Resource-Efficient Cross-Platform Verification with Modular Superconducting Devices

TL;DR

The paper addresses scalable benchmarking of modular quantum processors by comparing cross-platform verification protocols that estimate the inner product $\mathrm{tr}(\rho_A \rho_B)$ between distributed states. It implements two LOCC-based approaches (quantum-state tomography) and a quantum-communication-enabled Bell-basis scheme on a six-qubit device consisting of two three-qubit modules, demonstrating that LOCC methods scale exponentially with system size while Bell-basis measurements achieve sub-exponential, near-quadratic scaling. For $n=3$ qubits, Bell-basis measurements require about four times fewer repetitions than QST to reach a variance of $10^{-3}$, highlighting the utility of quantum links for scalable benchmarking. The work shows that inter-module Bell-basis measurements can significantly reduce resource overhead in modular architectures, supporting scalable certification as quantum processors grow beyond monolithic designs.

Abstract

Large-scale quantum computers are expected to benefit from modular architectures. Validating the capabilities of modular devices requires benchmarking strategies that assess performance within and between modules. In this work, we evaluate cross-platform verification protocols, which are critical for quantifying how accurately different modules prepare the same quantum state -- a key requirement for modular scalability and system-wide consistency. We demonstrate these algorithms using a six-qubit flip-chip superconducting quantum device consisting of two three-qubit modules on a single carrier chip, with connectivity for intra- and inter-module entanglement. We examine how the resource requirements of protocols relying solely on classical communication between modules scale exponentially with qubit number, and demonstrate that introducing an inter-module two-qubit gate enables sub-exponential scaling in cross-platform verification. This approach reduces the number of repetitions required by a factor of four for three-qubit states, with greater reductions projected for larger and higher-fidelity devices.

Figures (7)

• Figure 1: (a) Schematic of the device design showing the top modules and bottom carrier, with the qubit island (gaps where metal is removed are colored light red), qubit-qubit coupler (light blue), readout coupling pad (red), flux line (green), charge line (pink), and ground plane (gray). The vertical separation between the chips is 10. (b) Photo of the assembled device.
• Figure 2: (a) Empirical cumulative distribution function of device errors, with the error rates of single-qubit gates (pink square), two-qubit gates (double cyan circles), and readout in the qubit and qutrit manifold (red measurement icons). (b) State tomography result for a three-qubit GHZ state prepared on the three qubits of the left module. We show Pauli decomposition coefficients which are expected to be non-zero. Ideal results in the absence of noise are represented by the black outlined bars. The remaining coefficient with the largest absolute value is presented as "Other".
• Figure 3: Circuit diagram for the GHZ state preparation and the Bell-basis measurement (BBM) protocol. Measuring corresponding qubits on each module in the Bell basis results in two bitstrings $a_i$ and $b_i$. Analyzing these as described in the main text results in an estimate of the inner product between the states on each module. The red dotted line indicates where the quantum state tomography (QST) circuit would conclude, with measurements in a complete basis set.
• Figure 4: (a-d) Measured and calculated inner products for two $n$-qubit states, $\frac{1}{\sqrt{2}} \left( \ket{0}^{\otimes n} + \ket{1}^{\otimes n} \right)$ and $\frac{1}{\sqrt{2}} \left(\ket{0}^{\otimes n} + e^{i \varphi} \ket{1}^{\otimes n} \right)$, prepared on the left and right modules respectively. The overlap is calculated using quantum state tomography (QST, red symbols) and Bell-basis measurements (BBM, blue symbols). Statistical error bars are smaller than symbol sizes. (e) The number of measurements needed to estimate the inner product between two identically-prepared $\ket{\mathrm{GHZ}}_n^{(0)}$ states to a statistical variance of $10^{-3}$. The variance is calculated by taking 100 samples of varying size, with replacement, from a large dataset. The relationship between variance and the sample size is then interpolated for the y-axis values. For both plots, simulated data for the Bell-basis measurement protocol, using the median errors presented in Fig. \ref{['fig:main:performance']}a, are presented as light blue downward triangles. White downward triangles represent simulations with all error rates divided by ten. Exponential (for quantum state tomography and randomized measurements) and quadratic (for Bell-basis measurements) fitted curves are provided as dashed lines as a guide to the eye. Statistical error bars are smaller than symbol sizes. (f) The measured inner product for identically-prepared $\ket{\mathrm{GHZ}}_n^{(0)}$ states using Bell-basis measurements, quantum state tomography, and randomized measurements (RM).
• Figure A1: Measurement setup wiring diagram. See the text of Sec. \ref{['sec:appendix:measurementapparatus']} for details. This is exemplary for one qubit and one readout line, and does not display the whole setup.