refs + entropic uncertainty

codes/quantum/qubits/small_distance/small/stab_4_2_2.yml

@@ -46,7 +46,7 @@ features:
 realizations:
   - '\([[4,1,2]]\) subcode implemented using four-qubit graph state of photons \cite{arXiv:1404.5498}.'
   - 'Trapped-ion device by IonQ \cite{arXiv:1611.06946}.'
-  - 'Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM \cite{arXiv:1705.09259,arXiv:2110.04285}.'
+  - 'Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM \cite{arXiv:1705.09259,arXiv:1705.08957,arxiv:1806.02359,arXiv:2110.04285}.'
   - 'The subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) \cite{arXiv:1912.09410} and \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \cite{arXiv:2102.06132}, treated as a planar surface code, has been realized in superconducting-circuit devices.'
   - 'Logical gates between two copies of the subcode \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\), interpreted as lattice surgery between planar surface codes, realized in superconducting circuits \cite{arXiv:2006.03071}.'
   - 'Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits \cite{arXiv:2102.13071}.'

codes/quantum/qubits/small_distance/small/stab_5_1_3.yml

@@ -47,7 +47,7 @@ features:
     The depth of syndrome extraction circuits can be lowered by using past syndrome values \cite{arxiv:2305.00784}.'
 
 realizations:
-  - 'NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking \cite{arXiv:quant-ph/0101034}. Single-qubit logical gates \cite{arXiv:1208.4797}.'
+  - 'NMR: Implementation of perfect error correcting code on 5 spin subsystem of labeled crotonic acid for quantum network benchmarking \cite{arXiv:quant-ph/0101034}. Single-qubit logical gates \cite{arXiv:1208.4797}. Magic-state distillation using 7-qubit device \cite{arxiv:1103.2178}.'
   - 'Superconducting qubits \cite{arXiv:1907.04507}.'
   - 'Trapped-ion qubits: non-transversal CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 12-qubit device by Quantinuum \cite{arxiv:2208.01863}.'
   - 'Nitrogen-vacancy centers in diamond: fault-tolerant single-qubit Clifford operations \cite{arxiv:2108.01646}.'

codes/quantum/qubits/small_distance/small/steane.yml

@@ -42,7 +42,7 @@ description: |
 protection: 'The Steane code is a distance 3 code. It detects errors on 2 qubits, corrects errors on 1 qubit.'
 
 realizations:
-  - 'Trapped-ion qubits: seven-qubit device in Blatt group \cite{arXiv:1403.5426}, ten-qubit QCCD device by Quantinuum \cite{arXiv:2107.07505} (see APS Physics Synopsys \cite{doi:10.1103/Physics.14.184}). Fault-tolerant universal two-qubit gate set by Monz group \cite{arxiv:2111.12654}. Logical CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum \cite{arxiv:2208.01863}; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits.'
+  - 'Trapped-ion qubits: seven-qubit device in Blatt group \cite{arXiv:1403.5426}, ten-qubit QCCD device by Quantinuum \cite{arXiv:2107.07505} (see APS Physics Synopsis \cite{doi:10.1103/Physics.14.184}). Fault-tolerant universal two-qubit gate set by Monz group \cite{arxiv:2111.12654}. Logical CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum \cite{arxiv:2208.01863}; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits.'
   - 'Rydberg atom arrays: Lukin group \cite{arXiv:2112.03923}.'
 
 features:

codes/quantum/qubits/stabilizer/qubit_css.yml

@@ -69,6 +69,13 @@ description: |
 
   Usually, the chain complex \eqref{eq:chain} used in the construction comes from the chain complex associated with a cellulation of a manifold. When the manifold is a two-dimensional surface, its entire chain is used. Higher-dimensional manifolds allow for longer chain complexes, and one can use the three largest non-trivial vector spaces in its chain.
 
+  CSS codes saturate a type of \textit{error correction uncertainty relation} \cite[Thm. 3]{doi:10.1103/PhysRevLett.77.793}, which is a special case of an entropic uncertainty relation between a pair of bases \cite{doi:10.1007/BF01608825,doi:10.1103/PhysRevLett.50.631,doi:10.1103/PhysRevLett.60.1103}.
+  The code state \(\sum_{c\in C_{X}}|c\rangle\) can be expressed in terms of either basis states labeled by the code \(C_{X}\) or its dual, satisfying, with equality, the relation
+  \begin{align}
+    |C_{X}||C_{X}^{\perp}| \geq 2^{n}\,.
+  \end{align}
+
+
 #  The reverse mapping is as follows \cite{arXiv:1311.0885,arXiv:1807.09783}. Given a CSS code with parity check matrices \(H_X\) and \(H_Z\), let both boundary operators be \(\partial = H_Z^TUH_X\) for an arbitrary invertible matrix \(U\). The fact that the stabilizer generators commute ensures that the boundary operator satisfies \(\partial^2=0\), yielding a chain complex.
 #  Then, the normalizer of the stabilizers \({\mathsf{N}}(C_X)\) and \({\mathsf{N}}(C_Z)\) is \(\text{Ker}(\partial_2^T)\) and \(\text{Ker}(\partial_1)\), respectively.
 # Insert table linking manifold properties to QEC