From 3f8e38ea065e66e9a6ae81c49051818ba85f8d4b Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Thu, 9 Jan 2025 11:34:27 -0500 Subject: [PATCH] pure distance --- .../universally_optimal/univ_opt_q-ary.yml | 2 +- .../union_stabilizer/cws/cws.yml | 2 +- codes/quantum/qubits/qubits_into_qubits.yml | 24 ++++++++++--------- .../qubits/stabilizer/qubit_stabilizer.yml | 6 +++++ 4 files changed, 21 insertions(+), 13 deletions(-) diff --git a/codes/classical/q-ary_digits/universally_optimal/univ_opt_q-ary.yml b/codes/classical/q-ary_digits/universally_optimal/univ_opt_q-ary.yml index 63e32b334..0eb986ddf 100644 --- a/codes/classical/q-ary_digits/universally_optimal/univ_opt_q-ary.yml +++ b/codes/classical/q-ary_digits/universally_optimal/univ_opt_q-ary.yml @@ -11,7 +11,7 @@ name: 'Universally optimal \(q\)-ary code' introduced: '\cite{manual:{V. I. Levenshtein. Bounds for packings of metric spaces and some of their applications. Problemy Kibernet, 40 (1983), 43-110.},doi:10.1109/18.412678,doi:10.1007/BF00053379,preset:HPLevBounds,doi:10.1007/s10623-016-0286-4,doi:10.1109/18.915662,arxiv:1212.1913}' description: | - A binary or \(q\)-ary code that (weakly) minimizes all completely monotonic potentials on binary space \cite{arxiv:1212.1913}. + A binary or \(q\)-ary code that (weakly) minimizes all completely monotonic potentials on Hamming space \cite{arxiv:1212.1913}. All codes that attain the linear programming (LP) bound by Delsarte \cite{manual:{P. Delsarte, “Bounds for unrestricted codes, by linear programming,” Philips Research Reports, vol. 27, pp. 272–289, 1972}} are universally optimal \cite{arxiv:1212.1913}. Such codes are called \textit{LP universally optimal} or \textit{extremal}. diff --git a/codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/cws.yml b/codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/cws.yml index abe851fbb..5b20999eb 100644 --- a/codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/cws.yml +++ b/codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/cws.yml @@ -28,7 +28,7 @@ description: | protection: | Code distance \(\mathcal{Q} = ( \mathcal{G},\mathcal{C}) \) is upper bounded by the distance of the classical code \(\mathcal{C} \). - The \hyperref[code:qubits_into_qubits]{diagonal distance} is upper bounded by \(\delta + 1\), where \(\delta\) is the minimum degree of \(\mathcal{G}\) \cite{arxiv:2107.11286}. + The \hyperref[code:qubits_into_qubits]{pure distance} is upper bounded by \(\delta + 1\), where \(\delta\) is the minimum degree of \(\mathcal{G}\) \cite{arxiv:0712.1979,arxiv:2107.11286}. Some bounds on the distance are provided in Ref. \cite{arxiv:1108.5490}. features: diff --git a/codes/quantum/qubits/qubits_into_qubits.yml b/codes/quantum/qubits/qubits_into_qubits.yml index 736f097ff..8d323d788 100644 --- a/codes/quantum/qubits/qubits_into_qubits.yml +++ b/codes/quantum/qubits/qubits_into_qubits.yml @@ -25,11 +25,11 @@ protection: | As a result, qubit codes cannot tolerate adversarial errors on more than \((1-R)/4\) registers, where \(R = \log_2 K/n\) is the code rate. \subsection{Pauli-string error basis} + \label{topic:pauli} A convenient and often considered error set is the \textit{Pauli error} or \textit{Pauli string} basis. \begin{defterm}{Pauli strings} - \label{topic:pauli} For a single qubit, this set consists of products of powers of the Pauli matrices \begin{align} X=\begin{pmatrix}0 & 1\\ @@ -46,10 +46,6 @@ protection: | The Pauli error set is a unitary and Hermitian basis for linear operators on the multi-qubit Hilbert space that is orthonormal under the Hilbert-Schmidt inner product; it is a prototypical \hyperref[topic:nice-error-basis]{nice error basis}. The distance associated with this set is often the minimum weight of a Pauli string that implements a nontrivial logical operation in the code. - The minimum weight of a Pauli error that has a non-zero expectation value for some code basis state is called the \textit{diagonal distance} \cite{arxiv:0712.1979,arxiv:2107.11286} (see also pure distance \cite{arxiv:2107.14252}). - Codes whose distance is greater than the diagonal distance are \hyperref[topic:degeneracy]{degenerate}. - \hyperref[topic:degeneracy]{Degenerate} codes admit undetectable Pauli errors (i.e., errors whose projection into the codespace is nonzero) of weight less than the code distance (i.e., the projection satisfies the \term{Knill-Laflamme conditions}). - \subsection{Noise channels} A quantum channel that admits a set of Pauli strings as its Kraus operators is called a \textit{Pauli channel}, and such channels are typically more tractable than the more general, non-Pauli channels. @@ -57,10 +53,10 @@ protection: | Relevant non-Pauli channels are \hyperref[topic:ad]{AD} noise, erasure (which maps all qubit states into a third state \(|e\rangle\) outside of the qubit Hilbert space), and biased erasure (in which case only the \(|1\rangle\) qubit state is mapped to \(|e\rangle\)). Noise can be correlated in space or in time, with the latter being an example of a non-Markovian phenomenon \cite{arxiv:quant-ph/0505153,arxiv:2012.01894}. - \subsection{Quantum weight enumerators} + \subsection{Quantum weight enumerators and pure distance} + \label{topic:quantum-weight-enumerator} \begin{defterm}{Quantum weight enumerator} - \label{topic:quantum-weight-enumerator} Determining protection and bounds on code parameters can also be done using the code's Shor-Laflamme \textit{quantum weight enumerator} \cite{arxiv:quant-ph/9610040} (cf. \hyperref[topic:weight-enumerator]{weight enumerators}) \begin{align} \begin{split} @@ -82,14 +78,20 @@ protection: | It gives rise to quantum linear programming (LP) bounds \cite{arxiv:quant-ph/9611001,arxiv:quant-ph/9709049}; see the book \cite{preset:GottesmanBook}. The distance \(d\) of a code is the smallest \(j=d\) at which \(A_j \neq B_j\) \cite{arxiv:quant-ph/9906126}. - Such a code is called \textit{pure} if \(A_j = B_j = 0\) for all \(j < d\); otherwise, the code is called \textit{impure}. - \hyperref[topic:degeneracy]{Degeneracy} is sufficient but not necessary for impurity \cite{preset:GottesmanBook}. - - Other types of quantum weight enumerators are the Rains unitary enumerators \cite{ arXiv:quant-ph/9612015} and the \textit{Rains shadow enumerators} \cite{arxiv:quant-ph/9611001} (see also \cite{arxiv:quant-ph/0406063}), with the latter related to Bell sampling \cite{arxiv:2408.16914}. + A code is called \textit{pure} if \(A_j = 0\) for all \(1 < j < d\); otherwise, the code is called \textit{impure}. + The \textit{pure distance} \cite{arxiv:2107.14252} (a.k.a. diagonal distance \cite{arxiv:0712.1979}) \(d_{\smallsetminus}\) is the smallest \(1 < j=d_{\smallsetminus}\) at which \(A_j > 0\). + Codes for which \(d_{\smallsetminus} < d\) are impure, otherwise they are pure. + For impure codes, there exists a Pauli error of weight less than the code distance that has a non-zero expectation value with respect to a code state. + + Degenerate qubit codes are impure, but impure codes may not be degenerate \cite{preset:GottesmanBook}. + There are subtleties with defining \hyperref[topic:degeneracy]{degeneracy} for non-stabilizer qubit codes with even distance \cite{preset:GottesmanBook}. + + Other types of quantum weight enumerators are the Rains unitary enumerators \cite{arXiv:quant-ph/9612015} and the \textit{Rains shadow enumerators} \cite{arxiv:quant-ph/9611001} (see also \cite{arxiv:quant-ph/0406063}), with the latter related to Bell sampling \cite{arxiv:2408.16914}. These notions can be generalized to qudit codes and other error bases \cite{doi:10.1016/j.aam.2020.102085,arxiv:2211.02756,arxiv:2308.05152}. There are techniques to compute them for general codes \cite{arxiv:2308.05152}. Semidefinite programming (SDP) hierarchies and a quantum Delsarte bound have been developed \cite{arxiv:2408.10323}. + features: rate: 'Exact two-way assisted capacities have been obtained for the erasure and dephasing channels \cite{arxiv:1510.08863}. There are many bounds on the quantum capacity of the depolarizing channel (e.g., \cite{arxiv:quant-ph/0607039}); see review \cite{arxiv:1801.02019}.' transversal_gates: diff --git a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml index 638004796..cba180c27 100644 --- a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml +++ b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml @@ -107,6 +107,12 @@ protection: | Define the normalizer \(\mathsf{N(S)}\) of \(\mathsf{S}\) to be the set of all Pauli operators that commute with all \(S\in\mathsf{S}\). A stabilizer code can correct a Pauli error set \({\mathcal{E}}\) if and only if \(E^\dagger F \notin \mathsf{N(S)}\setminus \mathsf{S}\) for all \(E,F \in {\mathcal{E}}\). + There are subtleties with defining \hyperref[topic:degeneracy]{degeneracy} for non-stabilizer qubit codes with even distance \cite{preset:GottesmanBook}, but they are resolved for stabilizer codes. + A stabilizer code is a \hyperref[topic:degeneracy]{degenerate} with respect to \(\mathcal{E}\) if and only if \(E^\dagger F \in \mathsf{N(S)}\) for some Pauli strings \(E,F \in \mathcal{E}\). + As a distance-\(d\) code, a stabilizer code is degenerate if it admits a non-identity stabilizer whose weight is lower than the distance \cite{preset:GottesmanBook}. + Since that stabilizer is in the normalizer, a stabilizer code is degenerate if and only if it is \hyperref[topic:quantum-weight-enumerator]{impure}. + The \hyperref[topic:quantum-weight-enumerator]{pure distance} of a stabilizer code is the minimum weight of a non-identity stabilizer. + \begin{defterm}{Cleaning lemma} \label{topic:cleaning-lemma} If all logical operators act trivially on some subset of qubits in a stabilizer code, then any logical Pauli operator can be represented on the complementary qubit subset via a stabilizer.