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European Journal of Physics

A demonstration of contextuality using quantum computers

Amine Laghaout, Altay Dikme, Nicolas Reichel, Gunnar G. Bjork

Published Jun 17, 2022• 2 citations
Publisher

Abstract

Abstract Open-access, online quantum computers have shown significant improvements in the past decade. Although they still suffer from noise and scalability limitations, they do offer the possibility of experimenting with quantum circuits which would otherwise have required laboratory resources and prowesses beyond the reach of most students (and even researchers). In view of this, we revisit from the ground up the notion of contextuality and show that it can now be easily demonstrated on one of the IBM quantum computers. We showcase this with an implementation of the Peres–Mermin square which, despite the high error rates, manages to violate noncontextuality by almost 28 standard deviations.

DOI
10.1088/1361-6404/ac79e0

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Abstract Open-access, online quantum computers have shown significant improvements in the past decade. Although they still suffer from noise and scalability limitations, they do offer the possibility of experimenting with quantum circuits which would otherwise have required laboratory resources and prowesses beyond the reach of most students (and even researchers). In view of this, we revisit from the ground up the notion of contextuality and show that it can now be easily demonstrated on one of the IBM quantum computers. We showcase this with an implementation of the Peres–Mermin square which, despite the high error rates, manages to violate noncontextuality by almost 28 standard deviations.
We have implemented the six series of three commuting measurement of the Mermin-Peres magic square on an online, five qubit, quantum computer. The magic square tests if the measurements of the system can be described by physical realism (in the EPR sense) and simultaneously are non-contextual. We find that our measurement results violate any realistic and non-contextual model by almost 28 standard deviations. We also find that although the quantum computer we used for the measurements leaves much to be desired in producing accurate and reproducible results, the simplicity, the ease of re-running the measurement programs, and the user friendliness compensates for this fact.
Quantum systems show contextuality.More precisely, it is impossible to reproduce the quantum-mechanical predictions using a non-contextual realist model, i.e., a model where the outcome of one measurement is independent of the choice of compatible measurements performed in the measurement context.There has been several attempts to quantify the amount of contextuality for specific quantum systems, for example, in the number of rays needed in a KS proof, or the number of terms in certain inequalities, or in the violation, noise sensitivity, and other measures.This paper is about another approach: to use a simple contextual model that reproduces the quantum-mechanical contextual behaviour, but not necessarily all quantum predictions.The amount of contextuality can then be quantified in terms of additional resources needed as compared with a similar model without contextuality.In this case the contextual model needs to keep track of the context used, so the appropriate measure would be memory.Another way to view this is as a memory requirement to be able to reproduce quantum contextuality in a realist model.The model we will use can be viewed as an extension of Spekkens' toy model [Phys.Rev. A 75, 032110 (2007)], and the relation is studied in some detail.To reproduce the quantum predictions for the Peres-Mermin square, the memory requirement is more than one bit in addition to the memory used for the individual outcomes in the corresponding noncontextual model.
The question of a hidden variable interpretation of quantum contextuality in the Mermin-Peres square is considered. The Kochen-Specker theorem implies that quantum mechanics may be interpreted as a contextual hidden-variable theory. It is shown that such a hidden-variable description can be viewed as either contextual in the random variables mapping hidden states to observable outcomes or in the probability measure on the hidden state space. The latter view suggests that this apparent contextuality may be interpreted as a simple consequence of measurement disturbance, wherein the initial hidden state is altered through interaction with the measuring device, thereby giving rise to a possibly different final hidden-variable state from which the measurement outcome is obtained. In light of this observation, a less restrictive and, arguably, more reasonable definition of noncontextuality is suggested. To prove that such a description is possible, an explicit and, in this sense, noncontextual hidden-variable model is constructed, which reproduces all quantum theoretic predictions for the Mermin-Peres square. A critical analysis of some recent and proposed experimental tests of contextuality is also provided. Although the discussion is restricted to a four-dimensional Hilbert space, the approach and conclusions are expected to generalize to any Hilbert space.
The contextuality of quantum mechanics can be shown by the violation of inequalities based on measurements of well chosen observables. These inequalities have been designed separately for both discrete and continuous variables. Here we unify both strategies by introducing general conditions to demonstrate the contextuality of quantum mechanics from measurements of observables of arbitrary dimensions. Among the consequences of our results is the impossibility of having a maximal violation of contextuality in the Peres-Mermin scenario with discrete observables of odd dimensions. In addition, we show how to construct a large class of observables with a continuous spectrum enabling the realization of contextuality tests both in the Gaussian and non-Gaussian regimes.
A review is made of the field of contextuality in quantum mechanics. We study the historical emergence of the concept from philosophical and logical issues. We present and compare the main theoretical frameworks that have been derived. Finally, we focus on the complex task of establishing experimental tests of contextuality. Throughout this work, we try to show that the conceptualisation of contextuality has progressed through different complementary perspectives, before summoning them together to analyse the signification of contextuality experiments. Doing so, we argue that contextuality emerged as a discrete logical problem and developed into a quantifiable quantum resource.
Recently, various quantum computing and communication tasks have been implemented using IBM’s superconductivity-based quantum computers. Here, we show that the circuits used in most of those works were not optimized and obtain the corresponding optimized circuits. Optimized circuits implementable in IBM quantum computers are also obtained for a set of reversible benchmark circuits. With a clear example, it is shown that the reduction in circuit cost enhances the fidelity of the output state (with respect to the theoretically expected state in the absence of noise) as fewer gates and less circuit depth introduce fewer errors during evolution of the state. Further, considering Mermin inequality as an example, it is shown that the violation of the classical limit is enhanced when we use optimized circuits. Thus, the present approach can be used to identify a relatively weaker signature of quantumness and to establish quantum supremacy in a stronger manner.