638 research outputs found
Hybrid Codes
A hybrid code can simultaneously encode classical and quantum information
into quantum digits such that the information is protected against errors when
transmitted through a quantum channel. It is shown that a hybrid code has the
remarkable feature that it can detect more errors than a comparable quantum
code that is able to encode the classical and quantum information. Weight
enumerators are introduced for hybrid codes that allow to characterize the
minimum distance of hybrid codes. Surprisingly, the weight enumerators for
hybrid codes do not obey the usual MacWilliams identity.Comment: 5 page
Quantum weight enumerators
In a recent paper, Shor and Laflamme (see Phys. Rev. Lett., vol.78, p.1600-2, 1997) defined two "weight enumerators" for quantum error-correcting codes, connected by a MacWilliams transform, and used them to give a linear programming bound for quantum codes. We introduce two new enumerators which, while much less powerful at producing bounds, are useful tools nonetheless. The new enumerators are connected by a much simpler duality transform, clarifying the duality between Shor and Laflamme's enumerators. We also use the new enumerators to give a simpler condition for a quantum code to have specified minimum distance, and to extend the enumerator theory to codes with block size greater than 2
Quantum weight enumerators and tensor networks
We examine the use of weight enumerators for analyzing tensor network
constructions, and specifically the quantum lego framework recently introduced.
We extend the notion of quantum weight enumerators to so-called tensor
enumerators, and prove that the trace operation on tensor networks is
compatible with a trace operation on tensor enumerators. This allows us to
compute quantum weight enumerators of larger codes such as the ones constructed
through tensor network methods more efficiently. We also provide an analogue of
the MacWilliams identity for tensor enumerators.Comment: 21 pages, 3 figures. Sets up the tensor enumerator formalis
Linear programming bounds for quantum amplitude damping codes
Given that approximate quantum error-correcting (AQEC) codes have a
potentially better performance than perfect quantum error correction codes, it
is pertinent to quantify their performance. While quantum weight enumerators
establish some of the best upper bounds on the minimum distance of quantum
error-correcting codes, these bounds do not directly apply to AQEC codes.
Herein, we introduce quantum weight enumerators for amplitude damping (AD)
errors and work within the framework of approximate quantum error correction.
In particular, we introduce an auxiliary exact weight enumerator that is
intrinsic to a code space and moreover, we establish a linear relationship
between the quantum weight enumerators for AD errors and this auxiliary exact
weight enumerator. This allows us to establish a linear program that is
infeasible only when AQEC AD codes with corresponding parameters do not exist.
To illustrate our linear program, we numerically rule out the existence of
three-qubit AD codes that are capable of correcting an arbitrary AD error.Comment: 5 page
Semidefinite programming bounds on the size of entanglement-assisted codeword stabilized quantum codes
In this paper, we explore the application of semidefinite programming to the
realm of quantum codes, specifically focusing on codeword stabilized (CWS)
codes with entanglement assistance. Notably, we utilize the isotropic subgroup
of the CWS group and the set of word operators of a CWS-type quantum code to
derive an upper bound on the minimum distance. Furthermore, this
characterization can be incorporated into the associated distance enumerators,
enabling us to construct semidefinite constraints that lead to SDP bounds on
the minimum distance or size of CWS-type quantum codes. We illustrate several
instances where SDP bounds outperform LP bounds, and there are even cases where
LP fails to yield meaningful results, while SDP consistently provides tight and
relevant bounds. Finally, we also provide interpretations of the Shor-Laflamme
weight enumerators and shadow enumerators for codeword stabilized codes,
enhancing our understanding of quantum codes.Comment: 20 pages, 1 tabl
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