Efficient Characterization of Coherent and Correlated Low-Degree Noise in Layers of Gates

TL;DR

This work tackles scalable noise characterization for quantum gates by introducing a low-degree Pauli-channel ansatz and an efficient tomography protocol based on randomized Pauli state preparation and measurements. It achieves logarithmic sample complexity in the system size and provides explicit reconstruction guarantees in $\ell_2$ and diamond norms for the error channel $\mathcal{E}$, even when the hardware implements a gate layer $\mathcal{U}$, by performing the inversion in classical postprocessing. The method generalizes to layers with a polylogarithmic number of non-Clifford gates and includes strategies to bound variance and control resources, with numerical simulations validating the approach up to sizable qubit numbers. Practically, this enables efficient, SPAM-tolerant noise characterization and isolation of coherent noise and crosstalk in near-term quantum devices.

Abstract

We present a quantum process-tomography protocol based on a low-degree ansatz for the quantum channel, i.e. when it can be expressed as a fixed-degree polynomial in terms of Pauli operators. We demonstrate how to perform tomography of such channels with a logarithmic amount of effort relative to the size of the system, by employing random state preparation and measurements in the Pauli basis. We extend the applicability of the protocol to channels consisting of a layer of quantum gates with a polylogarithmic number of non-Clifford gates, followed by a low-degree noise channel. Rather than inverting the layer of quantum gates on the hardware-which would introduce additional errors-we instead carry out the inversion in classical postprocessing, while adding to the sample complexity a factor at most polynomial in system size. Numerical simulations support our theoretical findings and demonstrate the feasibility of our method.

Figures (7)

• Figure 1: Schematic summary of the proposed tomography protocol. A random set of single-qubit Pauli eigenstates is initialized, evolved through the noisy channel $\mathcal{C}_{\mathcal{U}}$ and measured in a random Pauli basis. The classical outcome of the measurements is then sent to a classical computer that uses them alongside the classical description of $\mathcal{U}$ (e.g., its matrix representation in the computational basis) to compute the estimated process matrix $\boldsymbol{\chi}^{\mathcal{E}_{\mathcal{U}}}$ by averaging over the function $\boldsymbol{G}^{\mathcal{U}}$ which converges entrywise to $\boldsymbol{\chi}^{\mathcal{E}_{\mathcal{U}}}$. The final output of the protocol is then the low-degree approximation of the noisy part of the implementation $\mathcal{E}_{\mathcal{U}}$.
• Figure 2: Numerical computation of the variance of some entries of $\boldsymbol{G}$ for an error channel with Kraus operators of the form \ref{['single_qubit']}. The computation is carried for the entries corresponding to $\vec{\alpha} = \vec{\beta} = (0,\ldots,0)$ (top panel) and $\vec{\alpha} = (0,\ldots,0)$, $\vec{\beta} = (x,0,\ldots,0)$ (bottom panel).
• Figure 3: Number of samples needed to reach a precision of $0.05$ in the reconstruction of the entry corresponding to $\vec{\alpha} = (\alpha_0,\ldots,0)$ and $\vec{\beta} = (\beta_0,\ldots,0)$ for (from top to bottom) $(\alpha_0,\beta_0) = (0,0),(1,1),(0,1)$. We report the results for two systems: (a) single iSWAP gate and (b) layer of iSWAP gates. We considered the convergence to be reached if for $500$ consecutive samples $\abs{\hat{\chi}_{\vec{\alpha}\vec{\beta}}-\chi_{\vec{\alpha}\vec{\beta}}} < 0.05$, where with $\hat{\chi}_{\vec{\alpha}\vec{\beta}}$ we mean the process matrix resulting from sampling. Every point in the plots is the result of an average over 10 independent samplings and the error bar shows the standard deviation from the mean. Below each plot is a representation of the corresponding circuit used in the numerics.
• Figure 4: Number of samples needed to reach a precision of $0.05$ in the reconstruction of the entry corresponding to $\vec{\alpha} = \vec{\beta} = (0,\ldots,0)$ for a layer of iSWAP gates. We considered the convergence to be reached if for $100$ consecutive samples $\abs{\hat{\chi}_{\vec{\alpha}\vec{\beta}}-\chi_{\vec{\alpha}\vec{\beta}}} < 0.05$, where with $\hat{\chi}_{\vec{\alpha}\vec{\beta}}$ we mean the process matrix resulting from sampling. In blue the data obtained using the optimized function $\textit{\bf{g}}^{\min}$ and in orange the data from using $\textit{\bf{g}}^{\mathrm{sh}}$. Every point in the plot is the result of an average over 10 independent samplings and the error bar shows the standard deviation from the mean.
• Figure 5: Variance of the entry of $\boldsymbol{G}$ when a layer of iSWAP gates is applied, followed by an error channel with error parameters decaying exponentially as defined in Eq. \ref{['err_ch_sq']}. The computation is carried for the entries corresponding to $\vec{\alpha} = \vec{\beta} = (0,\ldots,0)$ (top panel) and $\vec{\alpha} = (0,\ldots,0)$, $\vec{\beta} = (x,0,\ldots,0)$ (bottom panel).

Theorems & Definitions (12)

• Lemma 1
• proof
• Lemma 2: Bohnenblust--Hille inequality for trace nonincreasing maps
• proof
• Theorem 1
• proof
• Lemma 3
• proof
• Theorem 2
• proof