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arXiv:2305.10225 · Mathematical Structures in Computer Science

New and improved bounds on the contextuality degree of multi-qubit configurations

Axel Müller, Метод Санига, Alain Giorgetti, Henri de Boutray, Frédéric Holweck

Published Apr 1, 2024• 5 citations
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Abstract

Abstract We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.

DOI
10.1017/s0960129524000057

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Abstract We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.
We present a use of a SAT solver to decide the quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al (J. Phys. A: Math. Theor. 55 475301, 2022), but also arrived at a bunch of new noteworthy results. This poster describes the approach, and presents the results for a number of subspaces of symplectic polar spaces whose rank ranges from two to seven, as well as the proofs that were found with the help of these results. Working group: LVP
Abstract We introduce and describe a new heuristic method for finding an upper bound on the degree of contextuality and the corresponding unsatisfied (i.e. non-reproducible by any NCHV model) part of a quantum contextual configuration with three-element contexts (i.e. lines) located in a multi-qubit symplectic polar space of order two. While the previously used method based on a SAT solver was limited to three qubits, this new method is much faster and more versatile, enabling us to also handle four- to six-qubit cases. The method is illustrated by the structure of unsatisfied parts of the associated contextual configurations represented by quadrics and full spaces; these entail well-known finite geometries like the split Cayley hexagon of order two, the dual polar space of rank three and order two as well as certain incidence graphs of binary projective spaces of small dimensions.
In the intriguing realm of quantum physics, the concept of contextuality raises interesting questions about the nature of quantum systems. Since the commutation relation in the N-qubits Pauli group corresponds to the colinearity relation in the symplectic polar space W(2N − 1, 2), we study subgeometries of this space in order to discover new contextual quantum configurations. We have developed algorithms and a corresponding software in C language, that can effectively determine quantum contextuality and quantify it, by charting these symplectic polar spaces of various ranks. This tool allowed us to uncover new insights that go beyond the achievements of earlier work, such as that of de Boutray et al (J. Phys. A: Math. Theor. 55 475301, 2022). Our study spans across these geometric structures of ranks from two to seven, offering significant findings.In my talk I will explicitly illustrate the concept of quantum contextuality on certain wellknown finite-geometric configurations living in W(5,2), like the doily (and its two-spread), an elliptic quadric (and its spread of lines) and a hyperbolic quadric (and its associated Heawoodgraph); I will also consider configurations whose contexts are all the subspaces with a given dimension k < N.
The contextuality degree of a set of Pauli contexts is the least number of contexts that every noncontextual ±1\pm1±1 assignment violates. For the four-qubit hyperbolic quadric Q+(7,2)\mathcal{Q}^{+}(7,2)Q+(7,2), with 135135135 points and 157515751575 lines as contexts, Muller et al. conjectured from heuristic searches that the degree is 315315315 and that the violated contexts of every optimal assignment form a dual polar space DW(5,2)DW(5,2)DW(5,2). We prove both conjectures. The lower bound has a short proof by hand: every doily (GQ(2,2)GQ(2,2)GQ(2,2)) of the quadric contains ten Peres-Mermin grids covering each of its lines four times, so every assignment violates at least three of its fifteen lines, and averaging over the doilies gives 1575/5=3151575/5=3151575/5=315. The equality case is local: an assignment is optimal exactly when it violates one ruling of one grid in every doily. Using this as a SAT encoding, we show that there are exactly 240240240 optimal violated sets, one orbit of O+(8,2)\mathrm{O}^+(8,2)O+(8,2), each isomorphic to DW(5,2)DW(5,2)DW(5,2). The same argument gives the first lower bounds ⌈L/5⌉\lceil L/5\rceil⌈L/5⌉ for all quadrics with LLL lines in N≥4N\ge4N≥4 qubits. The degree theorem is kernel-checked in Lean 4, and the completeness of the enumeration is certified by a DRAT proof checked by the verified checker cake_lpr.
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 W n 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.
For N ≥ 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 [SdBHG21] 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.