We test the quantumness of IBM's quantum computer IBM Quantum System One in Ehningen, Germany. We generate generalised n-qubit GHZ states and measure Bell inequalities to investigate the n-party entanglement of the GHZ states. The implemented Bell inequalities are derived from non-adaptive measurement-based quantum computation (NMQC), a type of quantum computing that links the successful computation of a non-linear function to the violation of a multipartite Bell-inequality. The goal is to compute a multivariate Boolean function that clearly differentiates non-local correlations from local hidden variables (LHVs). Since it has been shown that LHVs can only compute linear functions, whereas quantum correlations are capable of outputting every possible Boolean function it thus serves as an indicator of multipartite entanglement. Here, we compute various non-linear functions with NMQC on IBM's quantum computer IBM Quantum System One and thereby demonstrate that the presented method can be used to characterize quantum devices. We find a violation for a maximum of seven qubits and compare our results to an existing implementation of NMQC using photons.
We test the quantumness of IBM's quantum computer IBM Quantum System One in Ehningen, Germany. We generate generalised n-qubit GHZ states and measure Bell inequalities to investigate the n-party entanglement of the GHZ states. The implemented Bell inequalities are derived from non-adaptive measurement-based quantum computation (NMQC), a type of quantum computing that links the successful computation of a non-linear function to the violation of a multipartite Bell-inequality. The goal is to compute a multivariate Boolean function that clearly differentiates non-local correlations from local hidden variables (LHVs). Since it has been shown that LHVs can only compute linear functions, whereas quantum correlations are capable of outputting every possible Boolean function it thus serves as an indicator of multipartite entanglement. Here, we compute various non-linear functions with NMQC on IBM's quantum computer IBM Quantum System One and thereby demonstrate that the presented method can be used to characterize quantum devices. We find a violation for a maximum of seven qubits and compare our results to an existing implementation of NMQC using photons.
Quantum correlations exhibit behaviour that cannot be resolved with a local hidden variable picture of the world. In quantum information, they are also used as resources for information processing tasks, such as measurement-based quantum computation (MQC). In MQC, universal quantum computation can be achieved via adaptive measurements on a suitable entangled resource state. In this paper, we look at a version of MQC in which we remove the adaptivity of measurements and aim to understand what computational abilities remain in the resource. We show that there are explicit connections between this model of computation and the question of non-classicality in quantum correlations. We demonstrate this by focusing on deterministic computation of Boolean functions, in which natural generalizations of the Greenberger-Horne-Zeilinger paradox emerge; we then explore probabilistic computation via, which multipartite Bell inequalities can be defined. We use this correspondence to define families of multi-party Bell inequalities, which we show to have a number of interesting contrasting properties.
Abstract Non-locality is not only one of the most prominent quantum features but can also serve as a resource for various information-theoretical tasks. Analysing it from an information-theoretical perspective has linked it to applications such as non-adaptive measurement-based quantum computing (NMQC). In this type of quantum computing the goal is to output a multivariate function. The success of such a computation can be related to the violation of a generalised Bell inequality. So far, the investigation of binary NMQC with qubits has shown that quantum correlations can compute all Boolean functions using at most 2 n − 1 qubits, whereas local hidden variables (LHVs) are restricted to linear functions. Here, we extend these results to NMQC with qutrits and prove that quantum correlations enable the computation of all three-valued logic functions using the generalised qutrit Greenberger–Horne–Zeilinger (GHZ) state as a resource and at most 3 n − 1 qutrits. This yields a corresponding generalised GHZ type paradox for any three-valued logic function that LHVs cannot compute. We give an example for an n-variate function that can be computed with only n + 1 qutrits, which leads to convenient generalised qutrit Bell inequalities whose quantum bound is maximal. Finally, we prove that not all functions can be computed efficiently with qutrit NMQC by presenting a counterexample.
Quantum correlations exhibit behaviour that cannot be resolved with a local hidden variable picture of the world. In quantum information, they are also used as resources for information processing tasks, such as measurement-based quantum computation (MQC). In MQC, universal quantum computation can be achieved via adaptive measurements on a suitable entangled resource state. In this paper, we look at a version of MQC in which we remove the adaptivity of measurements and aim to understand what computational abilities remain in the resource. We show that there are explicit connections between this model of computation and the question of non-classicality in quantum correlations. We demonstrate this by focusing on deterministic computation of Boolean functions, in which natural generalizations of the Greenberger–Horne–Zeilinger paradox emerge; we then explore probabilistic computation via, which multipartite Bell inequalities can be defined. We use this correspondence to define families of multi-party Bell inequalities, which we show to have a number of interesting contrasting properties.
Measurement-based quantum computation (or the one-way quantum computation) is a model in which a series of adaptive single qubit measurements upon an entangled resource state can produce classical output data equivalent to an arbitrary quantum logic circuit. Loosely speaking, the correlations in these measurements can be considered the resource which allows the computation to be performed. Bell inequalities demonstrate that quantum mechanics permits the outcomes of sets of measurements to be correlated in ways incompatible with the predictions of any local hidden variable theory, including classical physics. In the Bell inequality, and multi-party generalisations, a set of single-qubit measurements are performed, on spatially separated systems. Correlations in these measurements can act as a signature that no local hidden variable description of the experiment is possible. At first sight, we see superficial similarities between these settings. In both cases, non-classical correlations play a central role. However, there are also some key differences. Most importantly, measurement-based quantum computing requires adaptive measurements, at odds with the spatial-separated nature of Bell inequality experiments. Nevertheless some striking connections have been reported [1], [2] particularly between GHZ-type paradoxes and deterministic MBQC computations, and between CHSH-type Bell inequalities and non-deterministic MBQC correlations [1], [3]. In my talk I will survey these works and some recent results [3], [4] from my group at UCL, in which a number of connections between multi-party Bell inequalities and MBQC are explored. [1] J. Anders and D.E. Browne, "The Computational Power of Correlations", Physical Review Letters 102, 050502 (2009) [2] R. Raussendorf, "Quantum computation, discreteness, and contextuality", arxiv:0907.5449 [3] M. J. Hoban, E.T. Campbell, K. Loukopolous, D.E. Browne, "Non-adaptive measurement-based quantum computation and multi-party Bell inequalties", in preparation. [4] M. J. Hoban and D.E. Browne, in preparation.
This work provides quantitative tests of the extent of violation of two inequalities applicable to qubits coupled into Bell states, using IBM's publicly accessible quantum computers. Violations of the inequalities are well established. Our purpose is not to test the inequalities, but rather to determine how well quantum mechanical predictions can be reproduced on quantum computers, given their current fault rates. We present results for the spin projections of two entangled qubits, along three axes A, B, and C, with a fixed angle θ between A and B and a range of angles θ' between B and C. For any classical object that can be characterized by three observables with two possible values, inequalities govern relationships among the probabilities of outcomes for the observables, taken pairwise. From set theory, these inequalities must be satisfied by all such classical objects; but quantum systems may violate the inequalities. We have detected clear-cut violations of one inequality in runs on IBM's publicly accessible quantum computers. The Clauser-Horne-Shimony-Holt (CHSH) inequality governs a linear combination S of expectation values of products of spin projections, taken pairwise. Finding S > 2 rules out local, hidden variable theories for entangled quantum systems. We obtained values of S greater than 2 in our runs prior to error mitigation. To reduce the quantitative errors, we used a modification of the error-mitigation procedure in the IBM documentation. We prepared a pair of qubits in the state |00〉, found the probabilities to observe the states |00〉, |01〉, |10〉, and |11〉 in multiple runs, and used that information to construct the first column of an error matrix M. We repeated this procedure for states prepared as |01〉, |10〉, and |11〉 to construct the full matrix M, whose inverse is the filtering matrix. After applying filtering matrices to our averaged outcomes, we have found good quantitative agreement between the quantum computer output and the quantum mechanical predictions for the extent of violation of both inequalities as functions of θ'.
The predictions of quantum mechanics cannot be resolved with a completely classical view of the world. In particular, the statistics of space-like separated \nmeasurements on entangled quantum systems violate a Bell inequality [Bell1964]. \nWe put forward a computational perspective on a broad class of Bell tests that study correlators, or the statistics of joint measurement outcomes. We associate \nparticular maps, or functions to particular theories. The violation of a Bell \ninequality then implies the ability to perform some functions, or computations \nthat classical, or more generally, local hidden variable (LHV) theories cannot. \nWe derive an in�finite class of Bell inequalities that establish a link to so-called "non-local games" [Cleve2004]. We then make the connection between \nRaussendorf and Briegel's formulation of Measurement-based Quantum Computing (MBQC) [Raussendorf2001], and these non-local games. Not only can we show that a quantum violation implies a computational advantage in this model, \nwe show that adaptive measurements are required to perform all quantum computations. \nFinally, we explore post-selection of data in Bell tests from both a practical and \nconceptual point-of-view, with particular consideration to so-called "loopholes". \nLoopholes allow LHV theories to simulate quantum correlations through post-selection. \nWe give a computational description of how loopholes can emerge in diff�erent post-selection scenarios. This motivates us to �find a form of post-selection \nthat does not lead to loopholes. Central again to this discussion is the \ndescription of LHV theories in terms of computations. \nInterestingly, quantum correlators can be made more "non-classical" with this \nloophole-free post-selection. This method of post-selection also can simulate \ninformation processing tasks, such as MBQC, that have time-like separated components. \nThis opens up new avenues for the study of time-like tasks studied \nwithin the space-like separated scenario of the Bell test.