Classical shadows with arbitrary group representations

Abstract

Classical shadows (CS) has recently emerged as an important framework to efficiently predict properties of an unknown quantum state. A common strategy in CS protocols is to parametrize the basis in which one measures the state by a random group action; many examples of this have been proposed and studied on a case-by-case basis. In this work, we present a unified theory that allows us to simultaneously understand CS protocols based on sampling from general group representations, extending previous approaches that worked in simplified (multiplicity-free) settings. We identify a class of measurement bases which we call "centralizing bases" that allows us to analytically characterize and invert the measurement channel, minimizing classical post-processing costs. We complement this analysis by deriving general bounds on the sample-complexity necessary to obtain estimates of a given precision. Beyond its unification of previous CS protocols, our method allows us to readily generate new protocols based on other groups, or different representations of previously considered ones. For example, we characterize novel shadow protocols based on sampling from the spin and tensor representations of $\textsf{SU}(2)$, symmetric and orthogonal groups, and the exceptional Lie group $G_2$.

Figures (7)

• Figure 1: Schematic overview of our framework. From any choice of a group representation $R:G\to U(\mathcal{H})$ (and a choice of commuting subgroup $H\subseteq G$ with a non-degenerate simultaneous eigenbasis) we obtain a classical-shadows protocol such that a) the resulting measurement channel is constant on the isotypic components of operator space, $\mathcal{M}=\sum_{\lambda} a_\lambda^{H} \mathcal{P}_\lambda^{\mathcal{V}}$ (see Theorem \ref{['thm:mc']}), and therefore channel inversion $\mathcal{M}^{-1}=\sum_{\lambda}(a_\lambda^{H})^{-1} \mathcal{P}_\lambda^{\mathcal{V}}$ is exact, and b) the sample complexity is controlled by the visibility factors $a_\lambda^{H}$ via general variance bounds (see Theorem \ref{['thm:var']}). In this sense, our framework turns group representations into off-the-shelf shadows protocols.
• Figure 2: (a) The subspace $\mathcal{L}^D$ of $G$-diagonal operators are those which map the irreducible $G$-modules $\mathcal{H}^{\eta,i} \subseteq \mathcal{H}$ to themselves; these operator spaces are generally reducible and decompose into irreps. (b) An NDCSE shadows protocol produces unbiased estimates for the expectation values of the operators within the visible space $\mathcal{L}^\mathcal{V}\subseteq\mathcal{L}^D$. For a given operator, the variance of the corresponding estimator (and therefore sample-complexity of the protocol) is a function of the dimension of the irrep to which the operator belongs (see Theorem \ref{['thm:var']}).
• Figure 3: The (squared) shadow norm for Majorana monomials of fixed (even) degree for various values of $n$, when sampling uniformly from the distribution of fermionic Gaussian unitaries. As can be seen by applying Stirling's inequality to Eqs. \ref{['eq:fa']} and \ref{['eq:fgusn']}, monomials of degree $\sim n$ will possess an intractably large shadow norm of order $\mathcal{O}(2^n)$. Monomials of degree of order $\mathcal{O}(1)$ (mod $2n$), on the other hand, can be efficiently learnt.
• Figure 4: In the low-dimensional cases of $d=4,5$ we can visualize the representations that appear in the study of symplectic and orthogonal shadows by means of weight-diagrams (in general, for $\mathfrak{sp}_d$ and $\mathfrak{so}_d$ the diagrams will live in a ${\rm dim\ \mathfrak{h}^*}=\left \lfloor d/2 \right \rfloor$ dimensional space, so that the extent to which they clarify the situation decreases sharply with $d$). The fact that the geometry (in particular, lengths and angles) implied by the diagrams to exist on $\mathfrak{h}^*$ behaves sensibly follows from the general theory of representations of Lie algebras fulton1991representation. In the top row we plot the weights of the standard representations $\mathcal{H}$ of $\mathfrak{sp}_4$, $\mathfrak{so}_4$ and $\mathfrak{so}_5$, and in the following rows the decomposition into irreps of $\mathcal{L}^\mathcal{D}=\mathcal{L}$. Degeneracies are denoted by circles, and the centres of the diagrams represent the weight-zero spaces. In the $\mathfrak{sp}_4$ row we find $a_{{\rm Sym}^2\space\mathcal{H}}=2/10=1/5=a_W$ (with the equality as expected from Appendix \ref{['sec:symp']}); for $\mathfrak{so}_4$, $a_{\Lambda^2\space\mathcal{H}}=2/6\neq 1/9=a_U$, and for $\mathfrak{so}_5$, $a_{\Lambda^2\space\mathcal{H}}=2/10\neq 2/14=a_{U'}$. One can easily check that these agree with our general formulae.
• Figure 5: The elements of a spanning set of the third-order commutant of the orthogonal group defined by Eq. \ref{['eq:ortho_def']}. This figure is based on (and in fact slightly generalizes) Fig. 2 of Ref. west2025real, which depicts the set obtained under the choice $Q=\mathds{1}$. In this appendix we take $Q$ to be defined by Eq. \ref{['eq:q_def']}.

Theorems & Definitions (43)

• Definition 1
• Definition 2
• Lemma 1
• Definition 3
• Definition 4
• Theorem 1
• Theorem 2
• Lemma 2
• Theorem 3
• Lemma 3