Optimal symplectic Householder transformations for SR decomposition

Salam, A.; Alaidarous, Eman Salem; Farouk, A. El

doi:10.1016/j.laa.2008.02.029

Cited by 18 publications

(16 citation statements)

References 12 publications

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“…A transvection does not correspond to a single elementary Clifford operator. An important result that is involved in the proof of this fact is the following theorem from [20], [22], which we restate here for F 2m 2 since we will build on this result to state and prove Theorem 24. Next assume x, y s = 0.…”

Section: Symplectic Transvectionsmentioning

confidence: 99%

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Synthesis of Logical Clifford Operators via Symplectic Geometry

Rengaswamy

Calderbank

Pfister

et al. 2018

2018 IEEE International Symposium on Information Theory (ISIT)

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Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in C N×N as a partial 2m × 2m binary symplectic matrix, where N = 2 m . We state and prove two theorems that use symplectic transvections to efficiently enumerate all binary symplectic matrices that satisfy a system of linear equations. As an important corollary of these results, we prove that for an [[m, m − k]] stabilizer code every logical Clifford operator has 2 k(k+1)/2 symplectic solutions. The desired physical circuits are then obtained by decomposing each solution as a product of elementary symplectic matrices, each corresponding to an elementary circuit. Our assembly of the possible physical realizations enables optimization over the ensemble with respect to a suitable metric. Furthermore, we show that any circuit that normalizes the stabilizer of the code can be transformed into a circuit that centralizes the stabilizer, while realizing the same logical operation. However, the optimal circuit for a given metric may not correspond to a centralizing solution. Our method of circuit synthesis can be applied to any stabilizer code, and this paper provides a proof of concept synthesis of universal Clifford gates for the [[6, 4, 2]] CSS code. We conclude with a classical coding-theoretic perspective for constructing logical Pauli operators for CSS codes. Since our circuit synthesis algorithm builds on the logical Pauli operators for the code, this paper provides a complete framework for constructing all logical Clifford operators for CSS codes. Programs implementing the algorithms in this paper, which includes routines to solve for binary symplectic solutions of general linear systems and our overall circuit synthesis algorithm, can be found at https://github.com/nrenga/symplectic-arxiv18a.

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Section: Symplectic Transvectionsmentioning

confidence: 99%

“…Hence in this case F 2 F 1 F h2 satisfies x 1 F 2 = y 1 , x 2 F 2 = y 2 . For the case x 2 , y 2 s = 0 we again find a w 2 that satisfies x 2 , w 2 s = y 2 , w 2 s = 1 and set h 21 w 2 + y 2 , h 22 x 2 + w 2 . Then by Theorem 19 we clearly havex 2 F h21 F h22 = y 2 .…”

Section: Generic Algorithm For Synthesis Of Logical Clifford Opermentioning

confidence: 99%

Synthesis of Logical Clifford Operators via Symplectic Geometry

Rengaswamy

Calderbank

Pfister

et al. 2018

2018 IEEE International Symposium on Information Theory (ISIT)

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“…We obtain the algorithm Remark 1. The function sh2 in the body of the algorithm SRMSH may by replaced by the function osh2 (see [10,11]) which presents the best conditioning among all possible choices.…”

Section: End Endmentioning

confidence: 99%

A treatment of breakdowns and near breakdowns in a reduction of a matrix to upper J-Hessenberg form and related topics

Salam

Kahla

2019

Journal of Computational and Applied Mathematics

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The reduction of a matrix to an upper J-Hessenberg form is a crucial step in the SR-algorithm (which is a QR-like algorithm), structure-preserving, for computing eigenvalues and vectors, for a class of structured matrices. This reduction may be handled via the algorithm JHESS or via the recent algorithm JHMSH and its variants.The main drawback of JHESS (or JHMSH) is that it may suffer from a fatal breakdown, causing a brutal stop of the computations and hence, the SR-algorithm does not run. JHESS may also encounter near-breakdowns, source of serious numerical instability.In this paper, we focus on these aspects. We first bring light on the necessary and sufficient condition for the existence of the SR-decomposition, which is intimately linked to J-Hessenberg reduction. Then we will derive a strategy for curing fatal breakdowns and also for treating near breakdowns. Hence, the J-Hessenberg form may be obtained. Numerical experiments are given, demonstrating the efficiency of our strategies to cure and treat breakdowns or near breakdowns.In [12], the algorithm JHSH, based on an adaptation of SRSH, is introduced, for reducing a general matrix, to an upper J-Hessenberg form. Variants of JHSH, named JHOSH, JHMSH and JHMSH2 are then derived, motivated by the numerical stability. The algorithms JHESS as well as JHSH and its variants have O(n 3 ) as complexity.The algorithm JHESS (and also SRDECO), as described in [2] may be subject of a fatal breakdown, causing a brutal stop of the computations. As consequence, the J-Hessenberg reduction can not be computed and the SR-algorithm does not run. Moreover, we demonstrate that the algorithm JHESS may breaks down while a condensed J Hessenberg form exists. These algorithms also may suffer from severe form of near-breakdowns, source of serious numerical instability.In this paper, we restrict ourselves to the study of such aspects, bringing significant insights on SRDECO and JHESS algorithms.We will show derive a strategy for curing fatal breakdowns and treating near breakdowns. To this aim, we first bring light on the SR-decomposition and SRDECO algorithm, in connection with the theory developed by Elsner in [5]. Then, a strategy for remedying to such breakdowns is proposed. The same strategy is used for treating the near-breakdowns. Numerical experiments are given, demonstrating the efficiency of our strategies to cure breakdowns or to treat near breakdowns.The remainder of this paper is organized as follows. Section 2, is devoted to the necessary preliminaries. In the section 3, the algorithms SRDECO, SRSH or SRMSH are presented. Then, we establish a connection between some coefficients of the current matrix produced by the SRDECO algorithm, when applied to a matrix A and the necessary and sufficient condition for the existence of the SR decomposition of A, as given in [5]. In section 4, we recall the algorithms JHESS and JHMSH. We present then an example, for which a fatal breakdown is meet, in JHESS algorithm (also for JHMSH), for reducing the matrix to an upper J-Hessenber...

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“…Unlike Householder QR decomposition, the new algorithm SRSH involves free parameters and advantages may be taken from this fact. It has been demonstrated how these parameters can be de-termined in an optimal way providing an optimal version [9] of the algorithm (SROSH). The error analysis and computational aspects of this algorithm have been studied [10].…”

Section: Introductionmentioning

confidence: 99%