unentangled_permutation_invariant

codes/quantum/oscillators/fock_state/constant_excitation/constant_excitation_permutation_invariant.yml

@@ -6,7 +6,7 @@
 code_id: constant_excitation_permutation_invariant
 physical: oscillators
 
-name: 'Ouyang-Chao constant-excitation permutation-invariant code'
+name: 'Ouyang-Chao constant-excitation PI code'
 short_name: 'Ouyang-Chao'
 introduced: '\cite{arxiv:1809.09801}'
 

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

@@ -5,7 +5,8 @@
 
 code_id: permutation_invariant
 
-name: 'Permutation-invariant code'
+name: 'Permutation-invariant (PI) code'
+short_name: 'PI'
 introduced: '\cite{arxiv:quant-ph/0304153}'
 
 description: |
@@ -18,7 +19,7 @@ protection: |
   Permutation invariant codes of distance \(d\) can protect against \(d-1\) deletion errors \cite{arxiv:2001.08405,arxiv:2004.00814,arxiv:2102.02494,arxiv:2102.03015}, i.e., erasures of subsystems 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}.
+  The GNU PI 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:

codes/quantum/properties/stabilizer/qldpc/qldpc.yml

@@ -61,6 +61,7 @@ protection: |
     \delta = \frac{d}{n} \leq \frac{1}{2} - \Omega\left(\frac{1}{r}\right)~.
   \end{align}
 
+
 features:
   rate: |
     Asymptotic scaling of \(k\) and \(d\) with \(n\) depends heavily on the code construction.
@@ -101,6 +102,7 @@ notes:
   - 'Infleqtion QLDPC package for estimating distance and creating various qubit and Galois-qudit QLDPC CSS codes \cite{manual:{Michael A. Perlin. qLDPC. https://github.com/Infleqtion/qLDPC, 2023.}}'
   - 'Links to code tables of notable QLDPC codes \cite{arxiv:2103.06309}.'
   - 'Reviews of QLDPC codes provided in Refs. \cite{doi:10.1109/ACCESS.2015.2503267,arxiv:2103.06309}.'
+  - 'There exist distance-dependent \cite[Thm. 1]{arxiv:2405.01332} and rate-dependent \cite[Thm. 3ii]{arxiv:2405.01332} lower bounds on the degree of entanglement of a qubit QLDPC code.'
 
 relations:
   parents:

codes/quantum/qubits/nonstabilizer/eth.yml

@@ -47,7 +47,7 @@ relations:
     - code_id: topological
       detail: 'ETH codewords, like topological codewords, are locally indistinguishable.'
     - code_id: qubit_permutation_invariant
-      detail: 'Several instances of ETH codes contain permutation-invariant qubit codewords.'
+      detail: 'Several instances of ETH codes contain PI qubit codewords.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/permutation_invariant/binary_dihedral_permutation_invariant.yml

@@ -7,7 +7,7 @@ code_id: binary_dihedral_permutation_invariant
 physical: qubits
 logical: qubits
 
-name: 'Binary dihedral permutation-invariant code'
+name: 'Binary dihedral PI code'
 introduced: '\cite{arxiv:2310.17652}'
 
 description: |
@@ -29,20 +29,20 @@ relations:
   parents:
     - code_id: qubit_permutation_invariant
     - code_id: j_gross
-      detail: 'Binary dihedral permutation-invariant codes can be interpreted as Clifford single-spin codes.'
+      detail: 'Binary dihedral PI codes can be interpreted as Clifford single-spin codes.'
   cousins:
     - code_id: small_distance_quantum
-      detail: 'The first and second families of binary dihedral permutation-invariant codes have distance three, and the third family has the member \(((27,2,5))\).'
+      detail: 'The first and second families of binary dihedral PI codes have distance three, and the third family has the member \(((27,2,5))\).'
     - code_id: combinatorial_permutation_invariant
-      detail: 'The \(Q_{3,1,2m-4,+}\) and \(Q_{3,1,2^m-4,+}\) combinatorial permutation-invariant codes reduce to the \(((2m+3,2,3))\) and \(((2^{m-1}+3,2,3))\) binary dihedral permutation-invariant codes, respectively \cite[Prop. 5.6]{arxiv:2310.05358}.'
+      detail: 'The \(Q_{3,1,2m-4,+}\) and \(Q_{3,1,2^m-4,+}\) combinatorial PI codes reduce to the \(((2m+3,2,3))\) and \(((2^{m-1}+3,2,3))\) binary dihedral PI codes, respectively \cite[Prop. 5.6]{arxiv:2310.05358}.'
     - code_id: xp_stabilizer
       detail: 'Binary dihedral permutation invariant codewords form error spaces of XP stabilizer codes.'
     - code_id: diagonal_clifford
-      detail: 'The \(((2^{r-1}+3,2,3))\) family of binary dihedral permutation-invariant codes realizes the same transversal gates as the \([[2^r-1,1,3]]\) quantum Reed-Muller codes, but require fewer qubits in almost all cases.'
+      detail: 'The \(((2^{r-1}+3,2,3))\) family of binary dihedral PI codes realizes the same transversal gates as the \([[2^r-1,1,3]]\) quantum Reed-Muller codes, but require fewer qubits in almost all cases.'
     - code_id: stab_49_1_5
-      detail: 'The \(((27,2,5))\) binary dihedral permutation-invariant code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[49,1,5]]\) triorthogonal code.'
+      detail: 'The \(((27,2,5))\) binary dihedral PI code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[49,1,5]]\) triorthogonal code.'
     - code_id: stab_15_1_3
-      detail: 'The \(((11,2,3))\) binary dihedral permutation-invariant code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[15,1,3]]\) quantum Reed-Muller code.'
+      detail: 'The \(((11,2,3))\) binary dihedral PI code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[15,1,3]]\) quantum Reed-Muller code.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/permutation_invariant/combinatorial_permutation_invariant.yml

@@ -7,11 +7,11 @@ code_id: combinatorial_permutation_invariant
 physical: qubits
 logical: qubits
 
-name: 'Combinatorial permutation-invariant code'
+name: 'Combinatorial PI code'
 introduced: '\cite{arxiv:2310.05358}'
 
 description: |
-  A member of a family of permutation-invariant quantum codes whose correction properties are derived from solving a family of combinatorial identities.
+  A member of a family of PI quantum codes whose correction properties are derived from solving a family of combinatorial identities.
   The code encodes one logical qubit in superpositions of \hyperref[topic:dicke]{Dicke states} whose coefficients are square roots of ratios of binomial coefficients.
 
 protection: |

codes/quantum/qubits/permutation_invariant/gnu/four_qubit_permutation_invariant.yml

@@ -9,7 +9,7 @@ name: 'Four-qubit single-deletion code'
 introduced: '\cite{arxiv:2001.08405,arxiv:2004.00814}'
 
 description: |
-  Four-qubit permutation-invariant code that is the smallest qubit code to correct one deletion error.
+  Four-qubit PI code that is the smallest qubit code to correct one deletion error.
 
   In terms of \hyperref[topic:dicke]{Dicke states}, a basis of logical codewords is
   \begin{align}
@@ -34,7 +34,7 @@ relations:
     - code_id: stab_4_2_2
       detail: 'A basis of codewords for the four-qubit single-deletion code consists of the \(|\overline{00}\rangle\) and \(|\overline{01}\rangle+|\overline{10}\rangle+|\overline{11}\rangle\)states of the four-qubit code.'
     - code_id: combinatorial_permutation_invariant
-      detail: 'The combinatorial permutation-invariant code \(Q_{1,1,1,-}\) is another example of a four-qubit code correcting a single deletion error \cite[Sec. 5.1]{arxiv:2310.05358}.'
+      detail: 'The combinatorial PI code \(Q_{1,1,1,-}\) is another example of a four-qubit code correcting a single deletion error \cite[Sec. 5.1]{arxiv:2310.05358}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/permutation_invariant/gnu/gnu_permutation_invariant.yml

@@ -7,12 +7,12 @@ code_id: gnu_permutation_invariant
 physical: spins
 logical: qubits
 
-name: 'GNU permutation-invariant code'
+name: 'GNU PI code'
 short_name: 'GNU'
 introduced: '\cite{arxiv:1302.3247,arxiv:1512.02469}'
 
 description: |
-  Permutation-invariant code whose codewords can be expressed as superpositions of \hyperref[topic:dicke]{Dicke states} with coefficients are square-roots of the binomial distribution.
+  PI code whose codewords can be expressed as superpositions of \hyperref[topic:dicke]{Dicke states} with coefficients are square-roots of the binomial distribution.
 
   In terms of \hyperref[topic:dicke]{Dicke states}, logical codewords for codes encoding a single qubit \cite{arxiv:1302.3247} are
   \begin{align}
@@ -29,24 +29,27 @@ 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}.'
 
+notes:
+  - 'The degree of entanglement in (non-concatenated) GNU codes scales at most logarithmically in their distance \cite[Appx. D]{arxiv:2405.01332}.'
+
 
 relations:
   parents:
     - code_id: qudit_gnu_permutation_invariant
       detail: 'Qudit GNU codes encoding logical qubits reduce to GNU codes.'
   cousins:
     - code_id: combinatorial_permutation_invariant
-      detail: 'Combinatorial permutation-invariant codes \(Q_{g,(m-1)/2,g-1,+}\) are GNU codes for odd \(m\) \cite[Prop. 5.4]{arxiv:2310.05358}.'
+      detail: 'Combinatorial PI codes \(Q_{g,(m-1)/2,g-1,+}\) are GNU codes for odd \(m\) \cite[Prop. 5.4]{arxiv:2310.05358}.'
     - code_id: bacon_shor
-      detail: 'GNU codes of length \((2t+1)^2\) result from projecting Bacon-Shor codes into the permutation-invariant qubit subspace \cite{arxiv:1302.3247}.'
+      detail: 'GNU codes of length \((2t+1)^2\) result from projecting Bacon-Shor codes into the PI qubit subspace \cite{arxiv:1302.3247}.'
     - code_id: hamiltonian
       detail: 'GNU codes lie within the ground state of ferromagnetic Heisenberg models without an external magnetic field \cite{arxiv:1904.01458}.'
     - code_id: approximate_qecc
       detail: 'GNU codes protect approximately against amplitude damping errors.'
     - code_id: binomial
       detail: 'Binomial codes and GNU codes related via the Holstein-Primakoff mapping \cite{doi:10.1103/PhysRev.58.1098} (see also \cite{doi:10.2307/3212170}). A qudit generalization of GNU codes can be obtained from qudit binomial codes \cite[Appx. C]{arxiv:1708.05010}.'
     - code_id: metopt
-      detail: 'GNU codes can be used to sense signals within the permutation-invariant subspace \cite{arxiv:2212.06285}.'
+      detail: 'GNU codes can be used to sense signals within the PI subspace \cite{arxiv:2212.06285}.'
 
 
 #    - code_id: gkp

codes/quantum/qubits/permutation_invariant/gnu/qudit_gnu_permutation_invariant.yml

@@ -7,13 +7,13 @@ code_id: qudit_gnu_permutation_invariant
 physical: spins
 logical: qudits
 
-name: 'Qudit GNU permutation-invariant code'
+name: 'Qudit GNU PI code'
 short_name: 'Qudit GNU'
 introduced: '\cite{arxiv:1604.07925}'
 
 description: |
-  Extension of the GNU permutation-invariant codes to those encoding logical qudits into physical qubits.
-  Codewords can be expressed as superpositions of \hyperref[topic:dicke]{Dicke states} with coefficients are square-roots of polynomial coefficients, with the case of binomial coefficients reducing to the GNU permutation-invariant codes.
+  Extension of the GNU PI codes to those encoding logical qudits into physical qubits.
+  Codewords can be expressed as superpositions of \hyperref[topic:dicke]{Dicke states} with coefficients are square-roots of polynomial coefficients, with the case of binomial coefficients reducing to the GNU PI codes.
 
 
 relations:

codes/quantum/qubits/permutation_invariant/gnu/ruskai.yml

@@ -11,7 +11,7 @@ name: '\(((9,2,3))\) Ruskai code'
 introduced: '\cite{arxiv:quant-ph/9906114}'
 
 description: |
-  Nine-qubit permutation-invariant code that protects against single-qubit errors as well as two-qubit errors arising from exchange processes.
+  Nine-qubit PI code that protects against single-qubit errors as well as two-qubit errors arising from exchange processes.
 
   In terms of \hyperref[topic:dicke]{Dicke states}, the codewords are
   \begin{align}
@@ -28,11 +28,11 @@ protection: |
 relations:
   parents:
     - code_id: gnu_permutation_invariant
-      detail: 'The \(((9,2,3))\) Ruskai code is a GNU permutation-invariant code \cite{arxiv:1302.3247}.'
+      detail: 'The \(((9,2,3))\) Ruskai code is a GNU PI code \cite{arxiv:1302.3247}.'
     - code_id: small_distance_quantum
   cousins:
     - code_id: shor_nine
-      detail: 'The \(((9,2,3))\) Ruskai code results from projecting the Shor code into the permutation-invariant qubit subspace \cite{arxiv:quant-ph/9906114}.'
+      detail: 'The \(((9,2,3))\) Ruskai code results from projecting the Shor code into the PI qubit subspace \cite{arxiv:quant-ph/9906114}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/permutation_invariant/icosahedral_permutation_invariant.yml

@@ -11,7 +11,7 @@ name: '\(((7,2,3))\) Pollatsek-Ruskai code'
 introduced: '\cite{arxiv:quant-ph/0304153,arxiv:2005.10910,arxiv:2305.07023}'
 
 description: |
-  Seven-qubit permutation-invariant code that realizes gates from the binary icosahedral group transversally.
+  Seven-qubit PI code that realizes gates from the binary icosahedral group transversally.
   Can also be interpreted as a spin-\(7/2\) single-spin code.
   The codespace projection is a projection onto an irrep of the binary icosahedral group \(2I\).
 
@@ -30,7 +30,7 @@ features:
 relations:
   parents:
     - code_id: combinatorial_permutation_invariant
-      detail: 'The Pollatsek-Ruskai code is equivalent to the \(Q_{2,1,2,-}\) combinatorial permutation-invariant code \cite[Sec. 5.2]{arxiv:2310.05358}.'
+      detail: 'The Pollatsek-Ruskai code is equivalent to the \(Q_{2,1,2,-}\) combinatorial PI code \cite[Sec. 5.2]{arxiv:2310.05358}.'
     - code_id: t_group
       detail: 'The \(((7,2,3))\) Pollatsek-Ruskai code admits a transversal representation of the twisted \(1\)-group \(2I\) \cite{arxiv:2402.01638}.'
     - code_id: j_gross

codes/quantum/qubits/permutation_invariant/qubit_permutation_invariant.yml

@@ -7,20 +7,20 @@ code_id: qubit_permutation_invariant
 physical: qubits
 # Qudit GNU codes don't encode logical qubits
 
-name: 'Permutation-invariant qubit code'
+name: 'PI qubit code'
 
 description: |
   Block quantum code defined on two-dimensional subsystems such that any permutation of the subsystems leaves any codeword invariant.
 
   \begin{defterm}{Dicke states}
   \label{topic:dicke}
-  For \(n\)-qubit block codes, an often used basis for the \(n+1\)-dimensional permutation-invariant subspace consists of the Dicke states \(|D^n_w\rangle\) -- normalized permutation-invariant states of \(w\) excitations, i.e., a normalized sum over all binary-string basis elements with \(w\) ones and \(n - w\) zeroes.
+  For \(n\)-qubit block codes, an often used basis for the \(n+1\)-dimensional PI subspace consists of the Dicke states \(|D^n_w\rangle\) -- normalized PI states of \(w\) excitations, i.e., a normalized sum over all binary-string basis elements with \(w\) ones and \(n - w\) zeroes.
   For example, the single-excitation Dicke state on three qubits is
   \begin{align}
     |D_{1}^{3}\rangle=\frac{1}{\sqrt{3}}\left(|001\rangle+|010\rangle+|100\rangle\right)~.
   \end{align}
-  The \(n+1\)-dimensional permutation-invariant space can be thought of as a standalone spin-\(n/2\) quantum system, yielding a way to convert between permutation-invatiant qubit codes and \(SU(2)\) spin codes.
-  A single-spin code for the \(SU(2)\) group correcting spherical tensors can be mapped into a permutation-invariant qubit code with an analogous distance \cite{arxiv:2304.08611}\cite[Thm. 1]{arxiv:2310.17652}.
+  The \(n+1\)-dimensional PI space can be thought of as a standalone spin-\(n/2\) quantum system, yielding a way to convert between permutation-invatiant qubit codes and \(SU(2)\) spin codes.
+  A single-spin code for the \(SU(2)\) group correcting spherical tensors can be mapped into a PI qubit code with an analogous distance \cite{arxiv:2304.08611}\cite[Thm. 1]{arxiv:2310.17652}.
   \end{defterm}
 
 protection: |

codes/quantum/qubits/permutation_invariant/unentangled_permutation_invariant.yml

@@ -0,0 +1,41 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: unentangled_permutation_invariant
+physical: qubits
+logical: qubits
+
+name: '\(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code'
+introduced: '\cite{arxiv:2405.01332}'
+
+description: |
+  PI distance-two code on \(n\geq4\) qubits whose degree of entanglement vanishes asymptotically with \(n\) \cite[Appx. D]{arxiv:2405.01332}.
+
+  In terms of \hyperref[topic:dicke]{Dicke states}, the codewords are
+  \begin{align}
+    \begin{split}
+      |0_{L}\rangle&=\sqrt{1-\frac{2}{n}}|D_{0}^{n}\rangle+\sqrt{\frac{2}{n}}|D_{n}^{n}\rangle\\
+      |1_{L}\rangle&=|D_{2}^{n}\rangle~.
+    \end{split}
+  \end{align}
+
+
+relations:
+  parents:
+    - code_id: qubit_permutation_invariant
+    - code_id: movassagh_ouyang
+      detail: 'The \(((n,1,2))\) PI code is a Movassagh-Ouyang Hamiltonian code constructed from a binary code consisting of all codewords of weight 0, 2, or \(n\) \cite[Appx. D]{arxiv:2405.01332}.'
+    - code_id: small_distance_quantum
+  cousins:
+    - code_id: quantum_concatenated
+      detail: 'The Bravyi-Lee-Li-Yoshida PI code can be concatenated to yield codes that have higher distance and that admit codewords with vanishing entanglement \cite[Appx. D]{arxiv:2405.01332}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-02-07'

codes/quantum/qubits/qubits_into_qubits.yml

@@ -122,6 +122,8 @@ features:
 notes:
   - 'There is a relation between one-way entanglement distillation protocols and QECCs \cite{arxiv:quant-ph/9604024}.'
   - 'See \href{https://github.com/qiskit-community/qiskit-qec}{Qiskit QEC framework} for realizing protocols on IBM machines.'
+  - 'There exists a distance- and rate-dependent lower bound on the degree of entanglement of a qubit code \cite[Thm. 3i]{arxiv:2405.01332}.'
+
 
 relations:
   parents:

codes/quantum/qubits/stabilizer/qubit_stabilizer.yml

@@ -133,6 +133,8 @@ notes:
   - 'Tables of bounds and examples of stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Ref. \cite{doi:10.1007/978-3-540-37634-7_13}, are maintained by M. Grassl at this \href{http://codetables.markus-grassl.de/}{website}.'
   - 'Stabilizer error-recovery circuits can be simulated efficiently using dedicated software (e.g., STIM \cite{arxiv:2103.02202}).'
   - '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}.'
+
 
 relations:
   parents:

codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface/surface.yml

@@ -126,6 +126,7 @@ features:
     - 'Markov-chain Monte Carlo \cite{arxiv:1302.2669}.'
     - 'Cellular automaton decoders \cite{doi:10.7907/AHMQ-EG82,arxiv:1406.2338,arxiv:1511.05579}; see also \cite{arxiv:1512.04528}.'
     - 'Neural network \cite{arxiv:1610.04238,arxiv:1802.06441,arxiv:2208.05758}, reinforcement learning \cite{arxiv:1810.07207,arxiv:2212.11890}, and transformer-based \cite{arxiv:2311.16082} decoders.'
+    - 'Lightweight low-latency look-up table (LILLIPUT) decoder for small surface codes \cite{arxiv:2108.06569}.'
     - 'Decoders can be augmented with a pre-decoder \cite{arxiv:2001.11427,arxiv:2208.04660}, which can allow for some processing to be done inside the cryogenic environment of the quantum system \cite{arxiv:2208.08547}.'
     - 'Sliding-window \cite{arxiv:2209.09219,arxiv:2209.08552} and parallel-window \cite{arxiv:2209.09219} parallelizable decoders, designed to overcome the backlog problem, can be combined with many inner decoders, such as MWPM or union-find.'
     - 'Modifications of BP:

codes/quantum/spins/single_spin/j_gross.yml

@@ -43,7 +43,7 @@ relations:
     - code_id: group_representation
       detail: 'Clifford spin codes are group-representation codes with \(G\) being a subgroup of \(SU(2)\) \cite{arxiv:2306.11621}.'
     - code_id: qubit_permutation_invariant
-      detail: 'Clifford codes for spins housing representations of \(SU(2)\) yield permutation-invariant qubit codes with non-trivial distance when the single spin-\(n/2\) is treated as the permutationally invariant subspace of \(n\) qubits via the \hyperref[topic:dicke]{Dicke-state mapping}.
+      detail: 'Clifford codes for spins housing representations of \(SU(2)\) yield PI qubit codes with non-trivial distance when the single spin-\(n/2\) is treated as the permutationally invariant subspace of \(n\) qubits via the \hyperref[topic:dicke]{Dicke-state mapping}.
       The subgroup of gates of a Clifford spin code is implemented transversally via this mapping \cite{arxiv:2304.08611}.'