] and in COMPASS disruptions (reported here)are the same as observed in those numerical works [2, 3]. However, there are some qualitative43rd EPS Conference on Plasma Physics P5.003similarities between them.The main COMPASS [4] diagnostics used for the analysis in the present work, are the threetoroidally separated arrays (A at 32.5◦, B at 212.5◦and C at 257.5◦from the vessel axis) ofMirnov coils (MCs), each with 24 MCs located poloidally. The MC arrays A and C, toroidallyseparated by 135◦, measure the change in poloidal magnetic flux, dBp/dt. The MC array B,toroidally separated by 180◦to the array A, measures the poloidal magnetic field, Bp. Thesemagnetic sensors have good responsivity to high frequency (up to 1 MHz)

Weak mutually unbiased bases

Quantum systems with variables in ${\mathbb Z}(d)$ are considered. The properties of lines in the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space of these systems, are studied. Weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases, is equal to $d^{-1/2}$ or alternatively to one of the $d_i^{-1/2},0$ (where $d_i$ is a divisor of $d$ apart from $d,1$). They are designed for the geometry of the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space, in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin. In the special case of prime $d$, there are no divisors of $d$ apart from $1,d$ and the weak mutually unbiased bases are mutually unbiased bases