Moments of quantum channel ensembles

TL;DR

This paper extends the theory of moment operators and t-designs from ensembles of unitaries to ensembles of quantum channels by introducing a channel-centric framework with the t-th order twirl $\widehat{\mathcal{T}}^{(t)}_{\mathcal{C}}$. It defines three reference ensembles—Haar over unitaries, channel-Haar (cHaar) via Stinespring dilations, and the Depolarize ensemble—and proves a hierarchy among their moment-operator norms, showing that noise tends to depolarize channel ensembles while certain non-unital processes can enhance their design properties. The authors derive exact moment operators for these ensembles, reveal a block-orthogonal localized-permutation basis that simplifies spectral analysis, and demonstrate how different noise models influence convergence toward channel designs, including concentration phenomena for expectation values. Through numerical experiments on noisy parametrized circuits, they illustrate depth- and noise-dependent transitions toward Haar-like or Depolarize-like behavior, highlighting a channel-design perspective on noisy barren plateaus. Overall, the work provides a rigorous, operationally meaningful framework for assessing, comparing, and leveraging ensembles of quantum channels in channel-design-aware quantum information tasks and near-term quantum technologies.

Abstract

Moments of ensembles of unitaries play a central role in quantum information theory as they capture the statistical properties of dynamics of systems with some form of randomness. Indeed, concepts such as approximate $t$-designs arise when comparing how close an associated moment operator of a given unitary ensemble is to that of another, reference ensemble. Despite the importance of moment operators, their properties have not been as explored for quantum channels. In this work we develop a theoretical framework to compute moment operators for ensembles of quantum channels, for all moment orders $t$, with a special focus on determining ensembles that can be used as points of reference. By deriving hierarchies between ensembles, via inequalities of their moment operator norms, we give them operational meaning, and define useful concepts such as that of channel $t$-designs. Finally, we perform theoretical and numerical studies which show that different types of noise can decrease the norm of the moment operators (e.g., depolarizing noise), as well as increase it (e.g., amplitude damping), and generalize noise-induced concentration phenomena to channel-design-induced phenomena. Along the way, we find a block-orthogonal basis for permutations, which greatly simplifies our analyses, and may be of independent interest.

Figures (10)

• Figure 1: Schematic representation of our results. (a) We present a framework to study the statistical properties of an experimental setup where, at every run, a channel is sampled from some set according to a given probability distribution. (b) Our formalism centers around the moment operator, as it encodes the statistical properties arising from evolving states according to the ensemble, and performing a measurement at its output; and is also the central object used to define $t$-designs.
• Figure 2: Circuits implementing the reference channel ensembles. (a) The Haar ensemble can be implemented by sending a state $\rho$ through a random unitary uniformly sampled from $\mathcal{U}({d_{\newline{}}})$. b) The cHaar ensemble can be implemented via its Stinespring dilation: Initialize the joint state of the tensor product Hilbert space $\mathcal{H}\otimes\mathcal{E}$ in $\rho\otimes {\lvert{\nu_\mathcal{E}}\rangle} {\langle{\nu_\mathcal{E}}\lvert}$, apply a joint unitary uniformly sampled from $\mathcal{U}(dd_\mathcal{E})$, and trace out the environment. c) The Depolarize ensemble can be applied with a circuit that swaps the state $\rho$ with the maximally mixed state $\frac{I_d}{d}$, and then traces out the registers containing $\rho$.
• Figure 3: Moment operators in a basis that block-diagonalizes the Haar moment operator as a permutation transfer matrix. We show the scaling of the $k$-concatenated $t$-th order moment operators for the Haar (top) and cHaar (bottom) ensembles for $d_{\mathcal{E}} = d_{\newline{}}^{2}$, $k=1$ (left) , $k=3$ (right), and $t=4$. In this basis, the $t!$ elements are sorted by their support, from the smallest identity (top-left corner), to the largest cycle (bottom-right corner). Colors represent the leading-order scaling $l$ with $1/d_{\newline{}}$ of elements $\tau_{\mathcal{C}}^{(t,k)} \sim \mathcal{O}(1/d_{\newline{}}^{l})$.
• Figure 4: Concatenation of random unitaries and noisy channels. We study the $t$-th moment operator of a noisy circuit where we concatenate $k$ layers of random unitaries sampled from $\mathcal{U}$ (blue) with a fixed noise channel $\mathcal{N}$ (brown).
• Figure 5: Effects of noise on $||\mathcal{T}_{\mathcal{C}}^{(2)}(\rho)||^2$. We study two parametrized quantum circuit architectures: a hardware-efficient ansätz (HEA) and a matchgate ansätz (MAT), on $n=7$ qubits and $L\in[L]$ layers. We take the variational parameters uniformly in $[0,2\pi]$, and we simulate the effect of four different types of single-qubit noise channels, namely, local depolarizing, amplitude damping, bit-flip and dephasing nose, with noise strengths $\gamma\in[0,0.3]$. The input states for the HEA and MAT circuits are $\rho_0$ and $\rho_+$, respectively. We also plot the value for the Haar, cHaar with $d_{\mathcal{E}}=d^2$ (i.e., the Lebesgue measure) and Depolarize reference ensembles. These simulations show that certain types of noise can decrease the Hilbert-Schmidt norm of the $2$-order moment operator (e.g., depolarizing), while others can increase it (e.g., amplitude damping).

Theorems & Definitions (47)

• Proposition 1: Properties of Reference Ensembles
• Theorem 1: Hierarchy of moment operator norms
• Theorem 2: Spectrum of cHaar Moment Operator
• Claim 1
• Claim 2
• Theorem 3: Noise-Induced Depolarize $t$-Designs
• Theorem 4: Variance bounds in terms of reference ensemble
• Proposition 2
• Theorem 5: Localized Basis for Permutations
• proof