Denoising and Extension of Response Functions in the Time Domain

Response functions of quantum systems, such as electron Green’s functions, magnetic, or charge susceptibilities, describe the response of a system to an external perturbation. They are the central objects of interest in field theories and quantum computing and measured directly in experiment. Response functions are intrinsically causal. In equilibrium and steady-state systems, they correspond to a positive spectral function in the frequency domain. Since response functions define an inner product on a Hilbert space and thereby induce a positive definite function, the properties of this function can be used to reduce noise in measured data and, in equilibrium and steady state, to construct positive definite extensions for data known on finite time intervals, which are then guaranteed to correspond to positive spectra.

Figure 1

Denoised quantum Monte Carlo data. Main panel: inchworm [49] quantum Monte Carlo of a steady-state transport problem (for data and parameters see Fig. 1 of [30]), real part (orange) and imaginary part (blue). Denoised data, real part (black) and imaginary part(red). Inset: closeup of the region near $t=2$ highlighting the effect of denoising.

Figure 2

Extension of the on-site Hubbard dimer Green’s function $G_{11\uparrow }^{>}(t)$. Shown are real (black) and imaginary (red) parts of input data up to $t=2$ (circles), the numerically computed extension from $t=2$ to $t=10$ (squares), and the analytically known Green’s function up to $t=10$ (lines).

Figure 3

Main panel: greater Green’s function $G_k^{>}(t)$ at $k=\pi /2$ of an 8-site Su-Schieffer-Heeger with $\delta =0.4$ and $\mu =-3$. The figure shows the raw data obtained from the ibm_auckland quantum computer (blue), as well as the PSD denoised data (orange), and PSD denoised and extended data (green). Inset: corresponding spectral functions. The Fourier transform used a damping factor $\tau =100$. The vertical dashed lines indicate the expected analytic frequencies.

Figure 4

Spectral function of the $G_{Z_0{Z}_0}(t)$ correlation function computed for a 16-site Heisenberg chain in the antiferromagnetic regime. The noisy data was polluted by Gaussian noise with $\sigma =0.5·G_{Z_0{Z}_0}(t=0)$.