From 8c62385adb4cba9549b81832dc687fc1dac4bf36 Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Thu, 9 Jan 2025 12:04:42 -0500 Subject: [PATCH] ~ --- .../properties/block/tensor_network/single_tensor/ame.yml | 4 +++- codes/quantum/qubits/qubits_into_qubits.yml | 4 ++-- codes/quantum/qubits/stabilizer/data_syndrome.yml | 3 +++ codes/quantum/qubits/stabilizer/qubit_stabilizer.yml | 4 ++-- codes/quantum/qubits/subsystem/subsystem_stabilizer.yml | 2 ++ 5 files changed, 12 insertions(+), 5 deletions(-) diff --git a/codes/quantum/properties/block/tensor_network/single_tensor/ame.yml b/codes/quantum/properties/block/tensor_network/single_tensor/ame.yml index 7f51df949..ab0792ccd 100644 --- a/codes/quantum/properties/block/tensor_network/single_tensor/ame.yml +++ b/codes/quantum/properties/block/tensor_network/single_tensor/ame.yml @@ -52,7 +52,9 @@ relations: - code_id: galois_grs detail: 'GRS codes can yield perfect tensors via a generalized Hermitian construction \cite{arxiv:1801.09623,arxiv:1812.04057}.' - code_id: quantum_secret_sharing - detail: 'Perfect tensors are useful for quantum secret sharing \cite{arxiv:1204.2289}.' + detail: 'Perfect tensors are useful for quantum secret sharing and state teleportation \cite{arxiv:1204.2289}.' + - code_id: qubit_stabilizer + detail: 'The codespace of a qubit stabilizer code with \hyperref[topic:quantum-weight-enumerator]{pure distance} \(d_{\smallsetminus}\) is a \((d_{\smallsetminus}-1)\)-uniform space.' # Begin Entry Meta Information diff --git a/codes/quantum/qubits/qubits_into_qubits.yml b/codes/quantum/qubits/qubits_into_qubits.yml index 8d323d788..411b2b735 100644 --- a/codes/quantum/qubits/qubits_into_qubits.yml +++ b/codes/quantum/qubits/qubits_into_qubits.yml @@ -77,11 +77,11 @@ protection: | The weight enumerator and its dual satisfy the \textit{quantum MacWilliams identity} \cite{arxiv:quant-ph/9610040}; see \cite[Ch. 7]{preset:GottesmanBook}. 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}. + The distance \(d\) of a code is the smallest \(1 0\). 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. + For impure codes, there exists a Pauli error of weight less than the \(d\) 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}. diff --git a/codes/quantum/qubits/stabilizer/data_syndrome.yml b/codes/quantum/qubits/stabilizer/data_syndrome.yml index dc09f7f5f..8a183a0aa 100644 --- a/codes/quantum/qubits/stabilizer/data_syndrome.yml +++ b/codes/quantum/qubits/stabilizer/data_syndrome.yml @@ -34,6 +34,9 @@ features: fault_tolerance: - 'Shor error correction \cite{arxiv:quant-ph/9605011,arxiv:quant-ph/9605031}, in which fault tolerance against syndrome extraction errors is ensured by simply repeating syndrome measurements \(\ell\) times, can be recast as a QDS code whose underlying matrix \(A\) is the identity matrix \(I_m\) repeated \(\ell\) times \cite{arxiv:1907.01393}.' +notes: + - 'QDS codes can be used to estimate physical Pauli noise up to their \hyperref[topic:quantum-weight-enumerator]{pure distance} \cite{arxiv:2107.14252}, and logical Pauli noise for any correctable physical noise \cite{arxiv:2209.09267}.' + relations: parents: diff --git a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml index cba180c27..ebcbccd8f 100644 --- a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml +++ b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml @@ -111,7 +111,7 @@ protection: | 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. + The \hyperref[topic:quantum-weight-enumerator]{pure distance} of a qubit stabilizer code is the minimum weight of a non-identity stabilizer. \begin{defterm}{Cleaning lemma} \label{topic:cleaning-lemma} @@ -212,7 +212,7 @@ notes: - 'See Quantum Codes qubit stabilizer database, maintained by N. Aydin, P. Liu, and B. Yoshino \cite{arxiv:2106.12065,arxiv:2108.03567}, at this \href{https://quantumcodes.info/}{website}.' - 'Entanglement purification protocols with qubit stabilizer codes are related to quantum key distribution (QKD) \cite{arxiv:quant-ph/0209091}. There is a correspondence between stabilizer codes and bilocal Clifford entanglement distillation circuits \cite{arxiv:2303.11465}.' - 'The overlap between any stabilizer codeword and any \(n\)-qubit product state is at most \(2/2^d\) \cite[Thm. 2]{arxiv:2405.01332}.' - - 'Qubit stabilizer codes can be used to estimate Pauli noise \cite{arxiv:2107.14252}.' + - 'Qubit stabilizer codes can be used to estimate physical Pauli noise up to their \hyperref[topic:quantum-weight-enumerator]{pure distance} \cite{arxiv:2107.14252}, and logical Pauli noise for any correctable physical noise \cite{arxiv:2209.09267}.' - 'The stabilizer formalism has been gamified \cite{arxiv:2405.06795}.' - 'Codes can be found via genetic algorithms \cite{arxiv:2409.13017}.' diff --git a/codes/quantum/qubits/subsystem/subsystem_stabilizer.yml b/codes/quantum/qubits/subsystem/subsystem_stabilizer.yml index 151dc503b..f4def206a 100644 --- a/codes/quantum/qubits/subsystem/subsystem_stabilizer.yml +++ b/codes/quantum/qubits/subsystem/subsystem_stabilizer.yml @@ -76,6 +76,8 @@ features: notes: - 'See Ref. \cite{arxiv:1009.2203} for algorithms and lists of possible tilings of particular subsystem codes.' + - 'Subsystem qubit stabilizer codes can be used to estimate logical Pauli noise for any correctable physical noise \cite{arxiv:2209.09267}.' + relations: parents: