Exact analytic toolbox for quantum dynamics with tunable noise strength

Abstract

We introduce a framework that allows for the exact analytic treatment of quantum dynamics subject to coherent noise. The noise is modeled via unitary evolution under a Hamiltonian drawn from a random-matrix ensemble for arbitrary Hilbert-space dimension $N$. While the methods we develop apply to generic such ensembles with a notion of rotation invariance, we focus largely on the Gaussian unitary ensemble (GUE). Averaging over the ensemble of ''noisy'' Hamiltonians produces an effective quantum channel, the properties of which are analytically calculable and determined by the spectral form factors of the relevant ensemble. We compute spectral form factors of the GUE exactly for any finite $N$, along with the corresponding GUE quantum channel, and its variance. Key advantages of our approach include the ability to access exact analytic results for any $N$ and the ability to tune to the noise-free limit (in contrast, e.g., to the Haar ensemble), and analytic access to moments beyond the variance. We also highlight some unusual features of the GUE channel, including the nonmonotonicity of the coefficients of various operators as a function of noise strength and the failure to saturate the Haar-random limit, even with infinite noise strength.

Figures (5)

• Figure 1: The amplitude $f(t)$ of the term $\mathop{\mathrm{tr}}\limits(A) \mathds{1} /2$ in the GUE-averaged single-qubit channel $\langle A_{\boldsymbol{g}}(t)\rangle$\ref{['eq:qubit onefold channel']}. For comparison, the dashed line indicates the Haar-random value $f(t)=1$, which is never realized by $\langle A_{\boldsymbol{g}}(t)\rangle$. For $t \gtrsim 4$, $f(t)$ saturates to $2/3$. + AI.
• Figure 2: Standard deviation of matrix elements $A_{mn}(t)$ of $A_G(t)$\ref{['eq:qubit channel']} for an initial operator $A= \boldsymbol{a} \cdot \boldsymbol{\sigma}$, in units of $a = \lvert\boldsymbol{a}\rvert$. Diagonal elements ($m=n$) appear in green and off-diagonal elements ($m \neq n$) appear in orange. The choices $A=Z$ (solid) and $A=X$ (dashed) bound the standard deviation for a general operator, corresponding to the shaded region; the dash-dotted curves represent the same standard deviations with $A$ averaged over the GUE, appropriately normalized (see also Sec. \ref{['subsec:variance chan form']}). Inset: Plot of a "typicality" diagnostic \ref{['eq:qubit typ def']}, which is the ratio of the standard deviation to the mean. We find that $\operatorname{typ}(A_{mn},t) \ll 1$ is small when $t \lesssim 1$, meaning that the average \ref{['eq:qubit onefold channel']} well approximates a typical instance of the noisy unitary \ref{['eq:qubit channel']}. For $t \gtrsim 3$, we find a somewhat typical regime with and $\operatorname{typ}(A_{mn},t) \sim 1$.
• Figure 3: Left: Channel amplitude $f(t)$\ref{['eq:f(t) def']} for Hilbert-space dimensions $N=2^k$ for $k=1,2,3,4$, which captures the strength of the $\mathop{\mathrm{tr}}\limits(A) \mathds{1} / N$ term in the average channel \ref{['eq:Avg Noisy Chan two terms']}. Right: The two-point SFF $\mathcal{R}_2(t)$\ref{['eq:SFF main']} normalized by its $t=0$ value of $N^2$, for the same values of $N=2^k$.
• Figure 4: Left: Standard deviation of matrix elements $A_{mn}(t)$ of $A_G(t)$\ref{['eq:noisyOp']}, for an initial operator $A$ sampled from the GUE with width $\sigma_A$. The standard deviation is the square root of the variance \ref{['eq:varMatElGeneralN']}, and is plotted in units of $\sigma_A$ for various $N = 2^k$; solid lines denote diagonal elements ($m=n)$ and dashed lines denote off-diagonal elements ($m \neq n$). Right: Plot of the typicality \ref{['eq:qubit typ def']} of diagonal (solid) and off-diagonal (dashed) matrix elements $A_{mn}(t)$. The typicality is the standard deviation divided by the square root of \ref{['eq:avg square GUE avg']}.
• Figure 5: Left: The three-point SFF $\mathcal{R}_{4,1}(t)$\ref{['eq:SFF3 main']} normalized by its $t=0$ value of $N^3$, for $N=2^k$ with $k = 1,2,3,4$. Right: The four-point SFF $\mathcal{R}_4(t)$\ref{['eq:SFF4 main']} normalized by its $t=0$ value of $N^4$, for the same values of $N=2^k$. Both functions bear similarities to $\mathcal{R}_2(t)$\ref{['eq:SFF main']} [see Fig. \ref{['fig:SFFsFigure']}].