Randomized benchmarking of a high-fidelity remote CNOT gate over a meter-scale microwave interconnect

TL;DR

This work demonstrates a SPAM-tolerant benchmarking framework for meter-scale superconducting interconnects, using TCQ-based modules linked by a 60 cm CPW and evaluated with SATD-based remote state transfer and a remote CNOT gate. Frame tracking enables robust, SPAM-resistant network benchmarking and two-qubit randomized benchmarking, yielding an EPS of 0.012 for state transfer and an EPG of 0.067 for the remote CNOT. The combination of SATD, precise detuning/coupling control, and leakage- and decoherence-aware error budgeting establishes a standard method for characterizing module-to-module links, a critical step toward scalable, multi-module superconducting processors. The results highlight practical pathways to improve fidelity (e.g., increasing TCQ–CPW coupling) and provide concrete Bell-state and entangling-gate benchmarks across a meter-scale microwave interconnect.

Abstract

High-fidelity, meter-scale microwave interconnects between superconducting quantum processor modules are a key technology for extending system size beyond constraints imposed by device manufacturing equipment, yield, and signal delivery. Although tomographic experiments have been used in previous demonstrations for benchmarking remote state transfer between modules, they do not reliably separate State Preparation and Measurement (SPAM) error from the error per state transfer. Recent developments based on randomized benchmarking provide a compatible theory for separating these two errors. In this work, we present a module-to-module interconnect based on Tunable-Coupling Qubits (TCQs) and benchmark, in a SPAM-error-tolerant manner enabled by a frame-tracking technique, a remote state transfer fidelity of 0.988 across a 60cm-long coplanar waveguide (CPW). The state transfer is implemented via a superadiabatic transitionless driving method, which suppresses intermediate excitation in the internal modes of the CPW. We further propose and construct a remote CNOT gate between modules, composed of local CZ gates in each module and remote state transfers, and report a gate fidelity of 0.933 using the randomized benchmarking method. The remote CNOT construction and benchmarking we present provide a way to fully characterize the module-to-module link operation and standardize reporting fidelity, analogous to randomized benchmarking protocols for other quantum gates.

Figures (9)

• Figure 1: (a) Circuit schematic consisting of the fixed-frequency data qubits $\mathrm{Q_{D1,2}}$ and the $l$ coupler module coupled via the tunable couplers $\mathrm{C_{L1,2}}$. The $l$ coupler module consists of two TCQs PhysRevLett.106.030502PhysRevB.84.184515zhang2017suppression, $Q_\mathrm{{C1,2}}$, capacitively coupled to the $60$ cm on-chip CPW. The TCQs consist of three electrodes connected in series by two SQUIDs. The CPW is coupled to the middle electrodes of the TCQs. (b) Schematics showing the eigenenergy diagram of the $l$ coupler module when the coupling is switched off. Light-blue horizontal lines represent the energies of the top or bottom transmon modes of the TCQs, which will split into blue lines corresponding to the symmetric and anti-symmetric modes. By tuning the flux biases of the TCQs, we can independently manipulate the top and bottom mode energies. When the top and bottom modes of the TCQs become degenerate with each other (TCQ is balanced), the symmetric modes have voltage nodes at the middle electrode of the TCQ and become decoupled from the CPW modes, shown as black horizontal lines. We call the symmetric modes of the TCQs $l$ qubits. (c) Schematics showing the eigenenergy diagram of the $l$ coupler module when the coupling is switched on. When the top and bottom modes of the TCQs are detuned from each other (TCQ is unbalanced), the symmetric modes start to couple to the CPW modes. The purple-colored region represents the Hilbert space used for state transfers, where the symmetric modes of the TCQs and the target CPW mode are degenerate with each other and form dark and bright modes, as shown in \ref{['eq:bright_dark']}. During state transfers between $l$ qubits, the flux biases of the TCQs are swept adiabatically between conditions (b) and (c).
• Figure 2: (a) Pulse sequence for the fixed-duration XY Ramsey experiments to estimate the $l$ qubit eigenfrequencies while independently sweeping the top and bottom flux pulse amplitudes $V_\mathrm{T,B}$. The second flux pulse and the echo microwave pulse are used to cancel out the $l$ qubit eigenfrequency shifts at the edges of the flux pulses. (b) Experimental results of (a). Contour lines correspond to the CPW mode frequencies. Green contour lines represent the target CPW mode used for state transfers. Black dotted lines represent the trace where the TCQs become symmetric. White dots indicate the idle bias condition. (c) Pulse sequence for the vacuum Rabi oscillation experiment to evaluate the coupling strength between the $l$ qubits and the target CPW mode ($m=50$). (d) Experimental results of (c). We sweep the top and bottom flux biases along the green contour lines in \ref{['fig:control_accuracy']} (a) to vary the couplings linearly. The top panels show the calibration error of the coupling strengths.
• Figure 3: (a) Quantum circuit for the network benchmarking helsen2023benchmarking. $\mathrm{C_i}$ represents a random single-qubit Clifford gate. $\mathrm{C_{inv}}$ represents the inverse Clifford operation to make the total sequence equal to the identity operation. Arrows $\mathrm{ST_{\pi/2}}$ in the circuit represent directional state transfer between the $l$ qubits via the CPW. Note that, although the $l$ qubit population is transferred in both directions in SATD-based state transfer, only in one direction is it transferred with higher fidelity via the dark state. (b) Quantum circuit for the two-qubit randomized benchmarking of the remote CNOT gate. $\mathrm{C_i}$ represents a random two-qubit Clifford gate between the data qubits. $\mathrm{C_{inv}}$ represents the inverse Clifford operation to make the total sequence equal to the identity operation. Each remote CNOT gate consists of three local CZ gates at each module and two directional state transfers between the modules. $\mathrm{Y_{\pm\pi/2}}$ represent the $\pm\pi/2$ rotation on the Pauli Y axis, respectively. (c) Network benchmarking on the $l$ qubits: probability of measuring $Q_\mathrm{L1}$ in the $\ket{0}$ state as a function of the number of Cliffords. We measure an error per state transfer (EPS) of 0.012. (d) Two-qubit randomized benchmarking on the data qubits for the remote CNOT gate from $Q_\mathrm{D1}$ to $Q_\mathrm{D2}$: probability of measuring the data qubits $Q_\mathrm{D1,2}$ in the $\ket{00}$ state as a function of the number of Cliffords. EPC and EPG represent the error per Clifford and the error per gate, respectively. We measure an EPG of 0.067.
• Figure S1: Partial circuit diagram of the $l$-coupler module consisting of a TCQ and a CPW coupled to each other capacitively.
• Figure S2: (a) Toy model 1: an $l$ coupler module consisting of tunable-frequency qubits with tunable couplings to a CPW, which is treated as a multi-mode resonator. (b) Eigenenergy diagram of the tunable-qubit $l$ coupler module during SATD-based state transfer. The horizontal and vertical axes represent time and eigenfrequencies, respectively. The horizontal purple line indicates the target CPW mode ($m=50$) used for the SATD-based state transfer, while the horizontal black lines represent the frequencies of the other CPW modes, referred to as spectator modes. Blue and green curves represent the tunable qubit frequencies of $Q_{\mathrm{L1}}$ and $Q_{\mathrm{L2}}$, respectively. Vertical black lines and red labels indicate distinct time regions, explained in the appendix. (c) Toy model 2: an $l$ coupler module consisting of tunable TCQs with fixed couplings to a CPW, treated as a multi-mode resonator. Each tunable TCQ consists of top and bottom qubit modes coupled with each other at a fixed rate. (d) Eigenenergy diagram of the TCQ-based $l$ coupler module during SATD-based state transfer. The horizontal and vertical axes represent time and eigenfrequencies, respectively. The horizontal purple line indicates the target CPW mode ($m=50$) used for the SATD-based state transfer. Red and orange curves correspond to the anti-symmetric modes of $Q_{\mathrm{C1}}$ and $Q_{\mathrm{C2}}$, respectively, while blue and green curves correspond to the symmetric modes ($l$ qubits) of $Q_{\mathrm{C1}}$ and $Q_{\mathrm{C2}}$. Light-colored curves represent the top and bottom transmon modes of the TCQs.