Experimental test of Hardy’s paradox on a five-qubit quantum computer Soumya Das and Goutam Paul Cryptology and Security Research Unit, R. C. Bose Centre for Cryptology and Security, Indian Statistical Institute, Kolkata 700 108, India. Email: soumya06.das@gmail.com, goutam.paul@isical.ac.in We test Hardy’s paradox of non-locality experimentally on the IBM five-qubit quantum computer for the first time. The quantum circuit is constructed on superconducting qubits corresponding to the original Hardy’s test of non-locality. Results confirmed the theory that any non-maximally entangled state of two qubits violates Hardy’s Equations, whereas any maximally entangled state and product state of two qubits do not exhibit Hardy’s non-locality. We also point out the difficulties associated with the practical implementation of any Hardy’s paradox based quantum protocol and propose three performance measures for any two qubits of any quantum computer. I. INTRODUCTION In 1935, the Einstein-Podolsky-Rosen (EPR) Para- dox raised the question about the completeness of the quantum theory [1] and claimed that nature should be described by any local-realistic theory. In 1964, the non-local characteristics of quantum theory was demon- strated by Bell’s Theorem [2]. Since then, a significant number of experiments have been conducted favoring the correctness of Bell’s Theorem [3–17]. For this reason, Bell type inequalities are used to differentiate between quantum physics and classical physics. A comprehen- sive review of Bell’s Theorem including theoretical and experimental aspects can be found in [18]. In [19], the authors have demonstrated non-locality without using inequalities for three and four qubits. In 1992, through a thought experiment, Hardy constructed the test of local-realism without using inequalities for two qubits. This is called Hardy’s test [20, 21]. It is known as the “Best version of Bell’s Theorem” as indicated by Mermin [22]. This test provides a direct contradiction between the predictions of a quantum theory and Local Hidden Variable (LHV) theory. Several experiments have been performed to demon- strate Hardy’s paradox using polarization, energy-time and orbital angular momentum of photons, entangled qubits, classical light and two-level quantum states [23– 32]. The applications of Hardy’s paradox includes De- vice Independent Randomness [33], Device Independent Quantum Key Distribution [34] and Quantum Byzantine Agreement [35]. In the case of superconducting qubits, Clauser-Horne- Shimony-Holt (CHSH) inequality and Greenberger- Horne-Zeilinger (GHZ) test are already performed in IBM five-qubit quantum computer [36]. IBM has given access to its quantum computer that uses superconduct- ing qubits in the cloud and this opens a new door for testing of quantum phenomenons for researchers. In [37], the author has implemented some protocols in quantum error correction, quantum arithmetic, quantum graph theory and fault-tolerant quantum computation in IBM five-qubit quantum computer. In [38], the authors have tested the theoretical predictions of entropic un- certainty relation with quantum side information (EUR- QSI) in IBM five-qubit quantum computer. Compressed quantum computation [39], Leggett-Garg test [40], Quan- tum cheque [41] are also recently performed in IBM five- qubit quantum computer. Though Mermin inequalities have been tested exper- imentally using photons and ion traps [42, 43], subse- quently the authors of [44] have tested three, four and five qubits Mermin polynomials in IBM five-qubit quantum computer. We have already discussed the works related to the experimental verification of Hardy’s non-locality. However, none of them used any real quantum computer using superconducting qubits. This motivates us to test the Hardy’s paradox for two qubits in IBM five-qubit quantum computer. II. HARDY’S TEST OF NON-LOCALITY Hardy’s test of non-locality for two qubits involves two distant parties (may be space-like separated), Alice and Bob. A physical system consisting of two subsystems is shared between them. Alice and Bob can freely measure and observe the measurement results of their own sub- systems. Let us consider, Alice can perform the test of measurement on her own subsystem by choosing freely one of the two {+1, −1}-valued random variables A 1 and A 2 . Similarly, Bob can also choose freely one of the two {+1, −1}-valued random variables B 1 and B 2 for mea- suring the subsystem in his possession. Hardy’s test of non-locality starts with the following set of joint probability equations. P (+1, +1|A 1 ,B 1 ) =0, (1) P (+1, −1|A 2 ,B 1 ) =0, (2) P (−1, +1|A 1 ,B 2 ) =0, (3) P (+1, +1|A 2 ,B 2 )=  0 for LHV theory, q for non-locality, (4) where q> 0. Here P (x,y|A,B) denotes the joint prob- ability of obtaining outcomes x and y given that A and B were the experimental choices made. If an experiment arXiv:1712.04925v1 [quant-ph] 13 Dec 2017