We consider a fundamental problem of \emph{mixedness testing}: Given $n$ copies of an $N$-qubit state $\rho$, determine whether $\rho = \mathbb{I}_d/d$ or $\|\rho-\mathbb{I}_d/d\|_1 \geq \varepsilon$ with high probability, where $d = 2^N$. In particular, we focus on performing this task in the practical setting of single-qubit measurements, where measurements are prepared independently on each qubit. We provide a nearly complete picture of single-qubit mixedness tesing by showing $n = \tilde{\Theta}\left(\sqrt{10}^N/\varepsilon^2\right)$. To establish our lower bound, we introduce a measurement-dependent lower bound framework for adaptive single-copy state certification. For the upper bound, we present a randomized Pauli basis measurement protocol, which relies on a new primitive for computationally efficient uniformity testing of correlation-concentrated distributions on the Boolean hypercube. In conjunction, we provide lower and upper bounds for mixedness testing with fixed Pauli measurement protocols.
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.· 4 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...
Learning the full Pauli profile of the virtually distilled quantum state $\rho^m/\text{tr}(\rho^m)$ has so far required exponentially many copies of $\rho$. We show that all $4^n$ squared Pauli moments $[\text{tr}(P\rho^m)]^2$ can be learned to additive error $\varepsilon$ with confidence $1-\delta$ from one $2m$-repli...
Si-Yuan Chen, Congcong Zheng, Kun Wang et al.· 0 citations
We study the two-weight decision version of quantum approximate counting: given oracle access to $x\in\{0,1\}^N$, distinguish $|x|=M$ from $|x|=M+\Delta$ with success probability $1/2+\zeta$. Using the multiplicative adversary method, we prove $\Omega\left(\max\left\{\zeta\sqrt{(N-M)(M+\Delta)}/\Delta,\sqrt{\zeta N/\De...
We consider the problem of approximate cloning of quantum states: given $n$ copies of an unknown state $\rho \in \mathbb{C}^{d \times d}$, prepare an $(n+k)$-copy state with high fidelity to $\rho^{\otimes (n+k)}$. Werner's pure state cloner is the optimal channel for the pure state case, and shows that $n = \Theta(kd/...
Marco Fanizza, Dmitry Grinko, Thilo Scharnhorst et al.· 0 citations
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