Violating Bell's inequality in gate-defined quantum dots

TL;DR

This work demonstrates a Bell-inequality violation in gate-defined silicon quantum dots by combining heralded initialization with gate-set tomography (GST) to suppress both SPAM and coherent errors. The authors achieve full two-qubit gate fidelities above $99\%$, and Bell-state generation fidelities around $97.17\%$ (Phi+) uncorrected, yielding Bell signals up to $S = 2.731 \pm 0.088$, well above the classical bound $S=2$. The experiments remain robust up to $1.1\,ferspace K$, and entanglement lifetimes exceed $100\,\mu s$ with dynamical decoupling extending coherence. GST provides a detailed error decomposition, highlighting idle dephasing as the main limitation and guiding strategies for real-time phase tracking and improved interfaces toward scalable, fault-tolerant silicon spin-qubit processors.

Abstract

Superior computational power promised by quantum computers utilises the fundamental quantum mechanical principle of entanglement. However, achieving entanglement and verifying that the generated state does not follow the principle of local causality has proven difficult for spin qubits in gate-defined quantum dots, as it requires simultaneously high concurrence values and readout fidelities to break the classical bound imposed by Bell's inequality. Here we employ heralded initialization and calibration via gate set tomography (GST), to reduce all relevant errors and push the fidelities of the full 2-qubit gate set above 99 %, including state preparation and measurement (SPAM). We demonstrate a 97.17 % Bell state fidelity without correcting for readout errors and violate Bell's inequality with a Bell signal of S = 2.731 close to the theoretical maximum of $2\sqrt{2}$. Our measurements exceed the classical limit even at elevated temperatures of 1.1 K or entanglement lifetimes of 100 $μs$.

Figures (14)

• Figure 1: Device and basic operation.a, Scanning electron micrograph of a device nominally identical to that used in this work. Active gate electrodes and the microwave antenna are highlighted with colours. An external d.c. magnetic field $B_0$ and the antenna-generated a.c. magnetic field $B_1$ are indicated with arrows. The system operates at $T = 0.1K$, unless otherwise specified. b, Transmission electron micrograph of the "active" region with schematics indicating the quantum dot and electron spin qubit formation at the Si/SiOx interface including exchange control. c, Charge stability diagram as a function of P1, P2 voltage detuning $\Delta V_\mathrm{G} = -\Delta V_\mathrm{P1} = \Delta V_\mathrm{P2}$ and the J gate voltage $V_\mathrm{J}$, showing six loaded electrons across the double-dot system. The operation points for readout (RO), single-qubit operation ($J_{\rm off}$) and two-qubit operation ($J_{\rm on}$) are labelled as star ($\star$), square ($\blacksquare$), and triangle ($\blacktriangle$), respectively. d, e, Probability of detecting a blockaded state, $P_\mathrm{blockade}$, after a microwave burst of fixed power and duration at different J gate voltages $V_\mathrm{J}$ when preparing a mixed odd state $\frac{1}{\sqrt{2}}(\ket{\downarrow\uparrow}+\ket{\uparrow\downarrow})$ (d) and a pure state $\ket{\downarrow\downarrow}$ (e). The power and duration of the microwave burst are roughly calibrated to a single-qubit $\pi$-rotation. The following experiments are conducted with $\ket{\downarrow\downarrow}$ initialization, unless otherwise specified. f, g, Q1 and Q2 single-qubit Rabi oscillations at $V_\mathrm{J}=0.71V$ as a function of pulse time $t_\mathrm{ESR}$, respectively. h, Decoupled controlled phase (DCZ) oscillations as a function of exchange time $t_\mathrm{exchange}$ and $V_\mathrm{J}$. i, $\mathrm{DCZ}$ exchange oscillation fingerprint for fixed exchange time $t_\mathrm{exchange} = 5µ s$ as a function of $\Delta V_\mathrm{G}$ and $V_\mathrm{J}$. Readout probability is unscaled in all data. Error bars represent the 95% confidence level.
• Figure 1: Breakdown of GST error channels. Infidelity contributions of individual error channels using pyGSTi on Larmor frequency feedback rate measurement series including the additionally phase corrected experiment (*) with fidelities of full 2-qubit gate set above 99%.
• Figure 2: Two-qubit benchmarking using GST.a, Gate infidelity as a function of the Larmor frequency feedback rate and number of GST sequences per feedback cycle. The star ($\star$) indicates additional phase corrections based on the previous GST results. The green dashed line indicates the commonly considered 99% threshold. The inset is a zoom-in of the black-dashed box. b,c, State preparation and measurement (SPAM) matrix, respectively. The insets show the respective theory matrix. d-f, Error magnitude of error components for the $\mathrm{XI}$, $\mathrm{IX}$, and $\mathrm{DCZ}$ gates from GST with (coloured bars) and without (uncoloured bars) additional phase correction. The average gate fidelity is given above each plot for the phase corrected GST measurement. The on-target X gate fidelity $F_\mathrm{X,Qi}$ can be calculated from the relevant error components. Hamiltonian errors contribute to the fidelity in second order, while stochastic errors contribute in first order. Error bars represent the 95% confidence level.
• Figure 2: Bell state tomography as a function of temperature. Density matrices of all four Bell states for temperatures a, 0.1K, b, 0.2K, c, 0.3K, d, 0.5K, e, 0.7K, f, 0.85K, g, 1.0K, and h, 1.1K extracted from quantum state tomography.
• Figure 3: Bell test.a, Protocol for conducting the Bell test in a gate-defined double-dot electron spin system. After preparation of a maximally entangled Bell state (i-iv), the quantum correlation is measured via (I) a direct parity measurement after rotation of each qubit to obtain the desired combination of projection axes in two bases, rotated by $\pi/4$, or (II) quantum state tomography. b, Schematic of the two projection basis $(\alpha, \alpha')$ and $(\beta, \beta')$ of the electron spin qubit in quantum dot 1 and 2, respectively. c, Histograms of RF-SET readout signal for all four Bell states in all possible combinations of axis projections at $T = 0.1K$. The data is fitted with a bimodal Gaussian distribution. The intersect of the two Gaussian curves is indicated by a dashed line defining the threshold for distinguishing odd (unblockaded) and even (blockaded) parity. d, Quantum state tomography results for all four Bell states at $T = 0.1K$. No corrections have been applied to compensate for initialization and readout errors. Insets indicate the theoretical density matrix of each Bell state. Error bars represent the 95% confidence level.