Projects related to the quantum certification

This Git repository lists projects related to our work around quantum certification.

Method of constructing a block design by Key and Moori

Paper abstract:
In the field of experimental mathematics, the programs used to obtain various results rarely follow good software engineering practices, making these results difficult to reproduce and evaluate. This article presents a formalization of a method for of combinatorial structures (called block systems) and a validation of their properties.

More information related to this paper can be found in the Designs part of this GitHub repository.

GT IQ 2019’s days

The annual meeting of GT IQ is organised in Besançon for 2019, more details can be found on GT-IQ’19 (fr)

Mermin Polynomials for Entanglement Evaluation in Grover’s algorithm and Quantum Fourier Transform

Paper abstract:
The entanglement of a quantum system can be valuated using Mermin polynomials. This gives us a means to study entanglement evolution during the execution of quantum al- gorithms. We first consider Grover’s quantum search algorithm, noticing that states during the algorithm are maximally entangled in the direction of a single constant state, which allows us to search for a single optimal Mermin operator and use it to evaluate entanglement through the whole execution of Grover’s algorithm. Then the Quantum Fourier Transform is also studied with Mermin polynomials. A different optimal Mermin operator is searched at each execution step, since in this case there is no single direction of evolution. The results for the Quantum Fourier Transform are compared to results from a previous study of entanglement with Cayley hyperdeterminant. All our computations can be replayed thanks to a structured and documented open-source code that we provide.

More information related to this paper can be found in the Mermin-eval part of this GitHub repository.

Entanglement and non-locality of four-qubits connected hypergraph states

Paper abstract:
We study entanglement and non-locality of connected four-qubit hypergraph states. One obtains the SLOCC classification from the known LU-orbits. We then consider Mermin’s polynomials and show that all four-qubit hypergaph states exhibit non-local behavior. “Finally, we implement some of the corresponding inequalities on the IBM Quantum Experience.

More information related to this paper can be found in the Mermin-hypergraph-states part of this GitHub repository.

Contextuality degree of quadrics in multi-qubit symplectic polar spaces

Paper abstract:
Quantum contextuality takes an important place amongst the concepts of quantum computing that bring an advantage over its classical counterpart. For a large class of contextuality proofs, aka. observable-based proofs of the Kochen-Specker Theorem, we formulate the contextuality property as the absence of solutions to a linear system and define for a contextual configuration its degree of contextuality. Then we explain why subgeometries of binary symplectic polar spaces are candidates for contextuality proofs. We report the results of a software that generates these subgeometries, decides their contextuality and computes their contextuality degree for some small symplectic polar spaces. We show that quadrics in the symplectic polar space Wn are contextual for n=3,4,5. The proofs we consider involve more contexts and observables than the smallest known proofs. This intermediate size property of those proofs is interesting for experimental tests, but could also be interesting in quantum game theory.

More information related to this paper can be found in the Magma-contextuality part of this GitHub repository.

Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank

Paper abstract:
We study certain physically-relevant subgeometries of binary symplectic polar spaces of small rank N , W(2N-1,2), when the points of these spaces are parametrized by canonical N -fold products of Pauli matrices and the associated identity matrix (i.e., N-qubit observables). Key characteristics of a subspace of such W(2N-1,2) are: the number of its negative lines, distribution of types of observables, character of the geometric hyperplane the subspace shares with the distinguished (non-singular) quadric of W(2N-1,2) and the structure of its Veldkamp space. W(3,2) features three negative lines of the same type and its W(1,2)’s are of five different types. W(5,2) is endowed with 90 negative lines of two types and its W(3,2)’s split into 13 types. 279 out of 480 W(3,2)’s with three negative lines are composite, i.e. they all originate from the two-qubit W(3,2) by selecting in the latter a geometric hyperplane and formally adding to each two-qubit observable, at the same position, the identity matrix if an observable lies on the hyperplane and the same Pauli matrix for any other observable. Further, given a W(3,2) and any of its geometric hyperplanes, there are other three W(3,2)’s possessing the same hyperplane. There is also a particular type of W(3,2)’s, a representative of which features a point each line through which is negative. Finally, W(7,2) is found to possess 1908 negative lines of five types and its W(5,2)’s fall into as many as 29 types. 1524 out of 1560 W(5,2)’s with 90 negative lines originate from the three-qubit W(5,2). Remarkably, the difference in the number of negative lines for any two distinct types of four-qubit W(5,2)’s is a multiple of four.

More information related to these papers can be found in the Magma-contextuality part of this GitHub repository.

Testing quantum contextuality of binary symplectic polar spaces on a Noisy Intermediate Scale Quantum Computer

Paper abstract:
The development of Noisy Intermediate Scale Quantum Computers (NISQC) provides for the Quantum Information community new tools to perform quantum experiences from an individual laptop. It facilitates interdisciplinary research in the sense that theoretical descriptions of properties of quantum physics can be translated to experiments easily implementable on a NISCQ. In this note I test large state-independent inequalities for quantum contextuality on finite geometric structures encoding the commutation relations of the generalized N-qubit Pauli group. The bounds predicted by Non-Contextual Hidden Variables theories are strongly violated in all conducted experiences.

More information related to this paper can be found in the Testing_contextuality part of this GitHub repository.

Computational studies of entanglement and quantum contextuality properties towards their formal verification

Thesis abstract:
Although current quantum computers are limited to the use of a few quantum bits, the foundations of quantum programing have been growing over the last 20 years. These foundations have been theorized as early as in the 80’s but the complexity of their implementation caused these leads to be out of reach until very recently. In this context, the objective of this thesis is to contribute to the adaptation of the methods of formal specification and deductive verification of classical programs to quantum programs. I thus present my contributions to the study of quantum properties with the end goal of formalizing them. I study in particular quantum entanglement and quantum contextuality. These properties allow to classify quantum states and protocols, and in particular to differentiate them from classical ones. My study of entanglement is based more specifically on the evolution of entanglement during two quantum algorithms: the Grover algorithm and the Quantum Fourier Transform. To quantify entanglement along those algorithms, I use Mermin’s polynomials, which have the advantage of being implementable in actual quantum computers. My study of contextuality, on the other hand, relies on finite geometries representing experiments, which are said to be contextual when no non-contextual classical theory can predict the results. These geometries are associated with the binary symplectic polar spaces. We study their structure, and eventually use this structure to get insights on quantum protocols using contextuality. The study of these properties led to interesting conjectures which we started to formalize in proof environments, such as Coq and Why3, but are left as perspective as these works have not reach a conclusion yet.

More information related to this paper can be found in the Computational_studies_of_entanglement_and_contextuality part of this GitHub repository.

Three-Qubit-Embedded Split Cayley Hexagon is Contextuality Sensitive

Paper abstract:
It is known that there are two non-equivalent embeddings of the split Cayley hexagon of order two into W(5, 2), the binary symplectic polar space of rank three, called classical and skew. Labelling the 63 points of W(5, 2) by the 63 canonical observables of the three-qubit Pauli group subject to the symplectic polarity induced by the (commutation relations between the elements of the) group, the two types of embedding are found to be quantum contextuality sensitive. In particular, we show that the complement of a classically- embedded hexagon is not contextual, whereas that of a skewly-embedded one is.

More information related to these papers can be found in the Magma-contextuality part of this GitHub repository.

Multi-qubit doilies: enumeration for all ranks and classification for ranks four and five

Paper abstract:
For $N \geq 2$, an $N$-qubit doily is a doily living in the $N$-qubit symplectic polar space. These doilies are related to operator-based proofs of quantum contextuality. Following and extending the strategy of Saniga et al. (Mathematics 9 (2021) 2272) that focused exclusively on three-qubit doilies, we first bring forth several formulas giving the number of both linear and quadratic doilies for any $N > 2$. Then we present an effective algorithm for the generation of all $N$-qubit doilies. Using this algorithm for $N=4$ and $N=5$, we provide a classification of $N$-qubit doilies in terms of types of observables they feature and number of negative lines they are endowed with. We also list several distinguished findings about $N$-qubit doilies that are absent in the three-qubit case, point out a couple of specific features exhibited by linear doilies and outline some prospective extensions of our approach.

The numerical results related to this paper can be found in the doilies part of this GitHub repository.

Phase sensitivity of entanglement in the Quantum Phase Estimation Algorithm

Paper abstract:
We study entanglement in the pre-QFT part of the Quantum Phase Estimation and the Quantum Counting algorithms. In particular we focus on the sensitivity of entanglement to the input value (the phase and the ratio of marked elements M/N) in some basic cases. One starts from numerical observations and deduce some general results in particular regarding the classes of entanglement.

More information related to this paper can be found in the qpea_and_qca part of this GitHub repository.

Implementing 2-qubit pseudo-telepathy games on noisy intermediate scale quantum computers

Paper abstract:
It is known that Mermin-Peres like proofs of quantum contextuality can furnish non- local games with a guaranteed quantum strategy, when classically no such guarantee can exist. This phenomenon, also called quantum pseudo-telepathy, has been studied in the case of the so-called Mermin Magic square game. In this paper we review in detail two different ways of implementing on a quantum computer such a game and propose a new Doily game based on the geometry of 2-qubit Pauli group. We show that the quantumness of these games are almost revealed when we play them on the IBM Quantum Experience, however the inherent noise in the available quantum machines prevents a full demonstration of the non-classical aspects.

More information related to this paper can be found in the quantum_game part of this GitHub repository.

Contextuality degree of multi-qubit configurations

All information related to this subject can be found in the contextualityDegree part of this GitHub repository.