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Lower Bounds on Stabilizer Rank

Shir Peleg, Amir Shpilka, Ben Lee Volk
2022 Quantum  
;(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(n) of Bravyi, Smith and Smolin \cite{BSS16}.  ...  This is the first non-trivial lower bound for approximate stabilizer rank.Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of F2n, and we use  ...  Acknowledgements The third author would like to thank Andru Gheorghiu for introducing him to the notion of stabilizer rank.  ... 
doi:10.22331/q-2022-02-15-652 fatcat:2jnjwsn3pjghjpniikcb4kqpp4

Lower Bounds on Stabilizer Rank

Shir Peleg, Ben Lee Volk, Amir Shpilka, Mark Braverman
2022
We prove a lower bound of Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(√n) of Bravyi, Smith and Smolin [Bravyi et al., 2016].  ...  This is the first non-trivial lower bound for approximate stabilizer rank.  ...  Our results: Improved Lower Bounds on Stabilizer Rank and Approximate Stabilizer Rank Our first result is an improved lower bound on χ(H ⊗n ). ▶ Theorem 1. χ(H ⊗n ) = Ω(n).  ... 
doi:10.4230/lipics.itcs.2022.110 fatcat:z56zdjhygnasxay2d5u5hoijey

Lower Bounds on Stabilizer Rank [article]

Shir Peleg, Amir Shpilka, Ben Lee Volk
2021
We prove a lower bound of $Ω(n)$ on the stabilizer rank of such states, improving a previous lower bound of $Ω(\sqrt{n})$ of Bravyi, Smith and Smolin (arXiv:1506.01396).  ...  This is the first non-trivial lower bound for approximate stabilizer rank.  ...  Acknowledgements The third author would like to thank Andru Gheorghiu for introducing him to the notion of stabilizer rank.  ... 
doi:10.48550/arxiv.2106.03214 fatcat:ppy3ppuytfbqrhfisriwbphera

New techniques for bounding stabilizer rank [article]

Benjamin Lovitz, Vincent Steffan
2022 arXiv   pre-print
(and simplified) proofs of the best-known asymptotic lower bounds on stabilizer rank and approximate stabilizer rank, up to a log factor.  ...  In this work, we present number-theoretic and algebraic-geometric techniques for bounding the stabilizer rank of quantum states.  ...  Lower bounds on stabilizer rank and approximate stabilizer rank In Section 3, we refine a number-theoretic theorem of Moulton to prove lower bounds on stabilizer rank and approximate stabilizer rank [  ... 
arXiv:2110.07781v2 fatcat:gxm5vsg2dvafhbwtrmxxnxvrzu

Ranking DMUs based on efficiency stability
English

Hosseinzadeh Lotfi Farhad, Jahanshahloo Gholamreza, Vaez-ghasemi Mohsen, Moghaddas Zohreh
2014 African Journal of Business Management  
The mentioned upper and lower bound of efficiency interval is obtained from the optimistic and pessimistic viewpoints.  ...  In this paper considering the above issues a method for ranking units based on efficiency intervals is presented. With an application the clarity of the proposed procedure will be demonstrated.  ...  But for unit 8, this deference is +2 which means the first rank, based on the upper bound, is better than that of the second one, which is based on the upper and lower efficiencies.  ... 
doi:10.5897/ajbm11.2008 fatcat:srnij2aijjfhlkd37qjq3vwxfe

Entanglement of stabilizer codewords [article]

Xiao-yu Chen, Li-zhen Jiang
2011 arXiv   pre-print
The entanglement upper and lower bounds are determined with the generators of code. The entanglement of dual-containing CSS codes, Gottesman codes and the related codes are given.  ...  The geometric measure, the logarithmic robustness and the relative entropy of entanglement are proved to be equal for a stabilizer quantum codeword.  ...  The entanglement lower bound based on bipartite entanglement is reduced to a formula of calculating the ,maximal rank of some matrices.  ... 
arXiv:1008.5337v3 fatcat:zx2cluaixfbidj3o2kecy6gnb4

Page 1488 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
In this paper it is shown that the minimal rank of a bounded perturbation F can increase if we re- strict on so-called feedback perturbations F = BE, where B is a fixed linear bounded operator.  ...  One main result of this paper follows: If S is bounded, then min{rank E| E finite-rank operator and min ind(T — AS + BE) = 0 for 4 € Q} < max{min ind(T —AS)| 4 € Q}+1, and by means of an example it is  ... 

Remote State Estimation with Smart Sensors over Markov Fading Channels [article]

Wanchun Liu, Daniel E. Quevedo, Yonghui Li, Karl Henrik Johansson, Branka Vucetic
2021 arXiv   pre-print
We observe that the stability region in terms of the packet drop probabilities in different channel states can either be convex or concave depending on the transition probability matrix 𝐌.  ...  Our numerical results suggest that the stability conditions for remote estimation may coincide for setups with a smart sensor and with a conventional one (which sends raw measurements to the remote estimator  ...  Thus, one can find η > 0 such that |[Z i ] j,k | 2 is asymptotically lower bounded by ηρ 2i (Z).  ... 
arXiv:2005.07871v2 fatcat:s6grq534hjaijptsf2cxp7fzou

Stability and Generalization of Bipartite Ranking Algorithms [chapter]

Shivani Agarwal, Partha Niyogi
2005 Lecture Notes in Computer Science  
In particular, we derive generalization bounds for bipartite ranking algorithms that have good stability properties.  ...  ; this is in contrast with previous generalization bounds for ranking, which are based on uniform convergence and in many cases cannot be applied to these algorithms.  ...  an instance x if f (x) > f (x ) and is considered to rank x lower than x if f (x) < f (x ).  ... 
doi:10.1007/11503415_3 fatcat:upnmlttjerfbrnpc2ek5zoykrq

Improved upper bounds on the stabilizer rank of magic states [article]

Hammam Qassim, Hakop Pashayan, David Gosset
2021 arXiv   pre-print
The improvement is obtained by establishing a new upper bound on the stabilizer rank of m copies of the magic state |T⟩=√(2)^-1(|0⟩+e^iπ/4|1⟩) in the limit of large m.  ...  We also provide improved upper bounds on the stabilizer rank of symmetric product states |ψ⟩^⊗ m more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford  ...  Though it is not our focus in this work, we note that existing lower bounds on the stabilizer rank of magic states are unsatisfying: the best known unconditional lower bound is χ(T ⊗m ) = Ω(m) [7] .  ... 
arXiv:2106.07740v2 fatcat:o5rfzp32rbgsdmic5v3556tjze

Improved upper bounds on the stabilizer rank of magic states

Hammam Qassim, Hakop Pashayan, David Gosset
2021 Quantum  
The improvement is obtained by establishing a new upper bound on the stabilizer rank of m copies of the magic state |T⟩=2−1(|0⟩+eiπ/4|1⟩) in the limit of large m.  ...  We also provide improved upper bounds on the stabilizer rank of symmetric product states |ψ⟩⊗m more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford  ...  Though it is not our focus in this work, we note that existing lower bounds on the stabilizer rank of magic states are unsatisfying: the best known unconditional lower bound is χ(T ⊗m ) = Ω(m) [7] .  ... 
doi:10.22331/q-2021-12-20-606 fatcat:jr2kutctwzfyznf4jul2gb2goa

Stabilizer rank and higher-order Fourier analysis [article]

Farrokh Labib
2022 arXiv   pre-print
This allows us to import tools from this theory to analyze the stabilizer rank of quantum states. Quite recently, in it was shown that the n-qubit magic state has stabilizer rank Ω(n).  ...  We observe that n-qudit stabilizer states are so-called nonclassical quadratic phase functions (defined on affine subspaces of 𝔽_p^n where p is the dimension of the qudit) which are fundamental objects  ...  We then prove that the lower bound for this rank is also a lower bound for the stabilizer rank.  ... 
arXiv:2107.10551v2 fatcat:zhcywii4pnf5lfiderquyfwaz4

Stabilizer rank and higher-order Fourier analysis

Farrokh Labib
2022 Quantum  
This allows us to import tools from this theory to analyze the stabilizer rank of quantum states.  ...  We observe that n-qudit stabilizer states are so-called nonclassical quadratic phase functions (defined on affine subspaces of Fpn where p is the dimension of the qudit) which are fundamental objects in  ...  We then prove that the lower bound for this rank is also a lower bound for the stabilizer rank.  ... 
doi:10.22331/q-2022-02-09-645 fatcat:gg3pbbsfmfa2xffjbxtdig5ray

Bounds on Distributional Treatment Effect Parameters using Panel Data with an Application on Job Displacement [article]

Brantly Callaway
2020 arXiv   pre-print
This paper develops new techniques to bound distributional treatment effect parameters that depend on the joint distribution of potential outcomes -- an object not identified by standard identifying assumptions  ...  such as selection on observables or even when treatment is randomly assigned.  ...  The upper bound on the QoTT comes from inverting the lower bound of the DoTT, and the lower bound on the QoTT comes from inverting the upper bound on the DoTT.  ... 
arXiv:2008.08117v1 fatcat:6abl57aonzgolj7mu3pps6l4by

Lower Bounds on the Mean-Squared Error of Low-Rank Matrix Reconstruction

Gongguo Tang, Arye Nehorai
2011 IEEE Transactions on Signal Processing  
Applying a Chapman-Robbins type Barankin bound allows us to derive lower bounds on the worst-case scalar MSE.  ...  However, the infinite scalar MSE is achieved only on a set of low-rank matrices with measure zero.  ...  An immediate corollary is the following lower bound on the scalar MSE.  ... 
doi:10.1109/tsp.2011.2161471 fatcat:wrhcfdmoyzhhvcirgfcddeps6i
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