524 research outputs found
Graphical Structures for Design and Verification of Quantum Error Correction
We introduce a high-level graphical framework for designing and analysing
quantum error correcting codes, centred on what we term the coherent parity
check (CPC). The graphical formulation is based on the diagrammatic tools of
the zx-calculus of quantum observables. The resulting framework leads to a
construction for stabilizer codes that allows us to design and verify a broad
range of quantum codes based on classical ones, and that gives a means of
discovering large classes of codes using both analytical and numerical methods.
We focus in particular on the smaller codes that will be the first used by
near-term devices. We show how CSS codes form a subset of CPC codes and, more
generally, how to compute stabilizers for a CPC code. As an explicit example of
this framework, we give a method for turning almost any pair of classical
[n,k,3] codes into a [[2n - k + 2, k, 3]] CPC code. Further, we give a simple
technique for machine search which yields thousands of potential codes, and
demonstrate its operation for distance 3 and 5 codes. Finally, we use the
graphical tools to demonstrate how Clifford computation can be performed within
CPC codes. As our framework gives a new tool for constructing small- to
medium-sized codes with relatively high code rates, it provides a new source
for codes that could be suitable for emerging devices, while its zx-calculus
foundations enable natural integration of error correction with graphical
compiler toolchains. It also provides a powerful framework for reasoning about
all stabilizer quantum error correction codes of any size.Comment: Computer code associated with this paper may be found at
https://doi.org/10.15128/r1bn999672
Group divisible designs, GBRDSDS and generalized weighing matrices
We give new constructions for regular group divisible designs, pairwise balanced designs, generalized Bhaskar Rao supplementary difference sets and generalized weighing matrices. In particular if p is a prime power and q divides p - 1 we show the following exist;
(i) GDD (2(p2+p+1), 2(p2+p+1), rp2,2p2, λ1 = p2λ, λ2 = (p2-p)r, m=p2+p+1,n=2), r_+1,2;
(ii) GDD(q(p+1), q(p+1), p(q-1), p(q-1),λ1=(q-1)(q-2), λ2=(p-1)(q-1)2/q,m=q,n=p+1);
(iii) PBD(21,10;K),K={3,6,7} and PDB(78,38;K), K={6,9,45};
(iv) GW(vk,k2;EA(k)) whenever a (v,k,λ)-difference set exists and k is a prime power;
(v) PBIBD(vk2,vk2,k2,k2;λ1=0,λ2=λ,λ3=k) whenever a (v,k,λ)-difference set exists and k is a prime power;
(vi) we give a GW(21;9;Z3)
A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes
Some combinatorial designs, such as Hadamard matrices, have been extensively
researched and are familiar to readers across the spectrum of Science and
Engineering. They arise in diverse fields such as cryptography, communication
theory, and quantum computing. Objects like this also lend themselves to
compelling mathematics problems, such as the Hadamard conjecture. However,
complex generalized weighing matrices, which generalize Hadamard matrices, have
not received anything like the same level of scrutiny. Motivated by an
application to the construction of quantum error-correcting codes, which we
outline in the latter sections of this paper, we survey the existing literature
on complex generalized weighing matrices. We discuss and extend upon the known
existence conditions and constructions, and compile known existence results for
small parameters. Some interesting quantum codes are constructed to demonstrate
their value.Comment: 33 pages including appendi
NMR Quantum Computation
In this article I will describe how NMR techniques may be used to build
simple quantum information processing devices, such as small quantum computers,
and show how these techniques are related to more conventional NMR experiments.Comment: Pedagogical mini review of NMR QC aimed at NMR folk. Commissioned by
Progress in NMR Spectroscopy (in press). 30 pages RevTex including 15 figures
(4 low quality postscript images
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