Probing many-body Bell correlation depth with superconducting qubits

Abstract

Quantum nonlocality describes a stronger form of quantum correlation than that of entanglement. It refutes Einstein's belief of local realism and is among the most distinctive and enigmatic features of quantum mechanics. It is a crucial resource for achieving quantum advantages in a variety of practical applications, ranging from cryptography and certified random number generation via self-testing to machine learning. Nevertheless, the detection of nonlocality, especially in quantum many-body systems, is notoriously challenging. Here, we report an experimental certification of genuine multipartite Bell correlations, which signal nonlocality in quantum many-body systems, up to 24 qubits with a fully programmable superconducting quantum processor. In particular, we employ energy as a Bell correlation witness and variationally decrease the energy of a many-body system across a hierarchy of thresholds, below which an increasing Bell correlation depth can be certified from experimental data. As an illustrating example, we variationally prepare the low-energy state of a two-dimensional honeycomb model with 73 qubits and certify its Bell correlations by measuring an energy that surpasses the corresponding classical bound with up to 48 standard deviations. In addition, we variationally prepare a sequence of low-energy states and certify their genuine multipartite Bell correlations up to 24 qubits via energies measured efficiently by parity oscillation and multiple quantum coherence techniques. Our results establish a viable approach for preparing and certifying multipartite Bell correlations, which provide not only a finer benchmark beyond entanglement for quantum devices, but also a valuable guide towards exploiting multipartite Bell correlation in a wide spectrum of practical applications.

Figures (6)

• Figure 1: Many-body Bell correlation and the variational detection approach.a, The hierarchy of quantum correlations, starting from entanglement and Einstein-Podolsky-Rosen (EPR) steering, and finally to Bell correlation. Bell correlation is the strongest quantum correlation, in the sense that whenever it is detected then the other two types of correlation are guaranteed. In quantum many-body systems, Bell correlation can be further characterized by its depth (indicated by yellow contours), which quantifies the minimal number of particles sharing genuine nonlocal correlations. b, A generic multipartite quantum system for detecting many-body Bell correlation, where site-resolved measurements $\mathcal{M}( \mathbf{x})$ ($\mathbf{x}:=x_1,\dots,x_{N}$ with $x_i$ being the direction of measurement performed on the $i$th party) are performed on all partites and the corresponding outputs $\mathbf{a}:=a_1,\dots,a_{N}$ are collected. The multipartite Bell inequality can be transformed into a many-body Hamiltonian by assigning each party the corresponding quantum observables. The detection of Bell correlation then becomes equivalent to finding low-energy states of the system with the energy exceeding the classical bound. c, A schematic illustration of the essential idea of probing the many-body Bell correlation with a variational quantum circuit (VQC). Starting from the initial state $|\Phi_\text{ini}\rangle$, the VQC iteratively updates variational parameters $\boldsymbol{\theta}$ to minimize the energy of the many-body system $E=\langle\mathcal{H}_B\rangle_{\boldsymbol{\theta}}$, leading to violations of the corresponding Bell inequality and detection of the correlation depth. d, A sketch of the sets of correlations. For a given measurement configuration set $\mathcal{X}:=\{\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_K\}$, the probability distribution of all possible outcomes, denoted collectively by $\vec{p}= \{P(\mathbf{a}\vert \mathbf{x})\}_{\mathbf{a};\mathbf{x}\in\mathcal{X}}$, is confined in the polytope (light orange), which denotes the set of all possible classical correlations assuming local realism ($\mathcal{C}$). A facet of the polytope denotes a tight multipartite Bell inequality (black solid line), outside which certifies Bell correlations ($\mathcal{Q}$).
• Figure 1: Observation of Bell correlations in a $\bold{1}$D chain.a, The variational quantum circuit for detecting the Bell correlations, where variational parameters are specified as $\theta$ in the virtual Z gate PhysRevA.96.022330. b, The optimization trajectories of the XXZ-type Hamiltonian with $\Delta=2$ and $\epsilon=0.95$.
• Figure 2: Detecting Bell correlation with $\bold{73}$ transmon qubits on a $\bold{2}$D honeycomb lattice.a, Device and qubit topology. The superconducting chip contains a square lattice of 11$\times$11 transmon qubits, from which we select 73 qubits arranged in a brick-wall (honeycomb) lattice, which consists of two triangular sublattices (denoted here by A and B, respectively). Based on the explicit form of the many-body Bell inequality (\ref{['BellHoneyComb']}), the system Hamiltonian is constructed by assigning different weights to the coupling terms between the nearest neighbor qubits of different orientations as colored by red, green, and blue, respectively. b, The variational quantum circuit designed for detecting Bell correlations. This variational ansatz is composed of three blocks, each of which begins with a layer of Hadamard gates, followed by a variational entangler which contains a layer of independent $R_z(\theta)$ gates and is sandwiched by two layers of controlled-NOT (CNOT) gates. For blocks with different colors, the CNOT gates are applied on qubit pairs connected by the links of the corresponding color as shown in a, with the circuit for the red block exemplified in the lower panel. c, Optimization trajectories for the XZ-type Hamiltonian $\mathcal{H}_B$ in Eq. \ref{['XZ-type Hamiltonian 2D']} with different coupling configurations characterized by $\epsilon$. As the variational parameters are updated iteratively, all energy values for different $\epsilon$ decrease below the classical bound $\beta_C$, indicating a violation of the corresponding Bell inequality \ref{['BellHoneyComb']}. The upper right inset shows the measured minimal energies and the classical bound (yellow dashed line) versus $\epsilon$ during the variational process. Here we show both the data with (circle) and without (square) readout corrections. Error bars represent the standard deviation with ten repetitions of experiments.
• Figure 2: Quantum circuits for measuring multi-qubit correlations.a, Circuit for measuring parity oscillation. We add a single-qubit gate $U_\gamma=\frac{\sqrt{2}}{2} \left( {1e^{-i\gamma}-e^{i\gamma}1} \right)$ to each qubit at the end of the variational circuit $U_{\text{exp}}\left(\theta \right)$ depicted in Figure 3(a) of the main text, which brings the axis defined by the operator $\mathcal{P}(\gamma)=\sin(\gamma) \sigma_y + \cos(\gamma) \sigma_x$ to the $z$ axis. Subsequently, we perform simultaneous measurements on all qubits and calculate the parity expectations with different $\gamma$ values. b, Circuit for measuring MQC. After applying the variational circuit $U_{\text{exp}}\left(\theta \right)$, we add a single-qubit $R_x(\pi)$ gate and $R_z(\phi)$ gate to each qubit followed by a reversal circuit $U^\dagger_{\text{exp}}\left(\theta \right)$. Then we measure the probability of the system being in the $\left|00\dots0\right\rangle$ state.
• Figure 3: Detection of Bell correlation depth.a, The variational quantum circuit with a hierarchical architecture designed for probing Bell correlation depth. The circuit is divided into many sub-circuits labelled I, II, III, etc., which are trained sequentially to avoid the problem of barren plateaus. Starting with two qubits, we train sub-circuit I to minimize the energy of the two-qubit system, following which we expand the system by involving two more qubits and train sub-circuit II to minimize the energy of the four-qubit system. The procedure goes on until all $24$ qubits are involved, during which the detected Bell correlation depth increases step by step. b, Measured parity expectation $\left\langle\mathcal{P}(\gamma)\right\rangle$ and multiple quantum coherence (MQC) $\mathcal{K}(\phi)$ data for the $24$-qubit Greenberger-Horne-Zeilinger (GHZ) state with (blue circles) and without (yellow squares) readout correction, alongside their corresponding Fourier spectrum ($\mathcal{P}_f(q)$ and $\mathcal{K}_f(q)$) from which the energy of the system $\langle \mathcal{H}_B\rangle$ with $\mathcal{H}_B$ shown in Eq. \ref{['eq:k-nonlocal_Hamiltonian_ev']} can be obtained. Error bars denote the standard deviation from five repetitions of the experiment. c, Optimization trajectory. During the training procedure, we assess the loss function -- the energy of the system -- based on parity measurement. The whole training procedure is separated into $13$ phases, with the first $12$ phases corresponding to the hierarchical training procedure and the last phase retraining the whole circuit again. The energy values with (blue circles) and without (orange squares) readout correction are presented for each phase, with error bars denoting the standard deviation from five repetitions of the experiment. Dashed lines indicate the $k$-nonlocal bounds (lower than which guarantees Bell correlation depth $k+1$) for the corresponding phase. The lower panel shows the energy differences relative to the $k$-nonlocal bounds, with the energies measured by the MQC method at the end of each training phase, which are less affected by readout errors (Methods and Supplementary Sec. II.C).