Dressed-state relaxation in coupled qubits as a source of two-qubit gate errors

TL;DR

The paper reveals a frequency-selective decoherence channel for two-qubit gates: noise at the dressed-state splitting $2g$—set by inter-qubit coupling—induces dressed-state relaxation that degrades gate fidelity. Extending concepts from $T_1$ and $T_{1\rho}$ to interacting qubits, it derives $\Gamma_g = \tfrac{1}{2} S_{\Delta}(2g)$ and shows gate errors scale with $\Gamma_g$ and gate duration as $\epsilon_{\text{avg}} = \tfrac{1}{5} \Gamma_g \tau$ (Pauli error: $\epsilon = \tfrac{1}{4} \Gamma_g \tau$). The authors validate the theory experimentally by engineering a band-limited noise spectrum (5–20 MHz) and measuring both single-qubit spin-locking-like relaxation and two-qubit dressed-state relaxation, finding linear dependence on injected noise power and agreement with the predicted scaling across a range of $g$ and $\tau$. The work emphasizes suppressing noise near $2g$ to improve entangling-gate fidelity and provides a spectroscopic tool for noise characterization applicable to dual-rail and singlet-triplet encodings. Overall, it offers a concrete, frequency-targeted decoherence mechanism and practical mitigation guidance for scalable quantum processors.

Abstract

Understanding error mechanisms in two-qubit gate operations is essential for building high-fidelity quantum processors. While prior studies predominantly treat dephasing noise as either Markovian or predominantly low-frequency, realistic qubit environments exhibit structured, frequency-dependent spectra. Here we demonstrate that noise at frequencies matching the dressed-state energy splitting--set by the inter-qubit coupling strength g--induces a distinct relaxation channel that degrades gate performance. Through combined theoretical analysis and experimental verification on superconducting qubits with engineered noise spectra, we show that two-qubit gate errors scale predictably with the noise power spectral density at frequency 2g, extending the concept of $T_{1ρ}$ relaxation to interacting systems. This frequency-selective relaxation mechanism, universal across platforms, enriches our understanding of decoherence pathways during gate operations. The same mechanism sets coherence limits for dual-rail or singlet-triplet encodings.

Figures (7)

• Figure 1: (a) Schematic diagram of a realistic noise spectrum $S(\omega)$ that includes $1/f$ noise, thermal noise, quantum noise, and spurs from electronics, modes, or microscopic defects. (b) Comparison of (left) the energy ($T_1$) relaxation process, (center) the rotating-frame ($T_{1\rho}$) relaxation process, and (right) the two-qubit dressed-state ($T_{1g}$) relaxation processes.
• Figure 2: (a) Noise generation scheme. A Gaussian noise with $\sim$90 MHz bandwidth generated from an arbitrary waveform generator is first sent through a bandpass filter with a passband from $\rm 5\,MHz$ to $\rm 20\,MHz$, then envelope-mixed with a gating signal for selectively turning on the noise in specific time windows, and finally combined with regular flux control signal before being sent to the qubit. (b) The measured $\Gamma_{\Omega}$ (squares) and $\Gamma_g$ (dots) extracted from the single-qubit spin-locking experiment (top-left panel) and the two-qubit dressed-state relaxation experiment (top-right panel), respectively, under different injected noise power $P_{\mathrm{out}}$. The shaded areas are the noise spectrum $S_V(\omega)$ measured with a spectrum analyzer and converted to qubit detuning fluctuations. Note that the noise is turned on only during the locking or interaction period. In the measurement protocol of the two-qubit dressed-state relaxation, we use a combination of single-qubit gates and a pre-calibrated square-root-of-iSWAP gate to prepare the qubits in the $\left|\tilde{1}\right\rangle=\frac{1}{\sqrt{2}}(\left|10\right\rangle+\left|01\right\rangle)$ state. The locking period (XX+YY interaction) is followed by a phase gate with varying phase $\theta$ in order to extract the spin polarization in a robust way against any imperfect locking condition. Note that the noise is only turned on during the locking period by the use of the gating signal to avoid disrupting state preparation and measurement. (c) $\Gamma_{\Omega}$ (blue) and $\Gamma_g$ (green) versus $P_{\mathrm{out}}$ at a fixed Rabi frequency $\Omega/2\pi= 2g/2\pi$ = 15 MHz from which we extract the rate-to-power ratios $\Gamma_{\Omega} / P_{\mathrm{out}} = 1.23\,(\mathrm{mW\cdot\mu s})^{-1}$ and $\Gamma_{g} / P_{\mathrm{out}} = 1.13\,(\mathrm{mW\cdot\mu s})^{-1}$.
• Figure 3: (a) Circuit schematic for verifying the fidelity reduction characterized by $\Gamma_g$ under the application of an fSim gate. In the XEB circuit, the noise injection is timed to coincide with the two-qubit gate and only applied on one qubit. (b) $\Delta \epsilon$ as a function of input noise power. Green dots represent $\Delta\epsilon$ from XEB experiments at a coupling strength of 7.69 MHz, with the solid green line showing the theoretical prediction. (c) $\Delta \epsilon$ over interaction time $\tau$ from XEB experiments at input noise powers of 5 mW (green dotted), 3 mW (yellow dotted), 2 mW (magenta dotted), and 1 mW (gray dotted). The solid lines denote theoretical results. (d) $\Delta\epsilon(P_{\rm out},\tau)-\Delta\epsilon(P_{\rm out},0)$ with the interaction duration of $\rm \tau=48\,ns$ as functions of Rabi frequency $\Omega$ and coupling strength $g$, respectively, under different input noise powers. The solid lines indicate the gate error $\Delta\epsilon$ induced by the measured $\Gamma_{\Omega}$, and the dots indicate the gate error measured by the XEB experiments.
• Figure 4: Statistical analysis of the AWG output noise. (a) The time-domain output signal of the AWG. (b) The probability density function (PDF) constructed from the voltage data shown in panel (a). The empirical distribution (green dots) is closely approximated by a Gaussian fit (black dashed line), confirming the stochastic nature of the output noise.
• Figure 5: Noise generation, characterization, and impact. (a) Schematic of the electronic setup for generating classical noise with programmable spectral and temporal profiles. (b) Measured noise power spectrum (averaged over 100 traces) showing the filtered band-limited profile. The dip near 6 MHz is an artifact of the measurement apparatus. (c) Processed spectrum obtained by applying a robust moving-window average (450-point window, selecting the 30th point from the maximum) to the data in (b). (d) Calculated power spectral density of frequency noise affecting the qubit (derived from Eq. (\ref{['eq:somega']})) for different injected noise powers $P_{\mathrm{out}}$. Dashed lines represent theoretical predictions based on the characterized noise injection chain.