Distribution of Non-Locality On Quantum Random Circuits

TL;DR

This work investigates how nonlocal resources distribute across states generated by quantum random circuits, contrasting universal (Clifford+$T$) and non-universal (Clifford) gate sets under realistic noise. By analyzing multipartite nonlocality via the Mermin and Svetlichny inequalities, along with entanglement and quantum-magic measures, it reveals that Clifford-only circuits concentrate resources in discrete values (GHZ-achievable), whereas universal gates span a broad range of nonlocality and entanglement in the ideal limit. The authors propose a nonlocality-based benchmarking protocol and validate it through experiments on IonQ and IQM QPUs, demonstrating its potential for device certification and cross-architecture comparisons. They also discuss foundational implications regarding nonlocality versus contextuality and emphasize that nonlocality alone may not fully capture quantum advantage in near-term devices. Overall, the paper provides a framework for assessing the richness of the reachable state space in QPUs and highlights practical calibration opportunities grounded in fundamental quantum correlations.

Abstract

In this work we explore how different types of resources are distributed among the states generated by quantum random circuits (QRC). We focus on multipartite non-locality, but we also analyze quantum correlations by appealing to different entanglement and non-classicality measures. We analyze the violation of Mermin and Svetlichny inequalities in order to get a glance at the distribution of nonlocality and genuine multipartite nonlocality. Next, we compare universal vs non-universal sets of gates, to gain insight into the problem of explaining quantum advantage. By comparing the results obtained with ideal (noiseless) vs noisy intermediate-scale quantum (NISQ) devices, we lay the basis of a certification protocol, which aims to quantify how robust is the resources distribution among the states that a given device can generate. We have implemented our non-locality-based benchmark on actual quantum processors with different architectures, in order to assess up to which point they are capable of reproducing the ideal results.

Figures (19)

• Figure 1: Schematic representation of IQM Garnet's connectivity, where the three qubits Svetlichny inequality is to be tested on qubits $1$, $2$, and $4$. Each arrow represents a possible spin measurement on the chosen qubits. There are two directions per qubit, described by blue or yellow arrows. Thus, there are eight correlation terms in total (see Eqs. \ref{['e:SvetlichnyA']} and \ref{['e:SvetlichnyB']} in Appendix \ref{['s:MerminAndSvetlichnyExplained']}), which are schematically depicted as eight orientation choices on the right box.
• Figure 2: Schematic representation of the random choice of gates on a five qubits circuit. We used the Amazon Braket SDK braket to develop a function that generates quantum random circuits (QRC). We first fix a set of elementary gates (as for example, those of the Clifford $+$T set). Then, on each layer of the circuit, we make a random selection of gates and apply them to randomly chosen qubits. In the schematic diagram, we show different types of one-qubit gates (boxes indicating H, T and S gates), together with two-qubit entangling gates (such as, for example, CNOT). After the action of each random circuit, the target qubits are left in an output state $\rho$ that depends on the circuit. Each circuit is equivalent to a random unitary operator $U_{\rho}$.
• Figure 3: Three-qubits construction of the Svetlichny inequality. From the prepared state $U_\rho$ (see Fig. \ref{['f:Circuit']}), local rotations $U_A,U_B,U_C$ and their primed settings define the eight tripartite correlators $E_i$ used to build the Svetlichny operator $O_3$ and evaluate $S_3=\langle O_3\rangle$. Each local rotation of a qubit with the subsequent measurement in the computational basis simulates a local spin measurement.
• Figure 4: Histograms of maximal violation of the Svetlichny inequality from three to five qubits, using $100{,}000$ randomly generated states. For the sake of completeness, we have also included the histogram corresponding to the violation of the CHSH inequality (two-qubit case). Different qubit numbers are indicated with colors. The vertical dashed lines denote the limit discarding hybrid local realism for each case. The percentage of states violating the inequality for each case is indicated in the top right. The universality of the Clifford $+$T set implies that the above random circuits should approximate those generated with unitary matrices distributed according to the Haar measure. We have obtained very similar results using both methods, which serves as a check of the correctness of our results (see Fig. \ref{['f:Histograms_Svetlichny']} in Appendix \ref{['s:Histograms']}).
• Figure 5: Histograms for the maximal violation degree of the Svetlichny inequality for states generated with random circuits using only Clifford gates from three to five qubits. For completeness, we have also included the histogram corresponding to the CHSH inequality. We display the relative weights of the values obtained with $100{,}000$ instances. Differently from the universal gate set case of Fig. \ref{['f:Histograms_condensed']}, the distribution of degrees of nonlocality is now concentrated in certain specific values. Notice that, in all cases, there exist states reaching the maximal violation value allowed by quantum theory. The existence of such states holds for an arbitrary number of qubits, given that the generalized GHZ states can be produced using only Clifford resources.