PI refs

codes/quantum/properties/block/block_quantum.yml

@@ -8,11 +8,15 @@ code_id: block_quantum
 name: 'Block quantum code'
 
 description: |
-  A code constructed in a physical space consisting of a tensor product of \(n\) identical subsystems, e.g., qubits, modular qudits, Galois qudits, or oscillators.
+  A code constructed in a multi-partite quantum system, i.e., a physical space consisting of a tensor product of \(n\) identical subsystems, e.g., qubits, modular qudits, Galois qudits, or oscillators.
 
 protection: |
   Block codes protect from errors acting on a few of the \(n\) subsystems. A block code with \textit{distance} \(d\) detects errors acting on up to \(d-1\) subsystems, and corrects erasure errors on up to \(d-1\) subsystems.
 
+  Noise models for block codes include \textit{stochastic noise}, in which every possible error is assigned a probability.
+  In the case of \textit{local stochastic noise}, the probability decreases rapidly (typically, exponentially) with the number of subsystems that an error acts on.
+  On the other hand, the \textit{adversarial noise} model consists of errors acting on at most a fixed number of subsystems.
+
 features:
   transversal_gates: '\textit{Transversal gates} are logical gates on block codes that can be realized as tensor products of unitary operations acting on subsets of subsystems whose size is independent of \(n\). When the subsets are of size one and the single-subsystem unitaries are identical, then the gates are sometimes called \textit{strongly transversal}.'
 

codes/quantum/properties/block/symmetric/permutation_invariant.yml

@@ -12,15 +12,22 @@ description: 'Block quantum code such that any permutation of the subsystems lea
 
 #Such a code is said to be stabilized by the symmetric group \(S_n\) on \(n\) elements, in a generalization of stabilizer codes to binary codes utilizing (nonabelian) group actions.'
 
-protection: 'Depends on the family. The GNU permutation-invariant family (parameterized by \(t\)) protects against arbitrary weight \(t\) qubit errors and approximately corrects spontaneous decay errors \cite{arXiv:1302.3247,doi:10.1103/PhysRevA.93.042340}. Other related codes protect against amplitude damping \cite{doi:10.1109/TIT.2019.2956142} while admitting a constant number of excitations, and against deletion errors \cite{arxiv:2102.02494,arXiv:2102.03015}.'
+protection: |
+  Permutation invariant codes of distance \(d\) can protect against \(d-1\) deletion errors \cite{arxiv:2102.02494,arXiv:2102.03015}, i.e., erasures of qubits at unknown locations.
+
+  Other protection depends on the code family.
+  The GNU permutation-invariant family (parameterized by \(t\)) protects against arbitrary weight \(t\) qubit errors and approximately corrects spontaneous decay errors \cite{arXiv:1302.3247,doi:10.1103/PhysRevA.93.042340}.
+  Other related codes protect against amplitude damping \cite{doi:10.1109/TIT.2019.2956142} while admitting a constant number of excitations.
 
 features:
 
   encoders:
     - 'With quantum harmonic oscillators (superconducting charge qubits in a ultrastrong coupling regime) in \(O(N)\) as in \cite{doi:10.1103/PhysRevA.99.012335}. Can be done in \(O(N^2)\) steps using quantum circuits \cite{arXiv:1904.07358}, or using geometric phase gates in \(O(N)\) \cite{arxiv:1908.01120}.'
 
   decoders:
-    - 'For a family of codes, using projection, probability amplitude rebalancing, and gate teleportation can be done in \(O(N^2)\) \cite{arXiv:2102.02494}.'
+    - 'Schur-Weyl-transform based decoder for qubit permutation-invariant codes \cite{arxiv:2212.06285}.
+    Here, one first measures the total angular momentum of consecutive pairs of qubits, and then its projection modulo some spacing.
+    Recovery can be performed by applying geometric phase gates \cite{arxiv:quant-ph/0111017} and the quantum Schur transform.'
 
 notes:
   - 'Can be constructed using real polynomials for high-dimensional qudit spaces \cite{doi:10.1016/j.laa.2017.06.031}.'

codes/quantum/spins/gnu_permutation_invariant.yml

@@ -12,15 +12,23 @@ short_name: 'GNU'
 introduced: '\cite{arXiv:1302.3247}'
 
 description: |
-  Can be expressed in terms of Dicke states where the logical states are
+  Can be expressed in terms of Dicke states whose coefficients are square-roots of the binomial distribution.
+  The logical states are
   \begin{align}
   |\overline{\pm}\rangle = \sum_{\ell=0}^{n} \frac{(\pm 1)^\ell}{\sqrt{2^n}} \sqrt{n \choose \ell} |D^m_{g \ell}\rangle~.
   \end{align}
   Here, \(m\) is the number of particles used for encoding \(1\) qubit, and \(g, n \leq m\) are arbitrary positive integers.
+  
   The state \(|D^m_w\rangle\) is a Dicke state -- a normalized permutation-invariant state on \(m\) spin-half systems with \(w\) excitations, i.e., a normalized sum over all basis elements with \(w\) ones and \(m - w\) zeroes.
+  Each Dicke state in the code can be \textit{shifted} by adding a shift \(s\) to both \(m\) and \(w\).
 
 protection: 'Depends on the family. One family which is completely symmetrized versions of Bacon-Shor codes (parameterized by \(t\)) protects against arbitrary weight-\(t\) spin errors. Additionally, codes with large enough length \((t+1)(3t+1)+t\) can approximately correct \(t\) spontaneous decay errors.'
 
+features:
+  decoders:
+    - 'For a family of shifted gnu codes, decoding can be done using projection, probability amplitude rebalancing, and gate teleportation in time \(O(n^2)\) \cite{arXiv:2102.02494}.'
+
+
 relations:
   parents:
     - code_id: spins_into_spins