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Synthesis of Logical Clifford Operators via Symplectic Geometry
TLDR
A mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes, and a proof of concept synthesis of universal Clifford gates for the well-known $k(k+1)/2) code is provided.
On Optimality of CSS Codes for Transversal T
TLDR
An algebraic approach is adopted to characterize all stabilizer codes for which transversaltransversal gates preserving the codespace are needed, using Ax’s theorem on residue weights of polynomials as a guide.
Unifying the Clifford hierarchy via symmetric matrices over rings
The Clifford hierarchy is a foundational concept for universal quantum computation (UQC). It was introduced to show that UQC can be realized via quantum teleportation, given access to certain
Belief propagation with quantum messages for quantum-enhanced classical communications
TLDR
This paper derives an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of “belief-propagation with quantum messages” (BPQM), and suggests that a BPQM receiver might attain the Holevo capacity of this BPSK-modulated pure-loss channel.
Logical Clifford Synthesis for Stabilizer Codes
TLDR
This article proposes a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes, and proves two theorems that use symplectic transvections to efficiently enumerate all binary symplectic matrices that satisfy a system of linear equations.
Finite-Length Analysis of Spatially-Coupled Regular LDPC Ensembles on Burst-Erasure Channels
TLDR
This paper develops an analysis of the finite length performance for a single burst per code word and no errors otherwise, and provides new tight lower bounds for the block erasure probability at finite block length and bounds on the coupling parameter for being asymptotically able to recover the burst.
Kerdock Codes Determine Unitary 2-Designs
TLDR
The design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new, which significantly simplifies the description of the design and its translation to circuits.
Classical Coding Problem from Transversal T Gates
TLDR
This paper characterize all stabilizer codes whose code subspaces are preserved under physical T and T† gates, and proves that CSS codes are optimal among non-degenerate stabilizers that support transversal T.
Spatially Coupled LDPC codes affected by a single random burst of erasures
TLDR
It is shown that in the limit of code length, codewords can be recovered successfully if the length of the burst is smaller than some maximum recoverable burst length.
Un-Weyl-ing the Clifford Hierarchy
TLDR
It can be easily seen that, up to multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group, which implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010).
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