From 7a5ad3530514528e0a9cd26ac0d27d20f3f6e594 Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Thu, 18 Jul 2024 15:44:37 -0400 Subject: [PATCH] ~ --- .../coherent_state/cat_concatenated.yml | 16 +++++++++------- .../coherent_state/two-legged-cat.yml | 4 ---- codes/quantum/oscillators/hybrid/hybrid_cat.yml | 10 +++++----- .../stabilizer/hyperplane/cv_cluster_state.yml | 4 ++-- .../oscillators/stabilizer/lattice/dfour_gkp.yml | 5 ++--- .../stabilizer/lattice/gkp-cluster-state.yml | 15 +++++++-------- .../stabilizer/lattice/gkp_concatenated.yml | 2 +- .../lattice/gkp_surface_concatenated.yml | 4 ++-- .../stabilizer/lattice/quantum_lattice.yml | 2 +- codes/quantum/qubits/qubit_concatenated.yml | 4 +--- .../ea_stabilizer/ea_galois_stabilizer.yml | 4 ++-- .../ea_stabilizer/ea_quantum_lcd.yml | 4 +--- 12 files changed, 33 insertions(+), 41 deletions(-) diff --git a/codes/quantum/oscillators/coherent_state/cat_concatenated.yml b/codes/quantum/oscillators/coherent_state/cat_concatenated.yml index 27cb64196..047f76786 100644 --- a/codes/quantum/oscillators/coherent_state/cat_concatenated.yml +++ b/codes/quantum/oscillators/coherent_state/cat_concatenated.yml @@ -10,10 +10,12 @@ name: 'Concatenated cat code' introduced: '\cite{arxiv:1409.6759}' description: | - A concatenated code whose outer code is a cat code. Most examples encode physical qubits of an inner stabilizer codes into the two-component cat code. + A concatenated code whose outer code is a cat code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another cat outer code. Most examples encode physical qubits of an inner stabilizer code into the two-component cat code. protection: | - The cat code suppressed dephasing errors exponentially with the size of the coherent states, so the inner code (e.g., a quantum repetition code \cite{arxiv:1904.09474,arxiv:1905.00450,arxiv:2009.10756,arxiv:2212.11927}) can be highly biased toward one type of noise while still ensuring good performance. + The cat code suppresses dephasing errors exponentially with the size of its coherent states, so the inner code (e.g., a quantum repetition code \cite{arxiv:1904.09474,arxiv:1905.00450,arxiv:2009.10756,arxiv:2212.11927,arxiv:2212.11927}) can be highly biased toward one type of noise while still ensuring good performance. + + A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode \cite{arxiv:2205.09767}. relations: @@ -21,7 +23,7 @@ relations: - code_id: qsc cousins: - code_id: quantum_repetition - detail: 'Cat codes have been concatenated with quantum repetition codes \cite{arxiv:1904.09474,arxiv:1905.00450,arxiv:2009.10756,arxiv:2012.04108,arxiv:2212.11927}.' + detail: 'Two-component cat codes have been concatenated with quantum repetition codes \cite{arxiv:1904.09474,arxiv:1905.00450,arxiv:2009.10756,arxiv:2012.04108,arxiv:2212.11927}.' - code_id: rotated_surface detail: 'Cat codes have been concatenated with rotated surface codes \cite{arxiv:2012.04108}.' - code_id: ldpc @@ -29,11 +31,11 @@ relations: - code_id: lhz detail: 'LHZ parity-codes have been concatenated with cat codes \cite{arxiv:2404.11332}.' - code_id: steane - detail: 'Steane codes have been concatenated with cat codes \cite{arxiv:0707.0327}.' + detail: 'Two-component cat codes concatenated with Steane and Golay codes are estimated to be fault tolerant against photon loss noise with rate \(\eta < 5\times 10^{-4}\) provided that \(\alpha > 1.2\) \cite{arxiv:0707.0327}.' - code_id: qubit_golay - detail: 'Quantum Golay codes have been concatenated with cat codes \cite{arxiv:0707.0327}.' - - + detail: 'Two-component cat codes concatenated with Steane and Golay codes are estimated to be fault tolerant against photon loss noise with rate \(\eta < 5\times 10^{-4}\) provided that \(\alpha > 1.2\) \cite{arxiv:0707.0327}.' + - code_id: self_correct + detail: 'A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode \cite{arxiv:2205.09767}.' # Begin Entry Meta Information diff --git a/codes/quantum/oscillators/coherent_state/two-legged-cat.yml b/codes/quantum/oscillators/coherent_state/two-legged-cat.yml index 5b8172a2b..7a8b6f58b 100644 --- a/codes/quantum/oscillators/coherent_state/two-legged-cat.yml +++ b/codes/quantum/oscillators/coherent_state/two-legged-cat.yml @@ -66,10 +66,6 @@ relations: detail: 'Two-legged cat and quantum repetition codes can be thought of as classical codes because they protect against only one type of noise. Two-legged cat codes (quantum repetition) codes suppress cavity dephasing (bit-flip) noise exponentially with \(|\alpha|^2\) (\(n\)). The stability offered by cat codes has been linked to several favorable properties of phases of matter associated with the repetition-code Hamiltonian \cite{arxiv:1804.11293,arxiv:2008.02816}.' - code_id: coherent_state_c-q detail: 'Two-component cat codes can be thought of as coherent-state c-q codes because they protect against only one type of noise and thus only reliably store classical information.' - - code_id: self_correct - detail: 'A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode \cite{arxiv:2205.09767}.' - - code_id: cat_concatenated - detail: 'A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode \cite{arxiv:2205.09767}.' # Begin Entry Meta Information diff --git a/codes/quantum/oscillators/hybrid/hybrid_cat.yml b/codes/quantum/oscillators/hybrid/hybrid_cat.yml index 306ef68f4..d2a7287de 100644 --- a/codes/quantum/oscillators/hybrid/hybrid_cat.yml +++ b/codes/quantum/oscillators/hybrid/hybrid_cat.yml @@ -11,10 +11,10 @@ name: 'Hybrid cat code' introduced: '\cite{arxiv:1112.0825,arxiv:2401.00450}' description: | - A hybrid qubit-oscillator code admitting codewords that are tensor products of either a cat or coherent state and a single-qubit (photon polarization) state. + A hybrid qubit-oscillator code admitting codewords that are tensor products of a single-qubit (e.g., photon polarization) state with either a cat state or a coherent state. - Codewords of the coherent-state version \cite{arxiv:1112.0825} are \(|\alpha\rangle|+\rangle\) and \(|-\alpha\rangle|-\rangle\), i.e., hyper-entangled states of the polarization \(|\pm\rangle\) and occupation-number degrees of freedom of a photon, with the latter being in a coherent state \(|\pm\alpha\rangle\). - Codewords of a cat-state version \cite{arxiv:1712.10206,arxiv:2401.00450} are \((\left|\alpha\right\rangle +\left|-\alpha\right\rangle )|+\rangle\) and \((\left|i\alpha\right\rangle -\left|-i\alpha\right\rangle )|-\rangle\) + Codewords of the coherent-state version \cite{arxiv:1112.0825} are \(|\alpha\rangle|+\rangle\) and \(|-\alpha\rangle|-\rangle\), i.e., hyper-entangled states \cite{doi:10.1080/09500349708231877} of the occupation-number and polarization degrees of freedom of a photon. + Codewords of the cat-state version \cite{arxiv:1712.10206,arxiv:2401.00450} are proportional to \((\left|\alpha\right\rangle +\left|-\alpha\right\rangle )|+\rangle\) and \((\left|i\alpha\right\rangle -\left|-i\alpha\right\rangle )|-\rangle\) features: fault_tolerance: @@ -28,12 +28,12 @@ relations: - code_id: oscillators detail: 'The physical Hilbert space of a hybrid qubit-oscillator code contains at least one oscillator.' cousins: - - code_id: qudits_into_oscillators - detail: 'A hybrid qudit-oscillator code with \(n_1=0\) is a qudit-into-oscillator code.' - code_id: cat detail: 'Hybrid cat codewords consist of a bosonic mode in either coherent or cat states.' - code_id: rbh detail: 'Hybrid cat codes can be concatenated with RBH codes \cite{arxiv:2401.00450}.' + - code_id: oscillators_concatenated + detail: 'Hybrid cat codes can be concatenated with RBH codes \cite{arxiv:2401.00450}.' diff --git a/codes/quantum/oscillators/stabilizer/hyperplane/cv_cluster_state.yml b/codes/quantum/oscillators/stabilizer/hyperplane/cv_cluster_state.yml index fee79d18e..1f8f467d5 100644 --- a/codes/quantum/oscillators/stabilizer/hyperplane/cv_cluster_state.yml +++ b/codes/quantum/oscillators/stabilizer/hyperplane/cv_cluster_state.yml @@ -20,9 +20,9 @@ description: | The exact analog cluster state is non-normalizable, so approximate constructs have to be considered. Analog cluster states are analog stabilizer states defined on a graph. - There is one nullifier \(\eta_j\) per graph vertex \(j\) of the form + There is one nullifier \(\hat{\eta}_j\) per graph vertex \(j\) of the form \begin{align} - \eta_j = \hat{p}_{j} - \sum_{k\in N(j)} V_{jk} \hat{x}_k~, + \hat{\eta}_j = \hat{p}_{j} - \sum_{k\in N(j)} V_{jk} \hat{x}_k~, \end{align} where the neighborhood \(N(j)\) is the set of vertices which share an edge with \(j\), and where \(V_{jk}\) is a weighed (real-valued) adjacency matrix of a graph \cite{arxiv:1912.06463}. diff --git a/codes/quantum/oscillators/stabilizer/lattice/dfour_gkp.yml b/codes/quantum/oscillators/stabilizer/lattice/dfour_gkp.yml index cf2a9b0e6..2f8b69043 100644 --- a/codes/quantum/oscillators/stabilizer/lattice/dfour_gkp.yml +++ b/codes/quantum/oscillators/stabilizer/lattice/dfour_gkp.yml @@ -19,12 +19,11 @@ features: relations: parents: - - code_id: multimodegkp + - code_id: gkp_concatenated + detail: 'The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code \cite{arxiv:2201.12337}. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via \term{Construction A}.' - code_id: qudits_into_oscillators cousins: - code_id: dfour - - code_id: gkp_concatenated - detail: 'The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code \cite{arxiv:2201.12337}. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via \term{Construction A}.' - code_id: quantum_repetition detail: 'The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code \cite{arxiv:2201.12337}. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via \term{Construction A}.' diff --git a/codes/quantum/oscillators/stabilizer/lattice/gkp-cluster-state.yml b/codes/quantum/oscillators/stabilizer/lattice/gkp-cluster-state.yml index fdea9b457..4cc3de822 100644 --- a/codes/quantum/oscillators/stabilizer/lattice/gkp-cluster-state.yml +++ b/codes/quantum/oscillators/stabilizer/lattice/gkp-cluster-state.yml @@ -14,28 +14,27 @@ alternative_names: - 'Hybrid cluster-state code' description: | - This code can be thought of as a generalized analog cluster state that is initialized in GKP (resource) states for some of its physical modes. - Alternatively, it can be thought of as an oscillator-into-oscillator GKP code whose encoding consists of initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i \theta_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(\theta_{jk}\). + Cluster-state code can consists of a generalized analog cluster state that is initialized in GKP (resource) states for some of its physical modes. + Alternatively, it can be thought of as an oscillator-into-oscillator GKP code whose encoding consists of initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i V_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(V_{jk}\). Provides a way to perform fault-tolerant MBQC, with the required number \(n-k\) of GKP-encoded physical modes determined by the particular protocol \cite{arxiv:1310.7596,arxiv:2010.02905,arxiv:1712.00294,arxiv:2104.03241}. - Logical Clifford gates are performed on the cluster state via a combination of linear-optical gates and homodyne measurements on subsets of vertices \cite{arxiv:quant-ph/0605198,arxiv:0903.3233}. Magic-state distillation is required for universal computation. GKP error correction can be naturally combined with CV measurement-based protocols since the performance of both is quantified by a squeezing parameter. features: encoders: - 'Initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i \theta_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(\theta_{jk}\).' general_gates: + - 'Logical Clifford gates are performed on the cluster state via a combination of linear-optical gates and homodyne measurements on subsets of vertices \cite{arxiv:quant-ph/0605198,arxiv:0903.3233}. Magic-state distillation is required for universal computation.' - 'Single-mode logical Clifford gates can be performed using Gaussian operations and measurements on a 1D GKP cluster state, while two-mode logical Clifford gates require a 2D cluster state. Magic-state distillation using photon-counting can be used for a non-Clifford logical \(\pi/8\) gate.' - 'Gate teleportation and error correction can be performed without active squeezing \cite{arxiv:2008.12791}.' + decoders: + - 'GKP error correction can be naturally combined with CV MBQC protocols since the performance of both is quantified by a squeezing parameter \cite{arxiv:1310.7596}.' threshold: - 'A lower bound on the squeezing required to obtain a particular error rate can be formulated in terms of the displacement noise strength. This in turn determines how much squeezing is required in order to be below threshold for a particular concatenated code. A threshold of \(10^{-6}\) yields a required squeezing of 20.5 dB \cite{arxiv:1310.7596}. Anti-squeezing does not affect the threshold \cite{arxiv:1903.02162}.' - fault_tolerance: - - 'First encoding demonstrating the possibility of fault-tolerant measurement-based computation with analog cluster states. A fault-tolerance threshold can be achieved by concatenating existing fault-tolerant schemes for qubit-based cluster-state encodings with the GKP code \cite{arxiv:1310.7596}.' - relations: parents: - - code_id: quantum_lattice - detail: 'A GKP CV-cluster-state code can be created by initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i \theta_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(\theta_{jk}\).' + - code_id: gkp-stabilizer + detail: 'A GKP CV-cluster-state code can be created by initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i V_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(V_{jk}\).' - code_id: qudits_into_oscillators cousins: - code_id: cv_cluster_state diff --git a/codes/quantum/oscillators/stabilizer/lattice/gkp_concatenated.yml b/codes/quantum/oscillators/stabilizer/lattice/gkp_concatenated.yml index 5a609d23e..cba351f26 100644 --- a/codes/quantum/oscillators/stabilizer/lattice/gkp_concatenated.yml +++ b/codes/quantum/oscillators/stabilizer/lattice/gkp_concatenated.yml @@ -10,7 +10,7 @@ name: 'Concatenated GKP code' introduced: '\cite{arxiv:1706.03011}' description: | - A concatenated code whose outer code is a GKP code. + A concatenated code whose outer code is a GKP code. In other words, a bosonic code that can be thought of as a concatenation of an arbitrary inner code and another bosonic outer code. Most examples encode physical qubits of an inner stabilizer code into the square-lattice GKP code. protection: | The analog syndrome information of the outer GKP code can improve protection of the inner code. As an example, concatenating a three-qubit quantum repetition code with GKP codes can correct some two-bit-flip errors \cite{arxiv:1706.03011}. diff --git a/codes/quantum/oscillators/stabilizer/lattice/gkp_surface_concatenated.yml b/codes/quantum/oscillators/stabilizer/lattice/gkp_surface_concatenated.yml index e9809be93..4b32b794f 100644 --- a/codes/quantum/oscillators/stabilizer/lattice/gkp_surface_concatenated.yml +++ b/codes/quantum/oscillators/stabilizer/lattice/gkp_surface_concatenated.yml @@ -10,7 +10,7 @@ name: 'GKP-surface code' introduced: '\cite{arxiv:1712.00294,arxiv:1810.00047}' description: | - A concatenated code whose outer code is a GKP code and whose inner code is a toric code \cite{arxiv:1810.00047}, surface code \cite{arxiv:1712.00294}, rotated surface code \cite{arxiv:1908.03579,arxiv:2101.03014,arxiv:2103.06994}, or XZZX surface code \cite{arxiv:2207.04383}. + A concatenated code whose outer code is a GKP code and whose inner code is a toric surface code \cite{arxiv:1810.00047}, rotated surface code \cite{arxiv:1712.00294,arxiv:1908.03579,arxiv:2101.03014,arxiv:2103.06994,arxiv:2303.04702}, or XZZX surface code \cite{arxiv:2207.04383}. @@ -35,7 +35,7 @@ relations: - code_id: toric detail: 'GKP codes have been concatenated with toric codes \cite{arXiv:1810.00047}.' - code_id: rotated_surface - detail: 'GKP codes have been concatenated with rotated surface codes \cite{arxiv:1908.03579,arxiv:2101.03014,arxiv:2103.06994}.' + detail: 'GKP codes have been concatenated with rotated surface codes \cite{arxiv:1712.00294,arxiv:1908.03579,arxiv:2101.03014,arxiv:2103.06994,arxiv:2303.04702}.' - code_id: xzzx detail: 'GKP codes have been concatenated with XZZX surface codes \cite{arxiv:2207.04383}.' - code_id: asymmetric_qecc diff --git a/codes/quantum/oscillators/stabilizer/lattice/quantum_lattice.yml b/codes/quantum/oscillators/stabilizer/lattice/quantum_lattice.yml index 790fdf3bc..c52b5aef2 100644 --- a/codes/quantum/oscillators/stabilizer/lattice/quantum_lattice.yml +++ b/codes/quantum/oscillators/stabilizer/lattice/quantum_lattice.yml @@ -44,7 +44,7 @@ relations: detail: 'Quantum lattice codewords can be written as superpositions of coherent states lying on a lattice in phase space \cite{arxiv:quant-ph/0008040,arxiv:1708.05010}.' cousins: - code_id: points_into_lattices - detail: 'Quantum lattice codes can be thought of as quantum lattice codes because they store information in quantum superpositions of points on a lattice in quantum phase space.' + detail: 'Quantum lattice codes can be thought of as quantum analogues of lattices because they store information in quantum superpositions of points on a lattice in quantum phase space.' - code_id: css detail: 'Quantum lattice codes defined on rectangular lattices are CSS codes. There is no known relation to chain complexes for such codes. diff --git a/codes/quantum/qubits/qubit_concatenated.yml b/codes/quantum/qubits/qubit_concatenated.yml index 5f9ad33f4..959a26996 100644 --- a/codes/quantum/qubits/qubit_concatenated.yml +++ b/codes/quantum/qubits/qubit_concatenated.yml @@ -10,9 +10,7 @@ logical: qudits name: 'Concatenated qubit code' description: | - A concatenated code whose outer code is a qubit code. In other words, a qubit code that can be thought of as a concatenation of a possibly non-qubit inner code and another qubit outer code. - - A combination of two qubit codes, an inner code \(C\) and an outer code \(C^\prime\), where the physical subspace used for the inner code consists of the logical subspace of the outer code. + A concatenated code whose outer code is a qubit code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another qubit outer code. An inner \(C = ((n_1,K,d_1))\) and outer \(C^\prime = ((n_2,2,d_2))\) qubit code yield an \(((n_1 n_2, K, d \geq d_1d_2))\) concatenated qubit code. Concatenating an \(((n,2,d))\) qubit code can be done recursively, with the \(r\)\textit{th level} of concatenation yielding an \(((n^r,2,d^r))\) code. diff --git a/codes/quantum/qudits_galois/ea_stabilizer/ea_galois_stabilizer.yml b/codes/quantum/qudits_galois/ea_stabilizer/ea_galois_stabilizer.yml index 38a4476f5..4ca50c26c 100644 --- a/codes/quantum/qudits_galois/ea_stabilizer/ea_galois_stabilizer.yml +++ b/codes/quantum/qudits_galois/ea_stabilizer/ea_galois_stabilizer.yml @@ -23,9 +23,9 @@ relations: - code_id: ea_galois_into_galois cousins: - code_id: galois_stabilizer - detail: 'Pure Galois-qudit codes can be used to make EA Galois-qudit stabilizer codes \cite{arxiv:1105.5872}\cite[Thm. 10]{arxiv:2010.07902}.' + detail: 'EA Galois-qudit stabilizer codes utilize additional ancillary Galois-qudits in a pre-shared entangled state, but reduce to Galois-qudit stabilizer codes when said qudits are interpreted as noiseless physical qudits. Pure Galois-qudit codes can be used to make EA Galois-qudit stabilizer codes \cite{arxiv:1105.5872}\cite[Thm. 10]{arxiv:2010.07902}.' - code_id: galois_grs - detail: 'Galios-qudit GRS codes can be used to construct EA Galois-qudit stabilizer codes \cite{arxiv:1606.00134,doi:10.1007/s11128-021-03028-w}.' + detail: 'Galois-qudit GRS codes can be used to construct EA Galois-qudit stabilizer codes \cite{arxiv:1606.00134,doi:10.1007/s11128-021-03028-w}.' # Begin Entry Meta Information diff --git a/codes/quantum/qudits_galois/ea_stabilizer/ea_quantum_lcd.yml b/codes/quantum/qudits_galois/ea_stabilizer/ea_quantum_lcd.yml index c1c63daff..161be59e1 100644 --- a/codes/quantum/qudits_galois/ea_stabilizer/ea_quantum_lcd.yml +++ b/codes/quantum/qudits_galois/ea_stabilizer/ea_quantum_lcd.yml @@ -19,10 +19,8 @@ features: relations: parents: - - code_id: ea_galois_into_galois + - code_id: ea_galois_stabilizer cousins: - - code_id: galois_stabilizer - detail: 'Pure Galois-qudit codes can be used to make EA QECCs \cite{arxiv:1105.5872}\cite[Thm. 10]{arxiv:2010.07902}.' - code_id: lcd - code_id: maximal_entanglement_galois_stabilizer detail: 'Asymptotically good maximal-entanglement EA Galois-qudit stabilizer codes can be constructed from LCD codes \cite{arxiv:1606.00134}.'