In this note, we consider tomography of $n$-qubit quantum states $|\psi \rangle$ that are promised to have stabilizer rank $k$ i.e., admit a stabilizer decomposition of $|\psi\rangle = \sum_{i=1}^k c_i |\phi_i\rangle$ where each $|\phi_i\rangle$ is a stabilizer state. We give a tomography protocol for such states up to trace distance $\varepsilon>0$ running in $\textsf{poly}(n,2^k,1/\varepsilon)$ time and thus can handle $k=O(\log n)$ efficiently. This follows from first showing that the stabilizer extent $\xi(|\psi\rangle)$, which is the minimal $\ell_1$ norm over the coefficients $\{c_i\}$ of any stabilizer decomposition of $|\psi\rangle$, can be bounded as $\xi(|\psi\rangle) \leq 2^{O(k)}$ and improving upon the previous best bound of $k^{O(k)}$. The protocol is then obtained by utilizing a tomography protocol for states with bounded stabilizer extent.
Consider the following optimization problem over stabilizer product states: given an $n$-qubit stabilizer state $\left|\psi\right\rangle$ and a set of single-qubit stabilizer states $S$, maximize $\left|\langle \psi | \phi_1, \ldots, \phi_n \rangle\right|^2$ over single-qubit stabilizer states $\left|\phi_i\right\rangl...
Daniel Grier, Hakop Pashayan, Luke Schaeffer· 0 citations
It is well-known that learning a pure $n$-qubit stabilizer state $|\psi\rangle$ both requires, and can be accomplished with, access to a number of copies of $|\psi\rangle$ linear in $n$. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that $L_\delta(n)$,...
Rebecca Chang, Matthias C. Caro, Martín Larocca et al.· 6 citations
We prove that, for an $n$-qubit system of dimension $d=2^n$, every state satisfying $\operatorname{Tr}(\rho^2)\le 1/(d-a_\ast)$, with $a_\ast=0.458327\cdots$, lies inside the stabilizer polytope and is therefore magic-free. Combining this result with general geometric properties of high-dimensional polytopes, we establ...
Suppose a Hamiltonian $H$ has a unique ground state $|\psi_0\rangle$ with an eigenvalue $E_0$, and we have an estimate $\tilde{E}_0$ such that $|\tilde{E}_0-E_0|\leq\delta$, and there is a gap of at least $3\delta$ between $E_0$ and all other eigenvalues. Suppose we have a unitary $A$ available that can produce a"guidi...
Stacey Jeffery, R. Somma, F. Witteveen et al.· 3 citations
We consider the problem of quantum 1-PCA: given copies of an unknown $n$-qubit mixed state, recover a classical description of its leading eigenvector. Our goal is to do so using non-adaptive and single-qubit measurements. For an $n$-qubit state with top eigenvalue $\lambda$ and spectral gap at least $\Delta>0$, we giv...
A state over time (stote) pairs a state $\rho$ with a channel $\mathcal{E}$ as the Hermitian, generally non-positive operator $(\rho\otimes\mathbf{1})\star J_\mathcal{E}$. Characterizing the convex cone $\widehat{\mathcal{Q}}$ these operators generate was posed as an open problem in~\cite{HoogstederCalsamigliaWinter202...
Matt Hoogsteder-Riera· 0 citations
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