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[error correction zoo]
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<h1>Glossary of terms</h1>
<dl class="glossary-defterm-list"> <dt class="glossary-defterm-term-name"><a href="/c/q-ary_cyclic#defterm-_5cX_28Xq_5cX_29X-ary-cyclic-to-polynomial_20Xcorrespondence"><span class="inline-math">\(q\)</span>-ary-cyclic-to-polynomial correspondence</a></dt><dd class="glossary-defterm-body"> Cyclic codes and their properties can be naturally formulated using the theory of polynomials. Cyclic codes correspond to ideals in a particular polynomial ring. Codewords <span class="inline-math">\(c_1 c_2 \cdots c_n\)</span> of a <span class="inline-math">\(q\)</span>-ary code can be thought of as coefficients in a polynomial <span class="inline-math">\(c_1+c_2 x+\cdots+c_n x^{n-1}\)</span> in the set of polynomials with <span class="inline-math">\(q\)</span>-ary coefficients, <span class="inline-math">\(\mathbb{F}_q[x]\)</span> with <span class="inline-math">\(\mathbb{F}_q=GF(q)\)</span>. Polynomials corresponding to codewords of a linear cyclic code form an ideal (i.e., are closed under multiplication and addition) in the ring <span class="inline-math">\(\mathbb{F}_q[x]/(x^n-1)\)</span> (i.e., the set of equivalence classes of polynomials congruent modulo <span class="inline-math">\(x^n-1\)</span>). Multiplication of a codeword polynomial <span class="inline-math">\(c(x)\)</span> by <span class="inline-math">\(x\)</span> in such a ring corresponds to a cyclic shift of the corresponding codeword string. <a href="/c/q-ary_cyclic#defterm-_5cX_28Xq_5cX_29X-ary-cyclic-to-polynomial_20Xcorrespondence"
> — view in context →</a>
</dd> <dt class="glossary-defterm-term-name"><a href="/c/binary_cyclic#defterm-binary-cyclic-to-polynomial_20Xcorrespondence">binary-cyclic-to-polynomial correspondence</a></dt><dd class="glossary-defterm-body"> Cyclic codes and their properties can be naturally formulated using the theory of polynomials. Cyclic codes correspond to ideals in a particular polynomial ring. Codewords <span class="inline-math">\(c_1 c_2 \cdots c_n\)</span> of a binary code can be thought of as coefficients in a polynomial <span class="inline-math">\(c(x)=c_1+c_2 x+\cdots+c_n x^{n-1}\)</span> in the set of polynomials with binary coefficients, <span class="inline-math">\(\mathbb{F}_2[x]\)</span> with <span class="inline-math">\(\mathbb{F}_2=GF(2)\)</span>. Polynomials corresponding to codewords of a linear cyclic code form an <span class="textit">ideal</span> (i.e., are closed under multiplication and addition) in the ring <span class="inline-math">\(\mathbb{F}_2[x]/(x^n-1)\)</span> (i.e., the set of equivalence classes of polynomials congruent modulo <span class="inline-math">\(x^n-1\)</span>). Multiplication of a codeword polynomial <span class="inline-math">\(c(x)\)</span> by <span class="inline-math">\(x\)</span> in such a ring corresponds to a cyclic shift of the corresponding codeword string. <a href="/c/binary_cyclic#defterm-binary-cyclic-to-polynomial_20Xcorrespondence"
> — view in context →</a>
</dd> <dt class="glossary-defterm-term-name"><a href="/c/qecc_finite#defterm-Knill-Laflamme_20Xconditions">Knill-Laflamme conditions</a></dt><dd class="glossary-defterm-body"> In a finite-dimensional Hilbert space, there are necessary and sufficient conditions for a code to successfully correct a set of errors. These are called the <span class="textit">Knill-Laflamme conditions</span> <a href="#citation-1" class="href-endnote endnote citation">[1]</a><a href="#citation-2" class="href-endnote endnote citation">[2]</a><a href="#citation-3" class="href-endnote endnote citation">[3]</a>. A code defined by a partial isometry <span class="inline-math">\(U\)</span> with code space projector <span class="inline-math">\(\Pi = U U^\dagger\)</span> can correct a set of errors <span class="inline-math">\(\{ E_j \}\)</span> if and only if <span class="display-math env-align">\begin{align}
\Pi E_i E_j^\dagger \Pi = c_{ij}\, \Pi\qquad\text{for all \(i,j\),}
\end{align}</span> where <span class="inline-math">\(c_{ij}\)</span> can be arbitrary numbers. <a href="/c/qecc_finite#defterm-Knill-Laflamme_20Xconditions"
> — view in context →</a>
</dd> <dt class="glossary-defterm-term-name"><a href="/c/css#defterm-CSS-to-homology_20Xcorrespondence">CSS-to-homology correspondence</a></dt><dd class="glossary-defterm-body"> CSS codes and their properties can be formulated in terms of homology theory, yielding a powerful correspondence between codes and chain complexes, the primary homological structures. There exists a many-to-one mapping from size three chain complexes to CSS codes <a href="#citation-4" class="href-endnote endnote citation">[4]</a><a href="#citation-5" class="href-endnote endnote citation">[5]</a><a href="#citation-6" class="href-endnote endnote citation">[6]</a><a href="#citation-7" class="href-endnote endnote citation">[7]</a> that allows one to extract code properties from topological features of the complexes. Codes constructed in this manner are sometimes called <span class="textit">homological CSS codes</span>, but they are equivalent to CSS codes. This mapping of codes to manifolds allows the application of structures from topology to error correction, yielding <a href="/c/generalized_homological_product" class="href-ref ref-code">various QLDPC codes</a> with favorable properties. <a href="/c/css#defterm-CSS-to-homology_20Xcorrespondence"
> — view in context →</a>
</dd></dl>
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<div id="endnotes" class="endnotes sectioncontent"><h2 class="heading-level-2">References</h2><dl class="enumeration citation-list"><dt id="citation-1">[1]</dt><dd>E. Knill, R. Laflamme, and L. Viola, “Theory of Quantum Error Correction for General Noise”, Physical Review Letters <span class="textbf">84</span>, 2525 (2000). <a href="https://doi.org/10.1103/PhysRevLett.84.2525" target="_blank" class="href-href">DOI</a>; <a href="https://arxiv.org/abs/quant-ph/9604034" target="_blank" class="href-href">quant-ph/9604034</a></dd><dt id="citation-2">[2]</dt><dd>J. Preskill. <span class="textit">Lecture notes on Quantum Computation.</span> (1997–2020) <a href="http://theory.caltech.edu/~preskill/ph219/" target="_blank" class="href-href">URL</a></dd><dt id="citation-3">[3]</dt><dd>M. A. Nielsen and I. L. Chuang, <span class="textit">Quantum Computation and Quantum Information</span> (Cambridge University Press, 2009). <a href="https://doi.org/10.1017/CBO9780511976667" target="_blank" class="href-href">DOI</a></dd><dt id="citation-4">[4]</dt><dd>A. Y. Kitaev, “Quantum computations: algorithms and error correction”, Russian Mathematical Surveys <span class="textbf">52</span>, 1191 (1997). <a href="https://doi.org/10.1070/RM1997v052n06ABEH002155" target="_blank" class="href-href">DOI</a></dd><dt id="citation-5">[5]</dt><dd>H. Bombin and M. A. Martin-Delgado, “Homological error correction: Classical and quantum codes”, Journal of Mathematical Physics <span class="textbf">48</span>, 052105 (2007). <a href="https://doi.org/10.1063/1.2731356" target="_blank" class="href-href">DOI</a>; <a href="https://arxiv.org/abs/quant-ph/0605094" target="_blank" class="href-href">quant-ph/0605094</a></dd><dt id="citation-6">[6]</dt><dd>Sergey Bravyi and Matthew B. Hastings, “Homological Product Codes”. <a href="https://arxiv.org/abs/1311.0885" target="_blank" class="href-href">1311.0885</a></dd><dt id="citation-7">[7]</dt><dd>Nikolas P. Breuckmann, “PhD thesis: Homological Quantum Codes Beyond the Toric Code”. <a href="https://arxiv.org/abs/1802.01520" target="_blank" class="href-href">1802.01520</a></dd></dl></div>
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