ampdamp

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

@@ -29,7 +29,7 @@ features:
     - 'Error-detecting \(CCZ\) and \(cSWAP\) gates using three-level ancilla \cite{arxiv:2212.11196}.'
 
   fault_tolerance:
-    - 'Dual-rail qubits can be used to convert leakage and amplitude damping noise into erasure noise \cite{arxiv:2208.05461}.'
+    - 'Dual-rail qubits can be used to convert leakage and amplitude damping noise into erasure noise \cite{arxiv:0710.1052,arxiv:2208.05461}.'
 
   threshold:
     - 'Between \(1.78\%\) and \(11.5\%\) with faulty photon detectors when repeatedly concatenating with the Steane code \cite{arxiv:quant-ph/0502101}.'
@@ -47,17 +47,18 @@ notes:
 
 
 relations:
-
   parents:
     - code_id: one_hot_quantum
   cousins:
     - code_id: quantum_concatenated
       detail: 'The KLM protocol, one of the first protocols for fault-tolerant quantum computation, utilizes concatenations of the dual-rail with a stabilizer code \cite{doi:10.1038/35051009}.
       Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code \cite{arxiv:2010.00538} that protects against \(d-1\) amplitude damping errors \cite{arxiv:1001.2356}.'
+    - code_id: ampdamp
+      detail: 'Dual-rail qubits can be used to convert leakage and amplitude damping noise into erasure noise \cite{arxiv:0710.1052,arxiv:2208.05461}. Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code \cite{arxiv:2010.00538} that protects against \(d-1\) amplitude damping errors \cite{arxiv:1001.2356}.'
     - code_id: quantum_parity
-      detail: 'An \([[8,1,2]]\) QPC correcting a single amplitude damping error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code \cite{arxiv:quant-ph/0103042,arxiv:quant-ph/0501184,arxiv:2010.00538}.'
+      detail: 'An \([[8,1,2]]\) QPC correcting a single amplitude damping error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code \cite{arxiv:quant-ph/0103042,arxiv:quant-ph/0501184,arxiv:2010.00538}. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) amplitude damping errors \cite{arxiv:1001.2356}.'
     - code_id: stab_4_2_2
-      detail: 'An \([[8,1,2]]\) QPC correcting a single amplitude damping error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code \cite{arxiv:quant-ph/0103042,arxiv:quant-ph/0501184,arxiv:2010.00538}.'
+      detail: 'An \([[8,1,2]]\) QPC correcting a single amplitude damping error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code \cite{arxiv:quant-ph/0103042,arxiv:quant-ph/0501184,arxiv:2010.00538}. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) amplitude damping errors \cite{arxiv:1001.2356}.'
     - code_id: cluster_state
       detail: 'The KLM protocol can be combined cluster states in various ways \cite{arxiv:quant-ph/0303008,arxiv:quant-ph/0402005,arxiv:quant-ph/0405157}.'
 

codes/quantum/oscillators/fock_state/fock_state.yml

@@ -20,6 +20,8 @@ protection: 'Code distance \(d\) is the minimum distance (assuming some metric)
 relations:
   parents:
     - code_id: qudits_into_oscillators
+    - code_id: ampdamp
+      detail: 'Fock-state codes are designed to protect against AD noise.'
   cousins:
     - code_id: bits_into_bits
       detail: 'Fock-state code distance is a natural extension of Hamming distance between binary strings.'

codes/quantum/oscillators/oscillators.yml

@@ -45,7 +45,9 @@ protection: |
 
   \subsection{Loss and gain operators}
 
-  An error set relevant to \hyperref[code:fock_state]{Fock-state bosonic} codes is the set of loss operators associated with the \textit{amplitude damping} (a.k.a. \textit{photon loss} or \textit{attenuation}) noise channel, a common form of physical noise in bosonic systems. For a single mode, loss operators are proportional to powers of the mode's annihilation operator \(a=(\hat{x}+i\hat{p})/\sqrt{2}\), where \(\hat x\) (\(\hat p\)) is the mode's position (momentum) operator, and with the power signifying the number of particles lost during the error. For multiple modes, error set elements are tensor products of elements of the single-mode error set. A definition of distance associated with this error set is the minimum weight of a loss error that implements a nontrivial logical operation in the code.
+  An error set relevant to \hyperref[code:fock_state]{Fock-state bosonic} codes is the set of loss operators associated with the \hyperref[code:ampdamp]{amplitude damping channel}, a common form of physical noise in bosonic systems. 
+  For a single mode, loss operators are proportional to powers of the mode's annihilation operator \(a=(\hat{x}+i\hat{p})/\sqrt{2}\), where \(\hat x\) (\(\hat p\)) is the mode's position (momentum) operator, and with the power signifying the number of particles lost during the error. 
+  For multiple modes, error set elements are tensor products of elements of the single-mode error set. 
 
   \subsection{Number-phase operators}
 

codes/quantum/properties/block/ampdamp.yml

@@ -0,0 +1,42 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: ampdamp
+# Qubit or bosonic
+
+name: 'Amplitude-damping (AD) code'
+
+description: |
+  Block quantum code on either qubits or bosonic modes that is designed to detect and correct qubit or bosonic amplitude (AD) errors, respectively.
+
+  The \textit{amplitude damping channel} is a bosonic channel that models loss of particles in the bosonic mode (a.k.a. \textit{photon loss} or \textit{attenuation}).
+  Its Kraus operators are proportional to powers of a mode's annihilation operator \(a\), with the power signifying the number of particles lost during the error,
+  \begin{align}
+    E_{\ell}=\left(\frac{\gamma}{1-\gamma}\right)^{\ell/2}\frac{a^{\ell}}{\sqrt{\ell!}}\left(1-\gamma\right)^{\hat{n}/2}\,,
+  \end{align}
+  where \(\gamma\in[0,1)\) is the noise rate \cite{doi:10.1088/0954-8998/1/2/005,arxiv:1708.05010}.
+  Restricting the channel to the first two Fock states \(\{|0\rangle,|1\rangle\}) yields the non-Pauli qubit amplitude damping channel, which requires protecting against the loss error \(E_1\propto X+iY\) instead of \(X\) and \(Y\) Pauli errors individually.
+  Both channels are called AD since the context makes clear which one is being referred to.
+
+protection: |
+  A definition of distance associated with amplitude damping is the minimum weight of a tensor products of non-zero loss errors (\(\ell\neq 0\)) that implement a nontrivial logical operation in the code.
+  Certain codes also have intrinsic protection against AD, such as constant-excitation codes or self-complementary codes.
+
+relations:
+  parents:
+    - code_id: block_quantum
+  cousins:
+    - code_id: qubit_css
+      detail: 'An \([[n,k,d_Z=t+1,d_X=2t+1]]\) qubit CSS code protects against \(t\) amplitude damping errors \cite[Sec. 8.7]{arxiv:quant-ph/9705052}\cite{arxiv:1001.2356}.'
+    - code_id: quantum_concatenated
+      detail: 'Concatenated quantum codes can protect against amplitude damping \cite{arxiv:1601.07423}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-07-14'

codes/quantum/properties/hamiltonian/constant_excitation.yml

@@ -47,8 +47,8 @@ relations:
         Any \([[n,k,d]]\) CSS code can be made into an \([[mn,k,>d]]\) CE code \cite{doi:10.1109/ISIT45174.2021.9518206}.
     - code_id: qsc
       detail: 'QSC (CE) codewords are superpositions of coherent states (Fock states) with the same energy.'
-    - code_id: qubit_stabilizer
-      detail: 'Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code \cite{arxiv:2010.00538}.'
+    - code_id: ampdamp
+      detail: 'Fock-state CE codes are protected from identical amplitude damping acting on all modes because the damping acts on all codewords identically \cite{doi:10.1103/PhysRevA.56.1114}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/nonstabilizer/jump.yml

@@ -27,6 +27,7 @@ notes:
 relations:
   parents:
     - code_id: qubits_into_qubits
+    - code_id: ampdamp
     - code_id: constant_excitation
   cousins:
     - code_id: stab_4_2_2

codes/quantum/qubits/nonstabilizer/self_complementary.yml

@@ -18,12 +18,14 @@ description: |
 
 
 protection: |
-  Self-complementary codes automatically protect against a single \(Z\) error \cite{arxiv:quant-ph/0701065}, and can protect against a single amplitude damping error \cite{arxiv:0907.5149}.
+  Self-complementary codes automatically protect against a single \(Z\) error \cite{arxiv:quant-ph/0701065}, and can protect against a single amplitude damping error \cite{arxiv:0712.2586,arxiv:0907.5149}.
 
 
 relations:
   parents:
     - code_id: qubits_into_qubits
+    - code_id: ampdamp
+      detail: 'Self-complementary quantum codes consisting of codewords that are sufficient spaced apart in the configuration space correct at least one amplitude damping error \cite[Thm. 2.5]{arxiv:0712.2586}\cite[Thm. 2]{arxiv:0907.5149}.'
   cousins:
     - code_id: binary_linear
       detail: 'A linear binary code is called \textit{self-complementary} if, for each codeword \(c\), its negation \(\overline{c}\) is also a codeword.'

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

@@ -37,6 +37,8 @@ relations:
   parents:
     - code_id: qudit_gnu_permutation_invariant
       detail: 'Qudit GNU codes encoding logical qubits reduce to GNU codes.'
+    - code_id: ampdamp
+      detail: 'GNU codes protect approximately against amplitude damping errors.'
   cousins:
     - code_id: combinatorial_permutation_invariant
       detail: 'Combinatorial PI codes \(Q_{g,(m-1)/2,g-1,+}\) are GNU codes for odd \(m\) \cite[Prop. 5.4]{arxiv:2310.05358}.'

codes/quantum/qubits/qubits_into_qubits.yml

@@ -48,7 +48,7 @@ protection: |
 
   A quantum channel whose Kraus operators are Pauli strings is called a \textit{Pauli channel}, and such channels are typically more tractable than general, non-Pauli channels.
   Relevant Pauli channels include dephasing noise and depolarizing noise (a.k.a. Werner-Holevo channel \cite{arxiv:quant-ph/0203003}).
-  Relevant non-Pauli channels are amplitude damping (which requires protection against \(X+iY\) instead of \(X,Y\) individually), erasure (which maps all qubit states into a third state \(|e\rangle\) outside of the qubit Hilbert space), and biased erasure (in which case only the \(|1\rangle\) qubit state is mapped to \(|e\rangle\)).
+  Relevant non-Pauli channels are \hyperref[code:ampdamp]{amplitude damping channel}, erasure (which maps all qubit states into a third state \(|e\rangle\) outside of the qubit Hilbert space), and biased erasure (in which case only the \(|1\rangle\) qubit state is mapped to \(|e\rangle\)).
   Noise can be correlated in space or in time, with the latter being an example of a non-Markovian phenomenon \cite{arxiv:2012.01894}.
 
   \subsection{Bounds on code parameters}

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

@@ -102,6 +102,8 @@ relations:
       detail: 'The \([[4,2,2]]\) code can be derived from the five-qubit code using a protocol that converts an \([[n,k,d]]\) code into an \([[n-1, k+1, d-1]]\) code \cite[Sec. 3.5]{arxiv:quant-ph/9705052}.'
     - code_id: approximate_qecc
       detail: 'The \([[4,1,2]]\) subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) approximately corrects a single amplitude damping error \cite{arxiv:quant-ph/9704002}.'
+    - code_id: ampdamp
+      detail: 'The \([[4,1,2]]\) subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) approximately corrects a single amplitude damping error \cite{arxiv:quant-ph/9704002}.'
     - code_id: binomial
       detail: '\([[4,1,2]]\) subcode consisting of \(|\overline{00}\rangle\) and any other codeword reduces to the \(0,2,4\) binomial code when the basis labels in each codeword are written as in base-ten.
       Such a mapping can be generalized \cite{manual:{Linshu Li, private communication, 2018}}.'
@@ -114,7 +116,7 @@ relations:
         Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826,arxiv:2402.09606}.'
         Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code \cite{arxiv:1604.04062}.
         The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code \cite{arxiv:1604.04062}.
-        An \([[8,1,2]]\) QPC correcting a single amplitude damping error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code \cite{arxiv:quant-ph/0103042,arxiv:quant-ph/0501184,arxiv:2010.00538}.
+        An \([[8,1,2]]\) QPC correcting a single amplitude damping error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code \cite{arxiv:quant-ph/0103042,arxiv:quant-ph/0501184,arxiv:2010.00538}. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) amplitude damping errors \cite{arxiv:1001.2356}.
     - code_id: cws
       detail: 'A \([[4,1,2]]\) subcode can be thought of as a CWS code
        \cite{arxiv:1407.2777}.'

codes/quantum/qubits/stabilizer/css/quantum_parity.yml

@@ -55,6 +55,8 @@ notes:
 relations:
   parents:
     - code_id: generalized_shor
+    - code_id: amdamp
+      detail: 'An \([[8,1,2]]\) QPC correcting a single amplitude damping error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code \cite{arxiv:quant-ph/0103042,arxiv:quant-ph/0501184,arxiv:2010.00538}. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) amplitude damping errors \cite{arxiv:1001.2356}.'
     - code_id: group_quantum_parity
       detail: 'A \([[m_1 m_2,1,\min(m_1,m_2)]]_G\) group-based QPC reduces to a QPC for \(G=\mathbb{Z}_2\).'
     - code_id: quantum_lego

codes/quantum/qubits/stabilizer/qubit_stabilizer.yml

@@ -181,6 +181,10 @@ relations:
       detail: 'Any stabilizer code can be single shot if sufficiently non-local high-weight stabilizer generators are used for syndrome measurements.  These can be obtained with a Gaussian elimination procedure \cite{arxiv:1805.09271}.'
     - code_id: t-designs
       detail: 'Stabilizer states on \(n\) qubits form complex projective 3-designs \cite{arxiv:1510.02767}, while the Clifford group is a unitary 3-design \cite{arXiv:1510.02619,arXiv:1510.02769}.'
+    - code_id: constant_excitation
+      detail: 'Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code \cite{arxiv:2010.00538} that protects against \(d-1\) amplitude damping errors \cite{arxiv:1001.2356}.'
+    - code_id: ampdamp
+      detail: 'Concatenating the dual-rail code with an \([[n,k,d]]\) stabilizer code yields an \([[2n,k,d]]\) constant-excitation code \cite{arxiv:2010.00538} that protects against \(d-1\) amplitude damping errors \cite{arxiv:1001.2356}.'
 
 
 # Begin Entry Meta Information