Merge branch 'main' into correct-typo

.github/workflows/build-and-deploy-site.yml

@@ -32,7 +32,7 @@ jobs:
       - uses: actions/setup-node@v3
         with:
           #cache: 'yarn'
-          node-version: '20'
+          node-version: '22'
 
       - name: 'Enable Yarn (corepack enable)'
         run: 'corepack enable'

.github/workflows/devsite-build-and-deploy.yml

@@ -61,7 +61,7 @@ jobs:
       - uses: actions/setup-node@v3
         with:
           #cache: 'yarn'
-          node-version: '20'
+          node-version: '22'
 
       - name: 'Enable Yarn (corepack enable)'
         run: 'corepack enable'

.github/workflows/quick-test-compile-zoo.yml

@@ -28,7 +28,7 @@ jobs:
       - uses: actions/setup-node@v3
         with:
           #cache: 'yarn'
-          node-version: '20'
+          node-version: '22'
 
       - name: Corepack enable
         run: 'corepack enable'

codes/classical/analog/lattice/points_into_lattices.yml

@@ -51,7 +51,7 @@ protection: |
 
   The \textit{lattice quantizer problem} is to find a lattice whose \textit{fundamental Voronoi cell} \(\Pi\), the Voronoi cell at the origin, has the smallest possible normalized second moment,
   \begin{align}
-    G(\Pi)=\frac{\frac{1}{n}\int_{\Pi}x\cdot x\,dx}{\text{Vol}(\Pi)^{1+2/n}}\,.
+    G(\Pi)=\frac{\frac{1}{n}\int_{\Pi}x\cdot x\,\textnormal{d}x}{\text{Vol}(\Pi)^{1+2/n}}\,.
   \end{align}
   Higher-dimensional lattices yield quantizers with lower normalized second moments than the 1D integer lattice \cite{manual:{P. L. Zador, Development and evaluation of procedures for quantiZing multivariate distributions, Ph.D. Dissertation, Stanford Univ., 1963},doi:10.1109/TIT.1982.1056490}.
 

codes/classical/bits/tanner/ldpc.yml

@@ -43,6 +43,7 @@ features:
     The smallest stopping set size can reach the minimum distance of the code \cite{doi:10.1109/TIT.2005.864441}.'
     - 'Ensembles of random LDPC codes under iterative decoders are subject to the \textit{concentration theorem} \cite{doi:10.1109/18.910577,doi:10.1109/18.910578}; see \cite[Thm. 21.7.1]{preset:HKSgraphs} for the case of the BEC.'
     - 'Reinforcement learning \cite{arxiv:2112.13934}.'
+    - 'Quantum-enhanced BP decoding \cite{arxiv:2412.08596}.'
 
 notes:
   - 'The potential of LDPC codes was noted by Margulis \cite{doi:10.1007/BF02579283}, but realized by the broader community \cite{doi:10.1049/el:19970362,doi:10.1109/18.748992} much later after their discovery by Gallager \cite{doi:10.1109/TIT.1962.1057683,manual:{R. Gallager, \emph{Low-density parity check codes}. 1963. PhD thesis, MIT Cambridge, MA.}}.'

codes/classical/properties/block/distributed_storage/lrc/lcc.yml

@@ -42,6 +42,8 @@ relations:
       detail: 'There are relations between LDCs and LTCs \cite{doi:10.1007/978-3-642-15369-3_50}.'
     - code_id: quantum_locally_recoverable
       detail: 'There is not a natural quantum version of LCCs \cite[Thm. 9]{arxiv:2311.08653}.'
+    - code_id: analog
+      detail: 'LCCs can also be defined over the real or complex numbers, and there are no complex 2-query LCCs \cite{arxiv:1009.4375}.'
 
 
 # Begin Entry Meta Information

codes/classical/properties/block/universally_optimal/t-designs.yml

@@ -18,7 +18,7 @@ description: |
 
   As such, a design can be used to determine the average of degree-\(\leq t\) polynomials \(p\) over \(X\),
   \begin{align}
-    \int_{X}dxp(x)={\textstyle \frac{1}{|D|}}\sum_{x\in D}p(x)~,
+    \int_{X}\textnormal{d}xp(x)={\textstyle \frac{1}{|D|}}\sum_{x\in D}p(x)~,
   \end{align}
   where the integral is over \(X\) (given some measure \(d x\)), while the sum is over the design \(D\subset X\).
   A \textit{weighted design} is a design for which each term \(p(x)\) in the above sum must be multiplied by a weight \(w(x)\) in order to be equal to the left-hand side.
@@ -42,7 +42,7 @@ description: |
 #  when restricted to act on distinct \(t\)-tuples; see \cite[Remarks 6-7]{arXiv:2404.14648}
 
 notes:
-  - 'See the handbook \cite{doi:10.1201/9781420010541} for tables of various designs.'
+  - 'See books \cite{manual:{Stroud, Arthur H. Approximate calculation of multiple integrals. Prentice Hall, 1971.},doi:10.1201/9781420010541} for tables of various designs.'
 
 relations:
   parents:

codes/classical/spherical/spherical.yml

@@ -66,6 +66,10 @@ notes:
   - 'See \cite{preset:EricZin,manual:{Sloane, N. J. A., R. H. Hardin, and W. D. Smith. "Tables of spherical codes." collaboration with RH Hardin, WD Smith and others. Published electronically at https://neilsloane.com/packings/ (2004).}} for more details and tables of optimal codes.'
   - 'See article \cite{doi:10.1007/BF03024331} for relations of spherical codes to other fields.'
 
+realizations:
+  - 'Spherical codes are relevant to modern Hopfield networks \cite{arxiv:2410.23126,arxiv:2402.13725}'
+
+
 relations:
   parents:
     - code_id: points_into_spheres

codes/quantum/groups/rotors/stabilizer/css/zero_pi.yml

@@ -23,7 +23,7 @@ description: |
   An alternative codeword basis in terms of angular position states is
   \begin{align}
   \begin{split}
-    |\overline{+}\rangle&=\intop_{U(1)}d\phi\left|\phi,\phi\right\rangle \\|\overline{-}\rangle&=\intop_{U(1)}d\phi\left|\phi,\phi+\pi\right\rangle~.
+    |\overline{+}\rangle&=\intop_{U(1)}\textnormal{d}\phi\left|\phi,\phi\right\rangle \\|\overline{-}\rangle&=\intop_{U(1)}\textnormal{d}\phi\left|\phi,\phi+\pi\right\rangle~.
   \end{split}
   \end{align}
 

codes/quantum/groups/topological/quantum_double.yml

@@ -17,7 +17,7 @@ description: |
   The physical Hilbert space has dimension \( |G|^E  \), where \( E \) is the number of  edges in the tessellation. The dimension of the code space is the number of orbits of the conjugation action of \( G \) on \( \text{Hom}(\pi_1(\Sigma),G) \), the set of group homomorphisms from the fundamental group of the surface \( \Sigma \) into the finite group \( G \) \cite{arxiv:1908.02829}. When \( G \) is Abelian, the formula for the dimension simplifies to \( |G|^{2g} \), where \( g \) is the genus of the surface \( \Sigma \).
 
   The codespace is the ground-state subspace of the quantum double model Hamiltonian, while local excitations are characterized by anyons.
-  Different types of anyons are labeled by irreducible representations of the group's quantum double algebra, \(D(G)\) (a.k.a. Drinfield center) \cite{arxiv:1006.5479}.
+  Different types of anyons are labeled by irreducible representations of the group's quantum double algebra, \(D(G)\) (a.k.a. Drinfield center) \cite{arxiv:1006.5479,arxiv:2310.19661}.
   Not all isomorphic non-Abelian groups give rise to different quantum doubles \cite{arxiv:math/0605530}.
 
   For non-Abelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model \cite{doi:10.1007/3-540-49208-9_31,arxiv:quant-ph/0306063,doi:10.1017/CBO9780511792908}.
@@ -66,11 +66,14 @@ relations:
       Quantum-double codes for non-Abelian groups \(G\) are dual to Hopf-algebra quantum-double codes for Hopf algebras based on \(\text{Rep}(G)\) under the Tannaka-Krein duality \cite{arxiv:0907.2670}\cite[Fig. 1]{arxiv:1006.5823}.'
   cousins:
     - code_id: hamiltonian
-      detail: 'Quantum double code Hamiltonians can be simulated, with the help of perturbation theory, by two-dimensional two-body Hamiltonians with non-commuting terms \cite{arxiv:1011.1942}.'
+      detail: 'Quantum double code Hamiltonians can be simulated, with the help of perturbation theory and the \([[4,1,1,2]]\) subsystem code, by two-dimensional two-body Hamiltonians with non-commuting terms \cite{arxiv:1011.1942}.'
     - code_id: oecc
       detail: 'Subsystem versions of quantum-double codes have been formulated \cite{doi:10.5446/35287}.'
     - code_id: yetter_gauge_theory
       detail: 'Restricting 2-gauge theory constructions to a 2D manifold and replacing the 2-group with a group reproduces the phase of the Kitaev quantum double model \cite{arxiv:1606.06639}.'
+    - code_id: bacon_shor_4
+      detail: 'Quantum double code Hamiltonians can be simulated, with the help of perturbation theory and the four-qubit subsystem code, by two-dimensional two-body Hamiltonians with non-commuting terms \cite{arxiv:1011.1942}.'
+
 
 # \cite{manual:{Prashant Kumar, Quantum Double Subsystem Codes, Quantum Error Correction conference, University of Southern California, 2011}}'
 # https://qserver.usc.edu/qec11/slides/Kumar_QEC11.pdf

codes/quantum/oscillators/fock_state/rotation/binomial.yml

@@ -38,6 +38,7 @@ features:
   general_gates:
     - 'Error-detecting \(CCZ\) and \(cSWAP\) gates for "0-2-4" code using three-level ancilla \cite{arxiv:2212.11196}.'
     - 'Single logical-qubit rotations \cite{arxiv:2408.12968}.'
+    - 'Amplitude-mixing error-transparent gates \cite{arxiv:2412.08870}.'
 
   decoders:
     - 'Photon loss and dephasing errors can be detected by measuring the phase-space rotation \(\exp\left(2\pi\mathrm{i} \hat{n} / (S+1)\right)\) and the check operator \(J_x/J\) in the spin-coherent state language, where \(J\) is the total angular momentum and \(J_x\) is the angular momentum in the \(x\) direction \cite{arxiv:1708.05010}. This type of error correction fails for errors that are products of photon loss/gain and dephasing errors. However, for certain \((N,S)\) instances of the binomial code, detection of these types of errors can be done.'

codes/quantum/oscillators/fock_state/rotation/chebyshev.yml

@@ -9,7 +9,9 @@ name: 'Chebyshev code'
 introduced: '\cite{arxiv:1811.01450}'
 
 description: |
-  Single-mode bosonic Fock-state code that can be used for error-corrected sensing of a signal Hamiltonian \({\hat n}^s\), where \({\hat n}\) is the occupation number operator. Codewords for the \(s\)th-order Chebyshev code are
+  Single-mode bosonic Fock-state code that can be used for error-corrected sensing of a signal Hamiltonian \({\hat n}^s\), where \({\hat n}\) is the occupation number operator. 
+  
+  Codewords for the \(s\)th-order Chebyshev code are
   \begin{align}
   \begin{split}
   \ket{\overline 0} &=\sum_{k \text{~even}}^{[0,s]} \tilde{c}_k \Ket{\left\lfloor M\sin^2\left( k\pi/{2s}\right) \right\rfloor},\\

codes/quantum/oscillators/fock_state/rotation/number_phase.yml

@@ -15,7 +15,9 @@ alternative_names:
 # Ouyang
 
 description: |
-  Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states \cite{doi:10.1088/0305-4470/19/18/030},
+  Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states \cite{doi:10.1088/0305-4470/19/18/030}.
+
+  Pegg-Barnett phase states are expressed in terms of Fock states as
   \begin{align}
   |\phi\rangle \equiv \frac{1}{\sqrt{2\pi}}\sum_{n=0}^{\infty} \mathrm{e}^{\mathrm{i} n \phi} \ket{n}.
   \end{align}
@@ -37,6 +39,10 @@ features:
   fault_tolerance:
     - 'Fault-tolerant computation schemes with number-phase codes have been proposed based on concatenation with Bacon-Shor subsystem codes \cite{arxiv:1901.08071}.'
 
+realizations:
+  - 'Motional degree of freedom of a trapped ion: state initialization \cite{arxiv:2412.04865}.'
+
+
 relations:
   parents:
     - code_id: bosonic_rotation

codes/quantum/oscillators/stabilizer/lattice/gkp.yml

@@ -49,7 +49,7 @@ features:
     It has been extended to utilize previously measured syndrome information \cite{arxiv:2312.07391}.'
 
 realizations:
-  - 'Motional degree of freedom of a trapped ion: square-lattice GKP encoding realized with the help of post-selection by Home group \cite{arxiv:1807.01033,arxiv:1907.06478}, followed by realization of reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ion electronic state \cite{arxiv:2010.09681}. State preparation also realized by Tan group \cite{arxiv:2310.15546}. Universal gate set, including a two-qubit entangling gate, realized by Tan group \cite{arxiv:2409.05455}.'
+  - 'Motional degree of freedom of a trapped ion: square-lattice GKP encoding realized with the help of post-selection by Home group \cite{arxiv:1807.01033,arxiv:1907.06478}, followed by realization of reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ion electronic state \cite{arxiv:2010.09681}. State preparation also realized by Tan group \cite{arxiv:2310.15546}. Universal gate set, including a two-qubit entangling gate, realized by Tan group \cite{arxiv:2409.05455}. State initialization and application to measuring displacements \cite{arxiv:2412.04865}.'
   - 'Microwave cavity coupled to superconducting circuits: reduced form of square-lattice GKP error correction, where displacement error syndromes are measured to one bit of precision using an ancillary transmon \cite{arxiv:1907.12487}. Subsequent paper by Devoret group \cite{arxiv:2211.09116} uses reinforcement learning for error-correction cycle design and is the first to go beyond break-even error-correction, with the lifetime of a logical qubit exceeding the cavity lifetime by about a factor of two (see also \cite{arxiv:2211.09319}). See Ref. \cite{arxiv:2111.07965} for another experiment. A feed-forward-free, i.e., fully autonomous protocol has also been implemented by Nord Quantique \cite{arxiv:2310.11400}. Qudit encodings with \(q=3,4\) have been realized, with logical error rates also beyond break even \cite{arxiv:2409.15065}.'
   - 'GKP states and homodyne measurements have been realized in propagating telecom light by the Furusawa group \cite{arxiv:2309.02306}.'
   - 'Single-qubit \(Z\)-gate has been demonstrated \cite{arxiv:1904.01351} in the single-photon subspace of an infinite-mode space \cite{arxiv:2310.12618}, in which time and frequency become bosonic conjugate variables of a single effective bosonic mode. In this context, GKP position-state wavefunctions are called Dirac combs or frequency combs.'

codes/quantum/oscillators/stabilizer/lattice/gkp_concatenated.yml

@@ -17,7 +17,7 @@ protection: |
 
 
 features:
-  rate: 'Recursively concatenating the \([[6,2,2]]\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel \cite{arxiv:1712.00294}. Concatenating Abelian LP codes with GKP codes can surpass the CSS Hamming bound \cite{arxiv:2111.07029}.'
+  rate: 'Recursively concatenating the \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel \cite{arxiv:1712.00294}. Concatenating Abelian LP codes with GKP codes can surpass the CSS Hamming bound \cite{arxiv:2111.07029}.'
 
   general_gates:
     - 'Linear-optical computation \cite{arxiv:2408.04126}.'
@@ -40,9 +40,9 @@ relations:
     - code_id: quantum_repetition
       detail: 'Concatenating a three-qubit quantum repetition code with GKP codes can correct some two-bit-flip errors \cite{arxiv:1706.03011} (see also \cite{arxiv:2212.11397}).'
     - code_id: stab_4_2_2
-      detail: 'Recursively concatenating the \([[6,2,2]]\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel \cite{arxiv:1712.00294}.'
+      detail: 'Recursively concatenating the \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel \cite{arxiv:1712.00294}.'
     - code_id: stab_6_2_2
-      detail: 'Recursively concatenating the \([[6,2,2]]\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel \cite{arxiv:1712.00294}.'
+      detail: 'Recursively concatenating the \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel \cite{arxiv:1712.00294}.'
     - code_id: abelian_lifted_product
       detail: 'GKP codes have been concatenated with Abelian LP codes \cite{arxiv:2111.07029} that are in turn based on QC-LDPC codes \cite{doi:10.1109/TIT.2004.831841}.'
     - code_id: quantum_parity

codes/quantum/oscillators/tiger/tiger.yml

@@ -25,7 +25,7 @@ description: |
 
   Using multi-index notation, a projected coherent state can be written in two ways,
   \begin{align}
-    |\boldsymbol{\alpha}\rangle_{\boldsymbol{\Delta}}^{H}&\propto\int d\boldsymbol{\phi}e^{i\boldsymbol{\phi}(H\hat{\mathbf{n}}-\boldsymbol{\Delta})}|\boldsymbol{\alpha}\rangle\\&\propto\sum_{H\mathbf{n}=\boldsymbol{\Delta}}\frac{\boldsymbol{\alpha}^{\mathbf{n}}}{\sqrt{\mathbf{n}!}}|\mathbf{n}\rangle~,
+    |\boldsymbol{\alpha}\rangle_{\boldsymbol{\Delta}}^{H}&\propto\int \textnormal{d}\boldsymbol{\phi}e^{i\boldsymbol{\phi}(H\hat{\mathbf{n}}-\boldsymbol{\Delta})}|\boldsymbol{\alpha}\rangle\\&\propto\sum_{H\mathbf{n}=\boldsymbol{\Delta}}\frac{\boldsymbol{\alpha}^{\mathbf{n}}}{\sqrt{\mathbf{n}!}}|\mathbf{n}\rangle~,
   \end{align}
   where \(\boldsymbol{\alpha}\) is a complex vector, \(\boldsymbol{\Delta}\) is an integer vector, and \(\boldsymbol{\phi}\) is a vector of phases iterating over the elements of the group generated by \(H\).
   Tiger codewords are of the above form, and their phase-space values \(\boldsymbol{\alpha}\) lie on a torus embedded in the complex sphere of fixed-energy coherent coherent states, satisfying \(|\alpha_j|^2 = 1\).

codes/quantum/oscillators/uncategorized/penrose.yml

@@ -14,7 +14,7 @@ description: |
 
   Letting \(|T\rangle\) be a Penrose tiling, the codeword corresponding to this tiling is a superposition of all points in the tiling's orbit under all Euclidean transformations,
   \begin{align}
-    |\overline{T}\rangle=\int dg|gT\rangle~,
+    |\overline{T}\rangle=\int \textnormal{d}g|gT\rangle~,
   \end{align}
   where \(g\) is a Euclidean transformation.
 

codes/quantum/properties/approximate_qecc.yml

@@ -150,7 +150,7 @@ features:
     - 'The \textit{Petz recovery map} a.k.a. the \textit{transpose map} \cite{doi:10.1007/BF01212345,doi:10.1093/qmath/39.1.97,arxiv:1810.03150}, a quantum channel determined by the codespace and noise channel, yields an infidelity of recovery that is at most twice away from the infidelity of the best possible recovery \cite{arxiv:quant-ph/0004088}. 
     The fidelity can be expressed exactly as a function of the \term{Knill-Laflamme conditions} \cite[Thm. 1]{arxiv:2401.02022}, and it can be used to derive a generalization of the \term{Knill-Laflamme conditions} for approximate QECCs \cite{arxiv:0909.0931}.
     Satisfaction of the \term{Knill-Laflamme conditions} is sufficient but not necessary for the Petz recovery map to be the optimal recovery, and a necessary and sufficient condition has been derived \cite{arxiv:2410.23622}.
-    The infidelity of a modified Petz recovery map under erasure can be bounded using the conditional mutual information \cite{arxiv:1410.0664,arxiv:1509.07127,arxiv:1610.06169}.
+    The infidelity of a modified Petz recovery map under erasure can be bounded using the conditional mutual information via the \textit{approximate Petz theorem} \cite{arxiv:1410.0664,arxiv:1509.07127,arxiv:1610.06169}.
     In the case of topological codes, the Petz infidelity is related to the topological entanglement entropy \cite{arxiv:2408.00857}.
     Modifications include the Petz-like decoder \cite{arxiv:2405.06051}.'
     - 'The Yoshida-Kitaev decoder for the Hayden-Preskill protocol \cite{arxiv:1710.03363} can be extended to general QECCs \cite{arxiv:2405.06051}.'

codes/quantum/properties/asymmetric_qecc.yml

@@ -53,6 +53,8 @@ relations:
       detail: 'Random Clifford deformation can improve performance of surface codes against biased noise \cite{arxiv:2201.07802,arxiv:2211.02116}.'
     - code_id: xysurface
       detail: 'XY surface codes perform well against biased noise \cite{arxiv:1708.08474}.'
+    - code_id: xyz_product
+      detail: 'XYZ product codes can be used to protect against biased noise \cite{arxiv:2408.03123}.'
     - code_id: xyz_color
       detail: 'XYZ color codes perform well against biased noise \cite{arxiv:2203.16534}.'
     - code_id: twisted_xzzx

codes/quantum/properties/block/block_quantum.yml

@@ -56,7 +56,7 @@ protection: |
 
   \begin{defterm}{Quantum GV bound}
   \label{topic:quantum-gv-bound}
-  The \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{doi:10.1109/TIT.2004.838088} (see also Refs. \cite{arxiv:quant-ph/9602022,arxiv:quant-ph/9906131,doi:10.1109/18.959288,doi:10.1016/j.jmaa.2007.08.023}) for Galois qudits states that a \hyperref[topic:quantum-weight-enumerator]{pure} \([[n,k,d]]_q\) Galois-qudit stabilizer code exists if 
+  The \hyperref[topic:quantum-gv-bound]{quantum GV bound} \cite{doi:10.1109/TIT.2004.838088} (see also Refs. \cite{arxiv:quant-ph/9602022,arxiv:quant-ph/9906131,doi:10.7907/m0xg-zs21,doi:10.1109/18.959288,doi:10.1016/j.jmaa.2007.08.023}) for Galois qudits states that a \hyperref[topic:quantum-weight-enumerator]{pure} \([[n,k,d]]_q\) Galois-qudit stabilizer code exists if 
   \begin{align}
     \frac{q^{n-k+2}-1}{q^{2}-1}>\sum_{j=1}^{d-1}(q^{2}-1)^{j-1}\binom{n}{j}~.
   \end{align}