Non-perturbative CPMG scaling and qutrit-driven breakdown under compiled superconducting-qubit control: a single-qubit study

Abstract

Decoherence in superconducting qubits emerges from the interplay of multilevel dynamics and structured environmental noise, yet perturbative models cannot capture all resulting signatures. Here, EmuPlat couples instruction-set-architecture-level waveform generation to the hierarchical equations of motion (HEOM) under $1/f$ non-Markovian pure dephasing. In the resulting non-perturbative regime -- where filter-function predictions become quantitatively uninformative -- CPMG scaling of a three-level superconducting transmon yields one calibration result, two physical findings, and one structural null. Y-CPMG exhibits axis-dependent scaling-law breakdown -- non-monotonic decoherence, partial coherence revival, and pronounced X--Y population asymmetry ($0.204$ vs ${<}\,0.01$) -- driven by third-level anharmonicity amplified by bath memory; X-CPMG maintains well-behaved power-law scaling with a finite-$n$ transient excess consistent with non-Markovian bath-memory effects. The structural null is equally informative: waveform-level differences -- Standard versus VPPU realizations -- remain undetectable across all coupling strengths, establishing that rotating-frame pure-dephasing coupling renders control-layer detail invisible to scaling observables. These findings define testable predictions, the most experimentally accessible requiring only qualitative verification.

Figures (12)

• Figure 1: Platform context for the paired waveform-realization comparison.(a) Four-level intermediate representation (IR) pipeline: a quantum circuit is progressively lowered through gate decomposition and pulse parametrisation to time-domain IQ waveform samples. (b) Full compilation pipeline: an OpenQASM circuit is lowered through transpilation, compilation, and pulse parametrisation to a PulseSequence. A WaveformGeneratorFactory fans out to two execution routes: the Standard route retains float64 analytical envelopes; the VPPU route applies ISA-level NCO mixing and DAC quantisation (int16). Both routes share the same front-end and solver interface, so the physics comparison isolates waveform realization rather than the rest of the software stack.
• Figure 2: Waveform spectral structure and $1/f$ bath overlap.a, Power spectral density of Standard (rectangular), VPPU-noDac, and VPPU+DAC (16/12/10/8-bit) waveforms, overlaid with the $1/f$ bath spectral density $J(\omega) = \eta/\omega$. The shaded region marks the spectral overlap zone [10 kHz--10 MHz]. b, Bhattacharyya coefficient (BC $\approx 0.60$) is invariant across 8--16 bit DAC resolution, confirming that the spectral coupling channel exists independently of DAC bit depth. Inset: time-domain $\mathrm{SQNR}_t$ vs. bit depth ($6.35\space dB/bit$, near theoretical $6.02\space dB/bit$).
• Figure 3: Burkard decoherence function and finite-time Gaussian-ratio diagnostic for the $1/f$ bath.a, Free induction decay $e^{-\Gamma(t)}$ computed from Burkard's exact analytical formula (Eq. \ref{['eq:burkard']}) for three coupling tiers spanning one decade in $\eta$. Dashed lines mark $T_2^*$ (171.7, 102.8, 58.0 ns). Rotating-frame HEOM validation confirms $<1.5$% deviation from these curves across all tiers. b, The finite-time ratio diagnostic $\mathrm{GC}_{\mathrm{ratio}}(t) = \Gamma(2t)/\Gamma(t)$ for Tier-1 starts near $\approx$4 (Gaussian-like) at short times and decreases monotonically toward $\approx$2 (exponential) at $t \gg T_2^*$, confirming that the $1/f$ bath produces intermediate, non-trivially non-Gaussian decoherence at the operational timescale---a key reason the filter function perturbative expansion (Fig. \ref{['fig:ff_breakdown']}) breaks down.
• Figure 4: First-order FF becomes quantitatively uninformative in the present $1/f$ window.a, FF predicts infidelity $\sim 10^{-13}$ across all tiers and waveform realizations, while HEOM measures $\sim$0.27---a $>$12-OOM gap at the Ramsey reference point. b, HEOM infidelity decreases with coupling (0.270$\to$0.248$\to$0.236, Tier-0/1/2) as stronger coupling drives more Gaussian-like decay; FF remains at $\sim 10^{-13}$.
• Figure 5: Even-$n$ CPMG scaling reveals non-perturbative regime characterisation and axis-dependent breakdown.a, Ideal-pulse HEOM reference in $\chi$-space ($\chi = -\ln(1-\varepsilon)$): 15 ns rectangular $\pi$ pulses ($d = 3$, even $n = 2$--$20$; uncontrolled comparison---pulse duration, $\tau$, and fitting range all differ from compiled data; see text for caveats); X-CPMG/XY-4 $\gamma_\mathrm{ideal} \approx 0.95$--$1.00$, Y-CPMG$\gamma_\mathrm{ideal} = 1.12$ (all $R^2 > 0.99$). b, Raw infidelity $\varepsilon(n) = 1 - W_n/W_0$ for even-$n$ compiled pulses ($n_\pi = 2$--$10$; 80 ns, $d = 3$); Y-CPMG (red) is non-monotonic. c, Decoherence function $\chi_n = -\ln(W_n/W_0)$ for the same data. X-CPMG scales monotonically ($\gamma_\mathrm{compiled} \approx 1.12$, $R^2 > 0.99$); Y-CPMG$\chi$ peaks at $n_\pi = 4$ then decreases---a coherence revival inconsistent with any power-law model. d, Y-CPMG population dynamics (Standard) showing persistent $\rho_{11} > \rho_{00}$ imbalance driven by $d = 3$ anharmonicity (gap relaxing from $0.29$ to $0.08$ over the sampled window), versus balanced X-CPMG ($|\rho_{00} - \rho_{11}| < 0.02$). e, Standard vs. VPPU paired comparison: X-CPMG$\chi$ values are indistinguishable ($\Delta\gamma \approx 0$, offset $\lesssim 0.002$), confirming waveform insensitivity in the Tier-0 window.