vbs

codes/quantum/properties/approximate_qecc.yml

@@ -9,7 +9,11 @@ name: 'Approximate quantum error-correcting code (AQECC)'
 short_name: 'AQECC'
 introduced: '\cite{arxiv:quant-ph/9704002,doi:10.1070/RM1997v052n06ABEH002155,arxiv:quant-ph/0307138,arxiv:quant-ph/0503139,arxiv:0907.4207,arxiv:0907.5391,arxiv:1706.09434}'
 
-description: 'Encodes quantum information so that it is possible to approximately recover that information from noise up to an error bound in recovery.'
+description: |
+  Encodes quantum information so that it is possible to approximately recover that information from noise up to an error bound in recovery.
+
+  Many families of approximate block quantum codes become exact in the \(n\to\infty\) limit (see children).
+  More generally, codes that become exact for some parameter values are called \textit{quasi exact} \cite{arxiv:2105.14777}.
 
 protection: |
   Many of the state fidelity conditions that hold exactly for \hyperref[code:qecc_finite]{(exact) QECCs} can be shown to hold up to some error \(\epsilon\) for approximate QECCs.

codes/quantum/properties/hamiltonian/hamiltonian.yml

@@ -18,7 +18,15 @@ description: |
   Ground states of infinite families geometrically local block-code Hamiltonians can sometimes be said to be a particular phase of (quantum) matter.
   A phase is a "region in some parameter space in which the ... states possess properties that can be distinguished from those in other phases" \cite{doi:10.1063/PT.3.1641}.
 
-protection: 'Often determined from the underlying physical properties of the Hamiltonian.'
+  A Hamiltonian whose ground states minimize the energy of each term is called a \textit{frustration-free} Hamiltonian.
+  A Hamiltonian whose terms commute and can be written as orthogonal projectors (i.e., with eigenvalues zero or one) is called \textit{commuting projector} Hamiltonian.
+
+
+protection: |
+  Often determined from the underlying physical properties of the Hamiltonian.
+
+  Ground states of many Hamiltonians can be easily written as tensor-network states or, in 1D, matrix product states (MPS).
+  A no-go theorem states that open-boundary MPS that form a degenerate ground-state space of a gapped local Hamiltonian yield codes with distance that is only constant in the number of qubits \(n\), so MPS excitation ansatze have to be used to achieve a distance scaling nontrivially with \(n\) \cite{arxiv:1902.02115} (see also Ref. \cite{arxiv:1407.3413}).
 
 features:
   encoders:
@@ -27,7 +35,7 @@ features:
 relations:
   parents:
     - code_id: qecc
-    
+
 
 # Begin Entry Meta Information
 _meta:

codes/quantum/qubits/nonstabilizer/eth.yml

@@ -16,13 +16,17 @@ alternative_names:
 # \cite{arxiv:1902.07714}
 
 description: |
-  An \(n\)-qubit approximate code whose codespace is formed by eigenstates of a translationally-invariant quantum many-body system which satisfies the Eigenstate Thermalization Hypothesis (ETH). ETH ensures that codewords cannot be locally distinguished in the thermodynamic limit. Relevant many-body systems include 1D non-interacting spin chains, Motzkin chains, or Heisenberg models.
+  An \(n\)-qubit approximate code whose codespace is formed by eigenstates of a translationally-invariant quantum many-body system which satisfies the Eigenstate Thermalization Hypothesis (ETH).
+  ETH ensures that codewords cannot be locally distinguished in the thermodynamic limit.
+  Relevant many-body systems include 1D non-interacting spin chains or frustration-free systems such as Motzkin chains and Heisenberg models.
 
   ETH requires that for ordered energy eigenstates \(|E_l\rangle\) and any local observable \(O\),
   \begin{align}
   |\langle E_l|O|E_l\rangle-\langle E_{l+1}|O|E_{l+1}\rangle|\leq\exp(-cn)
   \end{align}
-  for a constant \(c\). This implies that energy eigenstates around some energy \(\bar E\) are approximately locally indistinguishable from one another, as their reduced density matrices on any subsystem are both approximately thermal at energy \(\bar E\). In this way, global information is protected from local measurements by the environment as \(n\to\infty\).
+  for a constant \(c\).
+  This implies that energy eigenstates around some energy \(\bar E\) are approximately locally indistinguishable from one another, as their reduced density matrices on any subsystem are both approximately thermal at energy \(\bar E\).
+  In this way, global information is protected from local measurements by the environment as \(n\to\infty\).
 
 protection: 'Approximately protects against erasure errors at known locations. Translation invariance alone is sufficient for good approximate error-correcting properties in a many-body spectrum, including in integrable models \cite{arxiv:1710.04631}. The ETH code generated from the spectrum of the translation-invariant 1D Heisenberg spin chain \cite{arxiv:1710.04631} has recovery infidelity (against the erasure of a constant number of sites) scale as \(\epsilon_\text{worst}=O(1/n)\) \cite{arxiv:1902.07714}.'
 

codes/quantum/qubits/nonstabilizer/movassagh_ouyang.yml

@@ -25,15 +25,18 @@ relations:
   parents:
     - code_id: qubits_into_qubits
     - code_id: hamiltonian
-      detail: 'Movassagh-Ouyang codes reside in the ground space of a Hamiltonian.'
+      detail: 'Movassagh-Ouyang codes reside in the ground space of a Hamiltonian.
+      Justesen codes can be used to build a family of \(n\)-qudit Movassagh-Ouyang Hamiltonian codes encoding one logical qubit with linear distance.
+      These codes form the ground-state subspace of a frustration-free geometrically local Hamiltonian \cite{arxiv:2012.01453}.'
 
   cousins:
     - code_id: qubit_stabilizer
       detail: 'Many, but not all, Movassagh-Ouyang codes are stabilizer codes.'
     - code_id: bits_into_bits
       detail: 'Movassagh-Ouyang codes are constructed from classical binary codes.'
     - code_id: justesen
-      detail: 'Justesen codes can be used to build a family of \(n\)-qubit Movassagh-Ouyang Hamiltonian codes encoding one logical qubit with linear distance.'
+      detail: 'Justesen codes can be used to build a family of \(n\)-qudit Movassagh-Ouyang Hamiltonian codes encoding one logical qubit with linear distance.
+      These codes form the ground-state subspace of a frustration-free geometrically local Hamiltonian \cite{arxiv:2012.01453}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/nonstabilizer/mps.yml

@@ -7,29 +7,24 @@ code_id: mps
 physical: qubits
 logical: qubits
 
-name: 'Matrix-product state (MPS) code'
-short_name: 'MPS'
+name: 'Magnon code'
 introduced: '\cite{arxiv:1902.02115}'
 
-alternative_names:
-  - 'Magnon code'
-
 description: |
-  An \(n\)-qubit approximate code whose codespace of \(k=\Omega(\log n)\) qubits is efficiently described in terms of matrix product states (MPS) or Bethe ansatz tensor networks.
-  A no-go theorem states that open-boundary MPS that form a degenerate ground-state space of a gapped local Hamiltonian yield codes with distance that is only constant in the number of qubits \(n\), so MPS excitation ansatze have to be used to achieve a distance scaling nontrivially with \(n\).
+  An \(n\)-qubit approximate code whose codespace of \(k=\Omega(\log n)\) qubits is efficiently described in terms of particular matrix product states or Bethe ansatz tensor networks.
 
 protection: 'Distance \(d=\Omega(n^{1-\nu})\) for any \(\nu\in(0,1)\).'
 
 relations:
   parents:
     - code_id: qubits_into_qubits
     - code_id: hamiltonian
-      detail: 'MPS codewords are low-energy excited states of a local Hamiltonian.'
+      detail: 'Magnon codewords are low-energy excited states of the frustration-free Heisenberg-XXX model Hamiltonian \cite{arxiv:1902.02115}.'
     - code_id: approximate_qecc
-      detail: 'MPS codes approximately protect against erasures in the thermodynamic limit.'
+      detail: 'Magnon codes approximately protect against erasures in the thermodynamic limit.'
   cousins:
     - code_id: eth
-      detail: 'MPS codes have been shown to protect against non-geometrically local noise, while ETH codes protect only against erasures on geometrically local patches.'
+      detail: 'Magnon codes have been shown to protect against non-geometrically local noise, while ETH codes protect only against erasures on geometrically local patches.'
 
 
 # Begin Entry Meta Information

codes/quantum/qudits/nonstabilizer/vbs.yml

@@ -0,0 +1,39 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: vbs
+physical: qudits
+logical: qudits
+
+name: 'Valence-bond-solid (VBS) code'
+short_name: 'VBS'
+introduced: '\cite{arxiv:1910.00038,arxiv:2105.14777}'
+
+description: |
+  An \(n\)-qubit approximate \(q\)-dimensional qudit code family whose codespace is described in terms of \(SU(q)\) valence-bond-solid (VBS) \cite{doi:10.1007/978-3-662-06390-3_18} matrix product states with various boundary conditions.
+  The codes become exact when either \(n\) or \(q\) go to infinity.
+
+features:
+  transversal_gates:
+    - 'Two classes of (approximate) VBS codes have \(SU(q)\) transversal gates \cite[Tab. III]{arxiv:2105.14777}.'
+
+relations:
+  parents:
+    - code_id: qudits_into_qudits
+    - code_id: hamiltonian
+      detail: 'VBS codewords are eigenstates of the frustration-free VBS Hamiltonian \cite{arxiv:1910.00038,arxiv:2105.14777}.'
+    - code_id: approximate_qecc
+      detail: 'VBS codes approximately protect against erasures in the thermodynamic limit.'
+  cousins:
+    - code_id: covariant
+      detail: 'Two classes of (approximate) VBS codes have \(SU(q)\) transversal gates, i.e., are \(SU(q)\)-covariant \cite[Tab. III]{arxiv:2105.14777}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-05-27'