01015nas a2200133 4500008004100000245004800041210004800089260001400137490000700151520064300158100001600801700002700817856003700844 2019 eng d00aValidating and Certifying Stabilizer States0 aValidating and Certifying Stabilizer States c9/16/20190 v993 a

We propose a measurement scheme that validates the preparation of a target n-qubit stabilizer state. The scheme involves a measurement of n Pauli observables, a priori determined from the target stabilizer and which can be realized using single-qubit gates. Based on the proposed validation scheme, we derive an explicit expression for the worse-case fidelity, i.e., the minimum fidelity between the target stabilizer state and any other state consistent with the measured data. We also show that the worse-case fidelity can be certified, with high probability, using O(n) copies of the state of the system per measured observable.

1 aKalev, Amir1 aKyrillidis, Anastasios uhttps://arxiv.org/abs/1808.1078601552nas a2200109 4500008004100000245008900041210006900130520116300199100002701362700001601389856003701405 2018 eng d00aImplicit regularization and solution uniqueness in over-parameterized matrix sensing0 aImplicit regularization and solution uniqueness in overparameter3 a

We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-r positive semi-definite (PSD) matrices in Rn×n, with a sensing mechanism that satisfies the restricted isometry property (RIP). The algorithm we study is that of \emph{factored gradient descent}, where we model the low-rankness and PSD constraints with the factorization UU⊤, where U∈Rn×r. Surprisingly, recent work argues that the choice of r≤n is not pivotal: even setting U∈Rn×n is sufficient for factored gradient descent to find the rank-r solution, which suggests that operating over the factors leads to an implicit regularization. In this note, we provide a different perspective. We show that, in the noiseless case, under certain conditions, the PSD constraint by itself is sufficient to lead to a unique rank-r matrix recovery, without implicit or explicit low-rank regularization. \emph{I.e.}, under assumptions, the set of PSD matrices, that are consistent with the observed data, is a singleton, irrespective of the algorithm used.

1 aKyrillidis, Anastasios1 aKalev, Amir uhttps://arxiv.org/abs/1806.0204601398nas a2200169 4500008004100000245006100041210006000102260001500162520088000177100002701057700001601084700001801100700002601118700002701144700002001171856003701191 2017 eng d00aProvable quantum state tomography via non-convex methods0 aProvable quantum state tomography via nonconvex methods c2017/11/193 a

With nowadays steadily growing quantum processors, it is required to develop new quantum tomography tools that are tailored for high-dimensional systems. In this work, we describe such a computational tool, based on recent ideas from non-convex optimization. The algorithm excels in the compressed-sensing-like setting, where only a few data points are measured from a lowrank or highly-pure quantum state of a high-dimensional system. We show that the algorithm can practically be used in quantum tomography problems that are beyond the reach of convex solvers, and, moreover, is faster than other state-of-the-art non-convex approaches. Crucially, we prove that, despite being a non-convex program, under mild conditions, the algorithm is guaranteed to converge to the global minimum of the problem; thus, it constitutes a provable quantum state tomography protocol.

1 aKyrillidis, Anastasios1 aKalev, Amir1 aPark, Dohuyng1 aBhojanapalli, Srinadh1 aCaramanis, Constantine1 aSanghavi, Sujay uhttps://arxiv.org/abs/1711.02524