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arXiv:2201.08672 · Physical Review Letters

Fingerprints of Qubit Noise in Transient Cavity Transmission

Philipp M. Mutter, Guido Burkard

Published Jun 8, 2022• 9 citations
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Abstract

Noise affects the coherence of qubits and thereby places a bound on the performance of quantum computers. We theoretically study a generic two-level system with fluctuating control parameters in a photonic cavity and find that basic features of the noise spectral density are imprinted in the transient transmission through the cavity. We obtain analytical expressions for generic noise and proceed to study the cases of quasistatic, white and 1/f^{α} noise in more detail. Additionally, we propose a way of extracting the noise power spectral density in a frequency band only bounded by the range of the qubit-cavity detuning and with an exponentially decaying error due to finite measurement times. Our results suggest that measurements of the time-dependent transmission probability represent a novel way of extracting noise characteristics.

DOI
10.1103/physrevlett.128.236801

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Noise affects the coherence of qubits and thereby places a bound on the performance of quantum computers. We theoretically study a generic two-level system with fluctuating control parameters in a photonic cavity and find that basic features of the noise spectral density are imprinted in the transient transmission through the cavity. We obtain analytical expressions for generic noise and proceed to study the cases of quasistatic, white and 1/f^{α} noise in more detail. Additionally, we propose a way of extracting the noise power spectral density in a frequency band only bounded by the range of the qubit-cavity detuning and with an exponentially decaying error due to finite measurement times. Our results suggest that measurements of the time-dependent transmission probability represent a novel way of extracting noise characteristics.
Noise-induced decoherence is one of the main threats to large-scale quantum computation. In an attempt to assess the noise affecting a qubit, we go beyond the standard steady-state solution of the transmission through a qubit-coupled cavity in input-output theory by including dynamical noise in the description of the system. We solve the quantum Langevin equations exactly for a noise-free system and treat the noise as a perturbation. In the long-time limit the corrections may be written as a sum of convolutions of the noise power spectral density with an integration kernel that depends on external control parameters. Using the convolution theorem, we invert the corrections and obtain relations for the noise spectral density as an integral over measurable quantities. Additionally, we treat the noise exactly in the dispersive regime and again find that noise characteristics are imprinted in the long-time transmission in convolutions containing the power spectral density.
A significant challenge in the field of large-scale fault-tolerant quantum computation is the influence of noise. In addition to the influence of noise on individual qubits, the smaller additional effect of noise correlations is also of high significance because correlated errors pose a challenge for quantum error correction. We describe the dynamics of two cavity-coupled qubits that are subject to correlated noise, assuming that the qubits are affected by longitudinal noise and not coupled directly. We find that the cavity emission enables the characterization of the noise correlations and describe the cases of white noise, quasi-static noise, and Ornstein-Uhlenbeck noise. For a known frequency spectrum, the reconstruction of the noise correlation spectral density from the cavity emission is possible by averaging over many different noise realizations. We demonstrate that, in the case of white noise, the noise correlation effects scale with the fifth order of the cavity-qubit coupling constant and are thus strongly suppressed compared with the case of quasi-static noise, where they scale with the third order. Furthermore, we present a method for extracting the noise correlation spectral density from the cavity emission in the case where the underlying noise type remains unidentified. This can be achieved by applying the convolution theorem.
Characterizing noise in superconducting qubits is essential for improving coherence and gate performance. Conventional noise-sensing methods typically use the qubit itself as the sensor, which limits both accessible bandwidth and applicability during driven operation. Here, we measure qubit frequency noise by detecting the photon loss it induces in a coupled high-Q superconducting cavity. We use repeated mid-circuit qubit measurements with post-selection to separate this induced loss from intrinsic cavity decay. We validate the protocol using injected noise and show that the extracted loss scales as expected with the applied noise strength. We place an upper bound of 0.70×103 rad2/s0.70\times10^3\,\mathrm{rad}^2/\mathrm{s}0.70×103rad2/s on the intrinsic qubit frequency-noise power spectral density at the cavity-qubit detuning of 508 MHz. The protocol opens access to a higher-frequency spectral window than standard qubit-based spectroscopy and may enable noise characterization during strong driving.
We develop a theory describing the transient transmission through noisy qubit-resonator systems with quadratic interactions as are found in superconducting and nanomechanical resonators coupled to solid-state qubits. After generalizing the quantum Langevin equations to arbitrary qubit-resonator couplings, we show that only the cases of linear and quadratic couplings allow for an analytical treatment within standard input-output theory. Focussing on quadratic couplings and allowing for arbitrary initial qubit coherences, it is shown that noise characteristics can be extracted from input-output measurements by recording both the averaged fluctuations in the transmission probability and the averaged phase. Our results represent an extension to the field of transmission-based noise spectroscopy with immediate practical applications.
As qubit decoherence times are increased and readout technologies are improved, nonidealities in the drive signals, such as phase noise, are going to represent a crucial limitation to the fidelity achievable at the end of complex control pulse sequences. Although the effect of phase noise of reference oscillators on qubit performance has been studied previously, its interaction with realistic time-dependent control pulses and its contribution to fidelity degradation have not yet been investigated in sufficient detail, and remains a critical challenge. Here we study the impact on fidelity of phase noise affecting reference oscillators with the help of numerical simulations, which allow us to directly take into account the interaction between the phase fluctuations in the control signals and the evolution of the qubit state, thereby achieving a comprehensive understanding of the actual role played by the different spectral components of phase noise. In particular, we perform an analysis of the effect of the individual noise frequency contributions, providing a clear identification of the spectral regions that most critically impact fidelity and establishing their relative weight in the overall fidelity degradation. Our method is based on the generation of phase noise realizations consistent with a given power spectral density, that are then applied to the pulse carrier in simulations, with Qiskit-Dynamics, of the qubit temporal evolution. By comparing the final state obtained at the end of a noisy pulse sequence with that in the ideal case and averaging over multiple noise realizations, we estimate the resulting degradation in fidelity, and, exploiting an approximate analytical representation of a carrier affected by phase fluctuations, we shed new light on the nature of the different contributions, and provide an intuitive physical picture.
Fluctuations of the qubit frequencies are one of the major problems to overcome on the way to scalable quantum computers. Of particular importance are fluctuations with a correlation time that exceeds the decoherence time due to decay and dephasing by fast processes. The statistics of the fluctuations can be characterized by measuring the correlators of the outcomes of periodically repeated Ramsey measurements. We describe qubit dynamics during repeated measurements and evaluate the two-time correlator for noise from two-level systems and two- and three-time correlators for Gaussian noise. The explicit expressions for the correlators are compared with simulations. We find that, even though the three-time correlators for noise from two-level systems and for Gaussian noise are generally significantly different, already ten two-level systems can mimic Gaussian noise provided they are symmetric. This is not the case for asymmetric systems.