Quantum errorcorrecting 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 CN ×N as a 2m × 2m binary sym plectic matrix, where N = 2m. We show that for an [m, m − k] stabilizer code every logical Clifford operator has 2k(k+1)/2 symplectic solutions, and we enumerate them efficiently using symplectic transvections. The desired circuits are then obtained by writing each of the solutions as a product of elementary symplectic matrices. For a given operator, our assembly of all of its physical realizations enables optimization over them with respect to a suitable metric. 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 well known [6, 4, 2] code. Programs implementing our algorithms can be found at https://github.com/nrenga/symplecticarxiv18a.
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A Polynomial Degree Bound on Equations for Nonrigid Matrices and Small Linear Circuits
We show that there is an equation of degree at most poly( n ) for the (Zariski closure of the) set of the nonrigid matrices: That is, we show that for every large enough field 𝔽, there is a nonzero n 2 variate polynomial P ε 𝔽[ x 1, 1 , ..., x n, n ] of degree at most poly( n ) such that every matrix M that can be written as a sum of a matrix of rank at most n /100 and a matrix of sparsity at most n 2 /100 satisfies P(M) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer, and Landsberg [ 9 ] and improves the best upper bound known for this problem down from exp ( n 2 ) [ 9 , 12 ] to poly( n ). We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n 2 /200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [ 21 ] to construct lowdegree “universal” maps for nonrigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a lowdegree annihilating polynomial completes the proof. As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [ 11 ].
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 Award ID(s):
 1705028
 NSFPAR ID:
 10389722
 Date Published:
 Journal Name:
 ACM Transactions on Computation Theory
 Volume:
 14
 Issue:
 2
 ISSN:
 19423454
 Page Range / eLocation ID:
 1 to 14
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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