Predicting Magic from Very Few Measurements

Abstract

The nonstabilizerness of quantum states is a necessary resource for universal quantum computation, yet its characterization is notoriously demanding. Quantifying nonstabilizerness typically requires an exponential number of measurements and a doubly exponential classical post-processing cost to evaluate its standard monotones. In this work, we show that nonstabilizerness is, to a large extent, in the eyes of the beholder: it can be witnessed and quantified using any set of $m$ $n$-qubit Pauli measurements, provided the set contains anti-commuting pairs. We introduce a general framework that projects the stabilizer polytope onto the subspace defined by these observables and provide an algorithm that estimates magic from Pauli expectation values with runtime exponential in the number of measurements $m$ and polynomial in the number of qubits $n$. By relating the problem to a stabilizer-restricted variant of the quantum marginal problem, we also prove that deciding membership in the corresponding reduced stabilizer polytope is NP-hard. In particular, unless $\mathrm{P} = \mathrm{NP}$, no algorithm polynomial in $m$ can solve the problem in full generality, thus establishing fundamental complexity-theoretic limitations. Finally, we employ our framework to compute nonstabilizerness in different Hamiltonian ground states, demonstrating the practical performance of our method in regimes beyond the reach of existing techniques.

Figures (3)

• Figure 1: Illustration of some of the reduced polytopes for $m \in \{2,3\}$. We draw the polytopes in $[-1,1]^m$, with the axis labels as vertices of the frustration graph drawn in overlap. In (a), we show the two reduced polytopes coming from the two possible frustration graphs, where in red we show the corresponding reduced polytope if the extra condition $P_1P_2=-1$ is added in the commuting case. In (b), we highlight four possibilities: The fully commuting case, where the correlations span the full $[-1,1]^3$, the case where the commuting tuple also satisfies the constraint $P_1 P_2 P_3=-1$, the case of a graph of two edges, and the fully anti-commuting case, that reduces to the definition of $\mathrm{STAB}_1$.
• Figure 2: Parametric evolution of the ground-state expectation values and the reduced Robustness of Magic ($\text{RoM}_{\mathcal{M}}$) for the transverse-field Ising model (TFIM). The plot explicitly illustrates only the first quadrant of the observable plane. The shaded region delineates the boundary of the reduced stabilizer polytope associated with the local measurement set $\mathcal{M}=\{Z_1Z_2, X_1\}$. The solid curves denote the trajectory of the ground state, $\psi_{\text{gs}}(g)$, within the observable space $(\langle Z_1Z_2 \rangle, \langle X_1 \rangle)$ as the transverse field $g$ varies from $0$ to $\infty$ across varying system sizes ($n=3, 6, 9, 12$). As the system size $n$ increases, the maximum value of the reduced robustness concentrates around the critical point $g=1$.
• Figure 3: Comparison between the energy gap and the Reduced RoM measure for the ANNNI and XXZ models. For the ANNNI model, 10 qubits were used, while 9 qubits were used for XXZ model. Column (a) presents the results for the ANNNI model as function of the frustration $k$ and the transverse field $g$. Column (b) details the XXZ model in terms of the magnetic field $h$ and the anisotropy $\Delta$.

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• Theorem 3
• Proposition 4
• proof
• Theorem 5
• Corollary 6
• Definition 1
• Definition 2
• Theorem 7