Prog. Here, we use them to upper-bound the sample complexity of our compressed tomography scheme. (That is, we bound the errors due to estimating each Pauli expectation value from a finite number Then one can apply theorem 1 to obtain an error bound for our estimate of ρ. Math.

Authorization RequiredLog InOther OptionsBuy Article »Find an Institution with the Article »×Download & SharePDFExportReuse & PermissionsCiting Articles (18)Tweet×ImagesFigure 1Setup for the preparation and measurements of two-photon four-qubit states. Full-text · Article · Sep 2016 Richard KuengHuangjun ZhuDavid GrossRead full-textThe Clifford group fails gracefully to be a unitary 4-design"The Pauli group and Clifford group play a crucial role in quantum Second, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. Eisert, arXiv:1406.3632.Continuous matrix product state tomography of quantum transport experiments:J.

Rev. Our extension of DFE may be of more general interest, since it can be used to efficiently certify any estimate regardless of whether it was obtained using compressed sensing or J. Building on results from coding theory, we give strong evidence suggesting that these orbits actually constitute complex projective 5-designs.

Compressed sensing techniques—measuring an incomplete set of observables, and regularizing the solution to favor low-rank states—lead to much better results. Let be the set of all d2 Pauli operators, i.e. Our main result implies that stabilizer measurements are essentially optimal for distinguishing pure quantum states. Soc.

Phys. 15, 125004 (2013);(realization for mixed states)Quantum field tomography:A. Riofrío, and J. That is, the curve is completely flat as a function of m above a certain cutoff. Usually, one does both of these things.

Proof. These error bounds follow from the RIP as shown by Theorem 2.1 in [30], and a straightforward modification of Lemma 3.2 in [31] to bound the error in trace norm rather than Frobenius Here, our method has an advantage when the unknown quantum process has a small Kraus rank, meaning that it can be expressed using only a few Kraus operators. C Phys. We give various constructions of exact complex projective 4-designs or approximate 4-designs of arbitrarily high precision from Clifford orbits.

Then we have For all 1 ≤ j ≤ k ≤ r, we use DFE to obtain an estimate of the matrix element 〈j|ρ|k〉, up to some additive error jk that is bounded by a Surv. (1960 - present) Russian Acad. Lemma 1. Assume that there are states such that Then the minimax risk as defined in (17) fulfills Proof. Let X be a random variable taking values uniformly in [s]. Phys.

Semicond. (2009 - present) J. In our previous paper, these minimax lower bounds are proved under the trace regression model with Gaussian noise and the noise is assumed to have common variance. Meas. (1980 - 1992) Commun. Also, it seems that the upper bound for the current protocol with respect to r could potentially be improved by a factor of r with a more careful analysis. 5. Numerical simulationsWe

Phys. Liu, and J. Eisert, D. Thus, one can use compressed tomography to get an estimated density matrix , and use DFE to check whether agrees with the true state ρ.

These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. Physical Review Letters™ is a trademark of the American Physical Society, registered in the United States, Canada, European Union, and Japan. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rdlog²d) measurement settings, compared to standard methods that require d² settings. Phys. 14, 085004 (2012).(extension to 2D)A scalable maximum likelihood method for quantum state tomography:T.

Opt. Let t be the number of copies of ρ. Phys. (1934 - present) Res. Phys.

Indeed, with every quantum measurement one can associate a number -- its POVM norm constant -- that quantifies how much the distinguishability of quantum states degrades in the worst case as Circles represent results of Eq. (5), triangles for Eq. (3), and squares for Eq. (2).Reuse & PermissionsFigure 4Numerical results of compressed sampling based state tomography on 4-qubit states of rank r and purity P, Phys. 14 095022 Create citation alert DOI http://dx.doi.org/10.1088/1367-2630/14/9/095022 Buy this article in print Journal RSS feed Sign up for new issue notifications Abstract Intuitively, if a density operator has small rank, Technically, it turns out that in a precise sense, the 4th tensor power of the Clifford group affords only one more invariant subspace than the 4th tensor power of the unitary

Matlab code included with the source files Subjects: Quantum Physics (quant-ph) Journalreference: New J. Lett. (2004 - present) Mater. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. We proposed two methods for state reconstruction, by (1) minimizing entropy and (2) maximizing likelihood.

Eisert, arXiv:1406.3631.Towards experimental quantum field tomography with ultracold atoms:A. Now define the minimax risk where the infimum is over all estimation procedures on t copies with estimator and choice of observables given by Πi. In section 5, we detail our numerical investigations. has s non-zero matrix elements), it requires only O(s log d) settings.

Rev. G: Nucl. Fluoresc. (2013 - present) Metrologia (1965 - present) Modelling Simul. Somma, D.

A similar argument works for the matrix Lasso. 3. Lower boundsHow good are the sample complexities of our algorithms? Read our cookies policy to learn more.OkorDiscover by subject areaRecruit researchersJoin for freeLog in EmailPasswordForgot password?Keep me logged inor log in withPeople who read this publication also read:Article: Quantum information: Show, Our method is easy to implement, since it requires only the ability to prepare eigenstates of Pauli operators and measure Pauli observables. Print.

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Eng. (2009 - present) Inverse Problems (1985 - present) Izv. Now, suppose that the goal is to reconstruct a low-rank state using as few samples as possible.