Mermin's Inequalities of Multiple qubits with Orthogonal Measurements on IBM Q 53-qubit system

Entanglement properties of IBM Q 53 qubit quantum computer are carefully examined with the noisy intermediate-scale quantum (NISQ) technology. We study GHZ-like states with multiple qubits (N=2 to N=7) on IBM Rochester and compare their maximal violation values of Mermin polynomials with analytic results. A rule of N-qubits orthogonal measurements is taken to further justify the entanglement less than maximal values of local realism (LR). The orthogonality of measurements is another reliable criterion for entanglement except the maximal values of LR. Our results indicate that the entanglement of IBM 53-qubits is reasonably good when N<= 4 while for the longer entangle chains the entanglement is only valid for some special connectivity.


Introduction
Entanglement is a very unique feature of the quantum sciences and cannot be observed in the classical world. Coherence is another general property of waves and describes the correlation between the constituent parts. Both entanglement and coherence are the fundamental elements of the quantum world.
The Bell inequality 1 defines a boundary between quantum non-locality and local realism (LR) for double quantum states and Mermin 2 extended the results into an entanglement of a higher number of quantum states. Mermin's inequalities 2 is an excellent test for the entanglement property of a state for checking whether it violates LR. Quantum correlation of measurements for an entangled system always exist even the sub-systems of entangled states are physically separated far away. This entanglement provides a possibility to study the information of all entangled states simultaneously and gives a superior advantage of quantum computation than classical computation 3 .
There are various examples about the second quantum revolution with quantum computers 4 . The violation of Bell inequalities has been verified in atomic physics experiments 5,6 . Daniel Alsina et al. 7

has reported
Mermin's inequalities 2 on IBM 5-qubit quantum computer. The search algorithm for highly entangled states of multiple qubits have also been discussed 8,9 . Recently, IBM 53-qubit quantum computer (IBM Rochester) 10 is online for all kinds of experiments. However, the coherence time of their qubits can only support the realization of a noise intermediated state quantum (NISQ) 11 environment and very few things can be done in reality, so we should try to reduce the environmental noise 12 . Alternative way is to emulating quantum computers on classical platforms 13 . Until now only some simple examples were demonstrated on the IBM quantum computer system [14][15][16] . However, in near-term, algorithms like the quantum adiabatic optimization algorithms 17 and the variational quantum eigensolvers 18 have some chance of demonstrating quantum advantage 19 for a N-qubits system. In addition, there are also other applications like hash preimage attacks 20 and modeling viral diffusion 21 . Google Sycamore achieved quantum supremacy 22 and took only 200 seconds to perform certain problems that the Summit supercomputer need 10,000 years to compute.
The full understanding of the entanglement properties of a large number of qubits within state-of-art quantum computers, e.g., IBM Rochester, becomes critical for the real applications. Previous studies between quantum non-locality and classical LR are mainly focused on the static and theoretical situations for a small number of qubits, in particular for pure quantum states as Bell and Greenberger-Horne-Zeilinger (GHZ)-type states 23 . Only recently had experimental data been reported but even that were still not a fully dynamic measurement due to the limitation of coherence time and the small number of mixed entangled states 7,24,25 . Therefore, a full understanding of the entanglements and correlations between large number of qubits are important for a universal quantum computer. Here we propose to use orthogonal measurements of Mermin's inequalities 2 to study the influence of phase angles and the correlations of orthogonal measurements in GHZ-like states 7 . Also a longer chains of qubits are executed on IBM Q 53qubit quantum computer 10 with the nearest-neighboring connectivity. The entanglement and correlations of Mermin's polynomials from 2 to 7 qubits are carefully examined. The orthogonal measurements give a consistent result with the analytic results under NISQ.

Theory
There are some indicators to evaluate the performance of a quantum computer, for example, negativity 26 , quantum volume 27 , or Mermin's inequalities 2 . However, the major difference between classical physics and quantum physics mainly is still mainly the entanglement property. A 2-qubits system can use either Bell's inequality 1 or Mermin's inequalities 2 to check whether the system violates LR. For a larger number of qubits, it needs to study Mermin's polynominals 28 to understand the entanglement of N-qubit and their quantum subsystems. The goal of experimental test on IBM Rochester is to characterize the entanglement behavior of N-qubits, and GHZ-like state 7 can assure the Mermin's polynomials will be maximum. Thus, here a GHZ-like state with appropriate initial phases 29 is used to probe the entanglement of N-qubits on the IBM 53-qubits machine.
Operator X and operator Y represent Pauli-X gate and Pauli-Y gate respectively.
The GHZ-like state 7 is  is the phase angle and the expectation value of Mermin's polynomials are Please see supplement A for the details of the derivations for ⟨ ⟩ and ⟨ ′ ⟩.
The upper limit of LR is 2 For brevity, we normalize all ⟨ ⟩ and ⟨ ′ ⟩ values by ⟨ ⟩ and thus The maximal values of ̃ and ′ with the appropriate phase angle are listed in Table I (7), it is important to note there is a sum rule for ̃2 + ′ 2 = 2 −1 for quantum entanglement among N-qubits system. For ̃ and ′ , the orthogonal measurements within an entangled system are strongly correlated, this unique quantum sum rule is very different from the independent and isotropy nature of a classical system. Fig. 1: The connectivity of qubits within hexagonal lattice structure of IBM Q 53-qubits quantum computer (IBM Rochester).

Method
To study the entanglement of N-qubits, the adjacent qubits within the hexagonal structure of IBM Rochester are studied and all combinatory of N-qubits which are analyzed in this article are listed in supplement B. The connecting structure of IBM Rochester is shown in Fig. 1.
A single qubit initialized with |0⟩ state becomes the superposition state  To measure the output of the quantum circuit, H gate needs to be included before a measurement along σx.
To measure σy, both + and H gates are needed before measurement. The operation of + gate equates to a rotation of the state around the Z-axis with the angle −π/2, and this changes σx to σy. For example, XXYYY measurement for the 5-qubits is shown in Fig. 2(b). Here X and Y gates are Pauli-X and Pauli-Y gate respectively.

Experiment results
Combinatory qubits with different qubit-connectivity within the hexagonal structure of the IBM Rochester are measured as Mermin's polynomials up to a 7-qubits connectivity. As shown in Fig. 3, the results from  The red line is the N-qubits LR of the Mermin's inequalities and the measurement values are arranged in a descending order from left to right. We choose the initial phase angle as shown in Table I (6) and (7)). Intuitively for a classical system, the results of ⟨ ⟩ and ⟨ ′ ⟩ should be independent and isotropic. It can be observed that from Fig. 4(a), the orthogonal measurements of N-qubits for N ≤ 4, the data points are more or less along the ⟨ ⟩ axis even though the experimental values do not exceed LR. However, for a classical results, ⟨ ⟩ and ⟨ ′ ⟩ should be independent measurements and the outcome cannot be along the ⟨ ⟩ axis with ⟨ ′ ⟩~0. The data for 2, 3, and 4 qubits, almost all combination of qubits are close to the ⟨ ⟩ axis, even though data fall within the LR limit. This indicates the orthogonality of ⟨ ⟩ and ⟨ ′ ⟩ are reasonably good from the orthogonal criterion point of view. Therefore, the N-qubits for N ≤ 4 can be concluded as an entangle system but with a reduced entangled strength arising due to the variation of amplitudes of the superposition states of NISQ (see supplement C).
For Fig. 4(b), measurements for the 5, 6 and 7 qubits obviously deviate from the ⟨ ⟩ axis and quite symmetrically distribute along the diagonal direction of LR square. Only a handful of data outside the classical RL limit is observed. From the measurements of larger number of qubits, the probability of orthogonal measurements is almost the same even though the initial phase angle of the maximal value along the ⟨ ⟩ axis is applied. For N-qubits with N ≥ 5, ⟨ ⟩ and ⟨ ′ ⟩ are quite independent and isotropic as classical expectations and the data scatter along the diagonal direction. This indicates that for the N ≥ 5 qubits system, entanglement within the NISQ is largely destroyed. Therefore, we can conclude entanglement for large number of qubits in the IBM Rochester are not robust and might easily drop.
However, the IBM Rochester did show excellent entanglement properties for the N≤4 systems, but the entangled strength depends on the environment noise and varies with the connectivity of the qubits. For larger number of qubits, i.e., N≥5, the entanglement of N-qubits is not stable and the errors and standard variation become large probably due to the limited coherence time of NISQ. It can be concluded that for small number of qubits, i.e., 2, 3, and 4, entanglement is fine; but for large number of qubits, the results of entanglement are not robust. qubits from different connectivity of IBM 53-qubit Rochester system.

Discussion and conclusion
In summary an easy orthogonal measurement test between ⟨ ⟩ and ⟨ ′ ⟩ is proposed to examine the entanglement among the N-qubits when the maximal values of the Mermin's polynomials are within LR.
We have used GHZ-like states to study the entangled pairs on the IBM Rochester 53 quits quantum computer and most of the pairs we have studied either had the orthogonal properties between ⟨ ⟩ and ⟨ ′ ⟩ or violated LR of Mermin's inequalities Therefore, two qubits are indeed entangled under NISQ and the IBM Rochester quantum computer performs reasonably well with 2-qubit entanglement. The entanglement is fine even up to the 4-qubits but with a reduced entangled strength from noise environment. While for the N ≥ 5 qubits, only very few cases can be observed as entangled systems and also the entanglement strength fluctuates heavily for different combination of connectivity. Therefore, we can conclude that the IBM Rochester 53 qubits system is good for problems with usage of small entangled qubits under the NISQ environments. For large number of qubits, only some particular qubit connectivity can still remain robust in noise environments.

DATA AVAILABILITY
All data supporting the findings of this study are available from the authors upon request.

Conflict of Interest Statement
The authors declare no competing interests.
Correspondence and requests for materials should be addressed to C.-R. C.

Supplement C
Here the relation between the entangle strength and the radii in the ⟨ 2 ⟩ − ⟨ 2 ′ ⟩ plane are analytically calculated for the 2-qubit case. Two qubits states with an entangled parameter θ is defined as below: The entangled parameter θ represents the influence from noise environment. For θ = 4 , the state gets which is a GHZ-like state and minimal values zero is at either θ = 0 or 2 , which is the state without superposition.

Figure 1:
The connectivity of qubits within hexagonal lattice structure of IBM Q 53-qubits quantum computer (IBM Rochester).