From 2d5b41e936fe91bd93c42f4fe0fb19a4ab1e2133 Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Tue, 13 Aug 2024 10:53:45 -0400 Subject: [PATCH 1/5] refs/corr --- codes/classical/properties/block/block.yml | 2 +- codes/quantum/qubits/qubits_into_qubits.yml | 2 +- .../qubits/stabilizer/hermitian/stabilizer_over_gf4.yml | 4 ++-- 3 files changed, 4 insertions(+), 4 deletions(-) diff --git a/codes/classical/properties/block/block.yml b/codes/classical/properties/block/block.yml index 6b6bee836..9cd855264 100644 --- a/codes/classical/properties/block/block.yml +++ b/codes/classical/properties/block/block.yml @@ -19,7 +19,7 @@ description: | \begin{defterm}{Asymptotic notation} \label{topic:asymptotics} - We are often interested in how parameters of particular infinite block-code families scale with increasing block length \(n\), necessitating the use of asymptotic or Bachmann–Landau notation. The table below summarizes the notation used throughout the EC Zoo for relating functions \(f,g\) that both grow with \(n\). + We are often interested in how parameters of particular infinite block-code families scale with increasing block length \(n\), necessitating the use of asymptotic or Bachmann–Landau notation; see the book \cite{manual:{Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2022). Introduction to algorithms. MIT press.}}. The table below summarizes the notation used throughout the EC Zoo for relating functions \(f,g\) that both grow with \(n\). \begin{table} \begin{cells} \celldata{relation & behavior} diff --git a/codes/quantum/qubits/qubits_into_qubits.yml b/codes/quantum/qubits/qubits_into_qubits.yml index b1bebb97f..82274f316 100644 --- a/codes/quantum/qubits/qubits_into_qubits.yml +++ b/codes/quantum/qubits/qubits_into_qubits.yml @@ -92,7 +92,7 @@ features: transversal_gates: - 'A qubit code is \(U\)-\textit{quasi-transversal} if it can realize the logical gate \(U\) in the third level of the \term{Clifford hierarchy} using the physical gate \(C T^{\otimes n}\), where \(C\) is some Clifford gate \cite[Def. 4]{arxiv:1606.01904}.' general_gates: - - 'The normalizer of the \hyperref[topic:complementary-channel]{Pauli group} is the Clifford group; see Ref. \cite{arxiv:1310.6813} for generators, relations, and normal form. The Clifford group permutes Pauli operators amongst themselves, and, up to any phases, is equivalent to the symplectic group \(Sp(2n,\mathbb{Z}_2)\). + - 'The normalizer of the \hyperref[topic:complementary-channel]{Pauli group} is the Clifford group; see Refs. \cite{arxiv:quant-ph/0304125,arxiv:0811.0898,arxiv:1310.6813} for generators, relations, and normal form. The Clifford group permutes Pauli operators amongst themselves, and, up to any phases, is equivalent to the symplectic group \(Sp(2n,\mathbb{Z}_2)\). The combined Pauli and Clifford group cannot be expressed as a semidirect product of those two constituents \cite{arxiv:2406.09951}.' - 'Computing using Clifford gates only can be efficiently simulated on a classical computer, according to the \textit{Gottesman-Knill theorem} \cite{arxiv:quant-ph/9807006,manual:{E. Knill, private communication, ca. 1998.}}. Universal quantum computing can be achieved using Clifford gates and a single type of non-Clifford gate, such as the \(T\) gate \cite{arxiv:quant-ph/9503016}. diff --git a/codes/quantum/qubits/stabilizer/hermitian/stabilizer_over_gf4.yml b/codes/quantum/qubits/stabilizer/hermitian/stabilizer_over_gf4.yml index 46c357184..a20ef7434 100644 --- a/codes/quantum/qubits/stabilizer/hermitian/stabilizer_over_gf4.yml +++ b/codes/quantum/qubits/stabilizer/hermitian/stabilizer_over_gf4.yml @@ -38,8 +38,8 @@ protection: | features: transversal_gates: - - 'Transversal \(SH\) gates \cite[Sec. 8.2]{arxiv:quant-ph/9703048}.' - - 'The three-block transversal gate mapping each \(X \to XYZ\) and each \(Z \to ZXY\) implements a logical gate \cite{arxiv:quant-ph/9702029}\cite[Exam. 2]{arxiv:quant-ph/9703048}.' + - 'Transversal \(SH\) gates \cite[Sec. 8.2]{arxiv:quant-ph/9705052}.' + - 'The three-block transversal gate mapping each physical \(X \to XYZ\) and each \(Z \to ZXY\) implements a logical gate \cite{arxiv:quant-ph/9702029}\cite[Exam. 2]{arxiv:quant-ph/9703048}.' fault_tolerance: - 'Characterizing fault-tolerant multi-qubit gates under the \hyperref[topic:gf4-representation]{\(GF(4)\) representation} may involve characterizing all global automorphisms of some number of copies of a code that preserve the symplectic inner product \cite[pg. 9]{arxiv:quant-ph/9703048}.' From ccd3c8bfa26e0c118586bebba788f71d0961a5a5 Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Tue, 13 Aug 2024 11:13:33 -0400 Subject: [PATCH 2/5] ref --- codes/quantum/qubits/stabilizer/qubit_stabilizer.yml | 1 + 1 file changed, 1 insertion(+) diff --git a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml index bc0df0b0e..8bbc37ec3 100644 --- a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml +++ b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml @@ -119,6 +119,7 @@ features: - 'Deep neural-network probabilistic decoder \cite{arxiv:1705.09334}.' - 'Generalized belief propagation (GBP) \cite{arxiv:2212.03214} based on a classical version \cite{manual:{J. S. Yedidia, W. T. Freeman, and Y. Weiss, Generalized belief propagation, in NIPS, Vol. 13 (2000) pp. 689–695.}}.' - 'Integer optimization decoder \cite{arxiv:2008.10206}.' + - 'Autonomous Lindbladian based decoders for codes encoding a single logical qubit \cite{arxiv:2308.16233}.' - 'For codes encoding a single logical qubit, logical information can be extracted by single-qubit operations and classical communication \cite{arxiv:2308.14054}.' - 'Correlated decoding can improve performance of Clifford and non-Clifford entangling gates \cite{arxiv:2403.03272}.' - 'Detector graphs \cite{arxiv:2103.02202,arxiv:2303.15933} and detector error models \cite{arxiv:2407.13826} can be used to design syndrome extraction circuits and logical measurements.' From 540f9813fcc3d641048025f1d5f3f2d37725dfb8 Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Tue, 13 Aug 2024 15:20:30 -0400 Subject: [PATCH 3/5] ~ --- codes/classical/rings/over_zq/easy/pentacode.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/codes/classical/rings/over_zq/easy/pentacode.yml b/codes/classical/rings/over_zq/easy/pentacode.yml index e6f22044b..583f3c367 100644 --- a/codes/classical/rings/over_zq/easy/pentacode.yml +++ b/codes/classical/rings/over_zq/easy/pentacode.yml @@ -20,7 +20,7 @@ relations: - code_id: small_distance cousins: - code_id: best - detail: 'Codewords of the Best code can be obtained by applying the Gray map to the pentacode \cite[Sec. 2]{doi:10.1007/BF01388558}.' + detail: 'Codewords of the Best code can be obtained by applying the Gray map to the pentacode \cite[Sec. 2]{doi:10.1007/bf01388558}.' # Begin Entry Meta Information From 087ed62f7eb91e3f5acfc7ff19caa472599a0a99 Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Tue, 13 Aug 2024 16:06:05 -0400 Subject: [PATCH 4/5] AEL ref --- codes/quantum/properties/hamiltonian/qltc.yml | 2 +- codes/quantum/properties/stabilizer/qldpc/good_qldpc.yml | 4 ++-- .../stabilizer/qldpc/quantum_locally_recoverable.yml | 4 ++-- .../stabilizer/topological/surface/higher_d/hemicubic.yml | 2 +- 4 files changed, 6 insertions(+), 6 deletions(-) diff --git a/codes/quantum/properties/hamiltonian/qltc.yml b/codes/quantum/properties/hamiltonian/qltc.yml index 815541a1a..ea5a4440d 100644 --- a/codes/quantum/properties/hamiltonian/qltc.yml +++ b/codes/quantum/properties/hamiltonian/qltc.yml @@ -33,7 +33,7 @@ protection: | \textit{Soundness amplification} \cite[Thm. 1.2]{arxiv:2309.05541} can be used to obtain a constant-soundness (i.e., \(R = O(1)\)) QLTC family from a CSS family with a sub-constant value, with the former's locality being at most polynomial in \(1/R\). - AEL distance amplification can be used to convert an \([[n^{\prime},k,d,w]]\) soundness-\(R\) CSS LTC family into an \([[n=n^{\prime}+O(1),k,d=O(n)]]\) family with \(w\) and \(R\) differing by a factor at most polynomial in \(w\) and \(n/d\) \cite[Thm. 1.3]{arxiv:2309.05541}. + AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} can be used to convert an \([[n^{\prime},k,d,w]]\) soundness-\(R\) CSS LTC family into an \([[n=n^{\prime}+O(1),k,d=O(n)]]\) family with \(w\) and \(R\) differing by a factor at most polynomial in \(w\) and \(n/d\) \cite[Thm. 1.3]{arxiv:2309.05541}. notes: - 'It was shown in Ref. \cite{doi:10.1109/FOCS.2017.46} that existence of a QLTC with constant parameters would implies resolution of the \textit{No low-energy trivial states} (NLTS) conjecture \cite{arxiv:1301.1363} (see also \cite{arxiv:2311.09503}). diff --git a/codes/quantum/properties/stabilizer/qldpc/good_qldpc.yml b/codes/quantum/properties/stabilizer/qldpc/good_qldpc.yml index ad4cde0af..5bf1355db 100644 --- a/codes/quantum/properties/stabilizer/qldpc/good_qldpc.yml +++ b/codes/quantum/properties/stabilizer/qldpc/good_qldpc.yml @@ -31,7 +31,7 @@ description: | features: - rate: 'The codes'' rate and distance are both separated from zero as block length goes to infinity. AEL distance amplification can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound \cite[Corr. 5.3]{arxiv:2212.09935}.' + rate: 'The codes'' rate and distance are both separated from zero as block length goes to infinity. AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound \cite[Corr. 5.3]{arxiv:2212.09935}.' relations: parents: @@ -40,7 +40,7 @@ relations: - code_id: translationally_invariant_stabilizer detail: 'Chain complexes describing some good QLDPC codes can be ''lifted'' into higher-dimensional manifolds admitting some notion of geometric locality \cite{arxiv:2012.02249,arxiv:2309.16104}. Applying this procedure to good QLDPC codes yiels \([[n,n^{1-2/D},n^{1-1/D}]]\) lattice stabilizer codes in \(D\) spatial dimensions that saturate the \hyperref[topic:bpt-bound]{BPT bound}, up to corrections poly-logarithmic in \(n\) \cite{arxiv:2303.06755}.' - code_id: quantum_mds - detail: 'AEL distance amplification can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound \cite[Corr. 5.3]{arxiv:2212.09935}.' + detail: 'AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound \cite[Corr. 5.3]{arxiv:2212.09935}.' # Begin Entry Meta Information diff --git a/codes/quantum/properties/stabilizer/qldpc/quantum_locally_recoverable.yml b/codes/quantum/properties/stabilizer/qldpc/quantum_locally_recoverable.yml index 193a24a40..2442f2209 100644 --- a/codes/quantum/properties/stabilizer/qldpc/quantum_locally_recoverable.yml +++ b/codes/quantum/properties/stabilizer/qldpc/quantum_locally_recoverable.yml @@ -21,12 +21,12 @@ protection: | \end{align} implying that locality restricts the distance of the code. Random QLRCs with qudit dimension \(q = 2^{O(r)}\) achieve a relative distance that is \hyperref[topic:asymptotics]{order} \(O(1/r)\) below the bound \cite[Prop. 5]{arxiv:2311.08653}. - Codes constructed with the help of AEL distance amplification admit a gap of \hyperref[topic:asymptotics]{order} \(O(1/r^{1/4})\) \cite[Prop. 6]{arxiv:2311.08653}. + Codes constructed with the help of AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} admit a gap of \hyperref[topic:asymptotics]{order} \(O(1/r^{1/4})\) \cite[Prop. 6]{arxiv:2311.08653}. features: decoders: - - 'Codes constructed with the help of AEL distance amplification admit efficient decoders \cite{arxiv:2311.08653}.' + - 'Codes constructed with the help of AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} admit efficient decoders \cite{arxiv:2311.08653}.' relations: parents: diff --git a/codes/quantum/qubits/stabilizer/topological/surface/higher_d/hemicubic.yml b/codes/quantum/qubits/stabilizer/topological/surface/higher_d/hemicubic.yml index f6a8a3f65..2181da3c2 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/higher_d/hemicubic.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/higher_d/hemicubic.yml @@ -20,7 +20,7 @@ relations: cousins: - code_id: qltc detail: 'The hemicubic code family has asymptotically diminishing soundness that scales as \hyperref[topic:asymptotics]{order} \(\Omega(1/\log n)\), locality of stabilizer generators scaling as \hyperref[topic:asymptotics]{order} \(O(\log n)\), and distance of \hyperref[topic:asymptotics]{order} \(\Theta(\sqrt{n})\). - Soundness amplification and AEL distance amplification can also yield improvements in various parameters \cite[Table 3]{arxiv:2309.05541}. + Soundness amplification and AEL distance amplification \cite{doi:10.1109/SFCS.1995.492581,doi:10.1109/18.556669} can also yield improvements in various parameters \cite[Table 3]{arxiv:2309.05541}. Application of generalized distance balancing \cite{arxiv:2004.07935} to hemicubic codes using an asymptotically good classical code of length \(t\) yields \(O(1/(\log(n) t^2))\) soundness and \hyperref[topic:asymptotics]{order} \(\Theta(\sqrt{n}t)\) distance while maintaining locality scaling and at the expense of a dimension scaling as \hyperref[topic:asymptotics]{order} \(\Theta(t^2)\) \cite{arxiv:2305.00689}.' - code_id: distance_balanced detail: 'Application of generalized distance balancing \cite{arxiv:2004.07935} to hemicubic codes using an asymptotically good classical code of length \(t\) yields \(O(1/(\log(n) t^2))\) soundness and \hyperref[topic:asymptotics]{order} \(\Theta(\sqrt{n}t)\) distance while maintaining locality scaling and at the expense of a dimension scaling as \hyperref[topic:asymptotics]{order} \(\Theta(t^2)\) \cite{arxiv:2305.00689}.' From 61af19027cd54e4b6ab982c0693ce5a73b488195 Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Tue, 13 Aug 2024 16:27:38 -0400 Subject: [PATCH 5/5] topic:clifford --- codes/classical/analog/lattice/bw/barnes_wall.yml | 2 +- .../classical/spherical/group_orbit/sidelnikov.yml | 6 +++--- .../spherical/polytope/disphenoidal288cell.yml | 2 +- .../oscillators/coherent_state/clifford_group.yml | 2 +- .../oscillators/coherent_state/clifford_qsc.yml | 4 ++-- .../block/dynamic/monitored_random_circuits.yml | 4 ++-- .../block/tensor_network/holographic_tensor.yml | 2 +- .../quantum/qubits/dynamic/random/haar_random.yml | 4 ++-- codes/quantum/qubits/qubits_into_qubits.yml | 14 +++++++++++--- codes/quantum/qubits/small_distance/iceberg.yml | 2 +- .../qubits/small_distance/small/stab_5_1_2.yml | 4 ++-- .../stabilizer/hermitian/stabilizer_over_gf4.yml | 2 +- .../qubits/stabilizer/holographic/happy.yml | 2 +- .../magic/k-divisible/quantum_divisible.yml | 2 +- .../quantum/qubits/stabilizer/qubit_stabilizer.yml | 2 +- .../topological/color/3d_color/stab_15_1_3.yml | 2 +- .../2d_surface/stellated_dodecahedron_css.yml | 2 +- codes/quantum/spins/single_spin/j_gross.yml | 2 +- 18 files changed, 34 insertions(+), 26 deletions(-) diff --git a/codes/classical/analog/lattice/bw/barnes_wall.yml b/codes/classical/analog/lattice/bw/barnes_wall.yml index 6b5346a36..ff4bebc6c 100644 --- a/codes/classical/analog/lattice/bw/barnes_wall.yml +++ b/codes/classical/analog/lattice/bw/barnes_wall.yml @@ -15,7 +15,7 @@ description: | Member of a family of \(2^{m+1}\)-dimensional lattices, denoted as BW\(_{2^{m+1}}\), that are the densest lattices known. Members include the integer square lattice \(\mathbb{Z}^2\), \(D_4\), the Gosset \(E_8\) lattice, and the \(\Lambda_{16}\) lattice, corresponding to \(m\in\{0,1,2,3\}\), respectively. - Its automorphism group is the Clifford group \cite{arxiv:math/0001038,arxiv:0712.1939,arxiv:2404.17677}. + Its automorphism group is the \hyperref[topic:clifford]{real Clifford group} \cite{arxiv:math/0001038,arxiv:0712.1939,arxiv:2404.17677}. protection: | BW lattices in dimension \(2^{m+1}\) have a nominal coding gain of \(2^{m/2}\). diff --git a/codes/classical/spherical/group_orbit/sidelnikov.yml b/codes/classical/spherical/group_orbit/sidelnikov.yml index 0ca3f6660..292293890 100644 --- a/codes/classical/spherical/group_orbit/sidelnikov.yml +++ b/codes/classical/spherical/group_orbit/sidelnikov.yml @@ -12,10 +12,10 @@ introduced: '\cite{manual:{V. M. Sidelnikov, On a finite group of matrices and c description: | Slepian group-orbit code of dimension \(2^r\), approximate asympotic size \(2.38 \cdot 2^{r(r+1)/2+1}\), and distance \(1\). - Code is constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group \cite{arxiv:math/0001038}, onto the vector \((1,0,0,\cdots,0)\). + Code is constructed by applying elements of an index-two subgroup of the \hyperref[topic:clifford]{real Clifford group}, when taken as a subgroup of the orthogonal group \cite{arxiv:math/0001038}, onto the vector \((1,0,0,\cdots,0)\). This group is the automorphism group of BW lattice, and the resulting codes coincide with the optimal spherical codes for dimensions \(\{4,8,16\}\). - Taking the orbit under the entire real Clifford group yields spherical codes twice the points and with distance \(2-\sqrt{2}\). + Taking the orbit under the entire \hyperref[topic:clifford]{real Clifford group} yields spherical codes twice the points and with distance \(2-\sqrt{2}\). relations: parents: @@ -24,7 +24,7 @@ relations: detail: 'The orbit of any point under the real Clifford subgroup is a spherical 7-design \cite{doi:10.1023/A:1018723416627}, and some are 11-designs \cite{manual:{V. M. Sidelnikov, “Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial”, Algebra i Analiz, 11:4 (1999), 183–203; St. Petersburg Math. J., 11:4 (2000), 673–686}}.' cousins: - code_id: barnes_wall - detail: 'The automorphism group of BW lattices is a subgroup of index 2 of a real Clifford group \cite{manual:{V. M. Sidelnikov, On a finite group of matrices and codes on the Euclidean sphere (in Russian), Probl. Peredach. Inform. 33 (1997), 35–54 (1997); English translation in Problems Inform. Transmission 33 (1997), 29–44},doi:10.1109/ISIT.1997.613373} (see \cite{arxiv:math/0001038,arxiv:2404.17677} for an explanation).' + detail: 'The automorphism group of BW lattices is a subgroup of index 2 of a \hyperref[topic:clifford]{real Clifford group} \cite{manual:{V. M. Sidelnikov, On a finite group of matrices and codes on the Euclidean sphere (in Russian), Probl. Peredach. Inform. 33 (1997), 35–54 (1997); English translation in Problems Inform. Transmission 33 (1997), 29–44},doi:10.1109/ISIT.1997.613373} (see \cite{arxiv:math/0001038,arxiv:2404.17677} for an explanation).' # Begin Entry Meta Information _meta: diff --git a/codes/classical/spherical/polytope/disphenoidal288cell.yml b/codes/classical/spherical/polytope/disphenoidal288cell.yml index e6b5a7d4b..b7863cdb1 100644 --- a/codes/classical/spherical/polytope/disphenoidal288cell.yml +++ b/codes/classical/spherical/polytope/disphenoidal288cell.yml @@ -23,7 +23,7 @@ relations: detail: 'The disphenoidal 288-cell code forms a spherical 7-design \cite{doi:10.1109/ITW.2003.1216742}.' cousins: - code_id: sidelnikov - detail: 'The disphenoidal 288-cell code is a group-orbit code with the group being the real Clifford group in \(4\) dimensions.' + detail: 'The disphenoidal 288-cell code is a group-orbit code with the group being the \hyperref[topic:clifford]{real Clifford group} in \(4\) dimensions.' # Begin Entry Meta Information diff --git a/codes/quantum/oscillators/coherent_state/clifford_group.yml b/codes/quantum/oscillators/coherent_state/clifford_group.yml index 4b3b3fbcb..17fc53254 100644 --- a/codes/quantum/oscillators/coherent_state/clifford_group.yml +++ b/codes/quantum/oscillators/coherent_state/clifford_group.yml @@ -12,7 +12,7 @@ introduced: '\cite{arxiv:2302.11593}' description: | A \(((2^r,2,2-\sqrt{2},8))\) QSC for \(r \geq 2\) constructed using the real-Clifford subgroup-orbit code. - Logical constellations are constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group \cite{arxiv:math/0001038} to \(2\) different vectors on the complex sphere. + Logical constellations are constructed by applying elements of an index-two subgroup of the \hyperref[topic:clifford]{real Clifford group}, when taken as a subgroup of the orthogonal group \cite{arxiv:math/0001038} to \(2\) different vectors on the complex sphere. The code is known as the \textit{Witting code} for \(r=2\) because its two logical constellations form vertices of Witting polytopes. relations: diff --git a/codes/quantum/oscillators/coherent_state/clifford_qsc.yml b/codes/quantum/oscillators/coherent_state/clifford_qsc.yml index e6c35e82a..51ad8f3c1 100644 --- a/codes/quantum/oscillators/coherent_state/clifford_qsc.yml +++ b/codes/quantum/oscillators/coherent_state/clifford_qsc.yml @@ -11,12 +11,12 @@ name: 'Clifford group-representation QSC' introduced: '\cite{arxiv:2306.11621}' description: | - QSC whose projection is onto a copy of an irreducible representation of the single-qubit Clifford group \(2O\), taken as the binary octahedral subgroup of the group \(SU(2)\) of Gaussian rotations. + QSC whose projection is onto a copy of an irreducible representation of the single-qubit \hyperref[topic:clifford]{Clifford group} \(2O\), taken as the binary octahedral subgroup of the group \(SU(2)\) of Gaussian rotations. Its codewords consist of non-uniform superpositions of 48 coherent states. features: general_gates: - - 'The Clifford group \(2O\) can be realized via Gaussian rotations. + - 'The \hyperref[topic:clifford]{Clifford group} \(2O\) can be realized via Gaussian rotations. The \(T\) and \(CZ\) gates can be realized using quartic Kerr operations \cite{arxiv:2306.11621}.' diff --git a/codes/quantum/properties/block/dynamic/monitored_random_circuits.yml b/codes/quantum/properties/block/dynamic/monitored_random_circuits.yml index 4f1396e82..72c384a63 100644 --- a/codes/quantum/properties/block/dynamic/monitored_random_circuits.yml +++ b/codes/quantum/properties/block/dynamic/monitored_random_circuits.yml @@ -10,7 +10,7 @@ introduced: '\cite{doi:10.1103/PhysRevX.9.031009,doi:10.1103/PhysRevB.98.205136, description: | Error-correcting code arising from a monitored random circuit. Such a circuit is described by a series of intermittant random local projective Pauli measurements with random unitary time-evolution operators. - An important sub-family consists of \textit{Clifford monitored random circuits}, where unitaries are sampled from the Clifford group \cite{arxiv:1901.08092}. + An important sub-family consists of \textit{Clifford monitored random circuits}, where unitaries are sampled from the \hyperref[topic:clifford]{Clifford group} \cite{arxiv:1901.08092}. When the rate of projective measurements is independently controlled by a probability parameter \(p\), there can exist two stable phases, one described by volume-law entanglement entropy and the other by area-law entanglement entropy. The phases and their transition can be understood from the perspective of quantum error correction, information scrambling, and channel capacities \cite{arxiv:1903.05124,arxiv:1905.05195}. @@ -60,7 +60,7 @@ relations: - code_id: topological detail: 'Topological order can be generated in 2D monitored random circuits \cite{arxiv:2011.06595}.' - code_id: random_stabilizer - detail: 'An important sub-family of monitored random-circuit codes are the Clifford monitored random-circuit codes, where unitaries are sampled from the Clifford group \cite{arxiv:1901.08092}.' + detail: 'An important sub-family of monitored random-circuit codes are the Clifford monitored random-circuit codes, where unitaries are sampled from the \hyperref[topic:clifford]{Clifford group} \cite{arxiv:1901.08092}.' # Begin Entry Meta Information diff --git a/codes/quantum/properties/block/tensor_network/holographic_tensor.yml b/codes/quantum/properties/block/tensor_network/holographic_tensor.yml index 035021d2b..b240d02be 100644 --- a/codes/quantum/properties/block/tensor_network/holographic_tensor.yml +++ b/codes/quantum/properties/block/tensor_network/holographic_tensor.yml @@ -33,7 +33,7 @@ features: transversal_gates: - 'There exist holographic approximate codes with arbitrary transversal gate sets for any compact Lie group \cite{arxiv:2108.11402}. - However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the Clifford group \cite{arxiv:2103.13404}.' + However, for sufficiently localized logical subsystems of holographic stabilizer codes, the set of transversally implementable logical operations is contained in the \hyperref[topic:clifford]{Clifford group} \cite{arxiv:2103.13404}.' code_capacity_threshold: - 'The ideal holographic tensor-network code (perfect representation of AdS/CFT) should be able to protect a central bulk operator against erasures of half of the physical qubits on the boundary, in line with AdS-Rindler reconstruction \cite{arxiv:1503.06237}.' diff --git a/codes/quantum/qubits/dynamic/random/haar_random.yml b/codes/quantum/qubits/dynamic/random/haar_random.yml index 4b148a448..be66fff7a 100644 --- a/codes/quantum/qubits/dynamic/random/haar_random.yml +++ b/codes/quantum/qubits/dynamic/random/haar_random.yml @@ -31,9 +31,9 @@ relations: - code_id: random_circuit cousins: - code_id: local_haar_random - detail: 'Approximating the random projections through \(t\)-designs is necessary in order to make the protocol practical. Replacing with random Clifford gates is especially convenient since the Clifford group forms a unitary 2-design and produces stabilizer codes.' + detail: 'Approximating the random projections through \(t\)-designs is necessary in order to make the protocol practical. Replacing with random Clifford gates is especially convenient since the \hyperref[topic:clifford]{Clifford group} forms a unitary 2-design and produces stabilizer codes.' - code_id: t-designs - detail: 'Approximating the random projections through \(t\)-designs is necessary in order to make the protocol practical. Replacing with random Clifford gates is especially convenient since the Clifford group forms a unitary 2-design and produces stabilizer codes.' + detail: 'Approximating the random projections through \(t\)-designs is necessary in order to make the protocol practical. Replacing with random Clifford gates is especially convenient since the \hyperref[topic:clifford]{Clifford group} forms a unitary 2-design and produces stabilizer codes.' # Begin Entry Meta Information diff --git a/codes/quantum/qubits/qubits_into_qubits.yml b/codes/quantum/qubits/qubits_into_qubits.yml index 82274f316..0211afc07 100644 --- a/codes/quantum/qubits/qubits_into_qubits.yml +++ b/codes/quantum/qubits/qubits_into_qubits.yml @@ -92,8 +92,16 @@ features: transversal_gates: - 'A qubit code is \(U\)-\textit{quasi-transversal} if it can realize the logical gate \(U\) in the third level of the \term{Clifford hierarchy} using the physical gate \(C T^{\otimes n}\), where \(C\) is some Clifford gate \cite[Def. 4]{arxiv:1606.01904}.' general_gates: - - 'The normalizer of the \hyperref[topic:complementary-channel]{Pauli group} is the Clifford group; see Refs. \cite{arxiv:quant-ph/0304125,arxiv:0811.0898,arxiv:1310.6813} for generators, relations, and normal form. The Clifford group permutes Pauli operators amongst themselves, and, up to any phases, is equivalent to the symplectic group \(Sp(2n,\mathbb{Z}_2)\). - The combined Pauli and Clifford group cannot be expressed as a semidirect product of those two constituents \cite{arxiv:2406.09951}.' + - | + \begin{defterm}{Clifford group} + \label{topic:clifford} + The Clifford group is the normalizer of the \hyperref[topic:pauli]{Pauli group}. + The group consists of the Pauli group as well as elements that permute Pauli operators amongst themselves. + Up to any phases and Pauli strings, the group is equivalent to the symplectic group \(Sp(2n,\mathbb{Z}_2)\). + See Refs. \cite{arxiv:quant-ph/0304125,arxiv:0811.0898,arxiv:1310.6813,preset:GottesmanBook} for generators, relations, and normal form. + The group cannot be expressed as a semidirect product of the Pauli and symplectic groups \cite{arxiv:2406.09951}. + Restricting the group to real-valued elements yields the \textit{real Clifford group}. + \end{defterm} - 'Computing using Clifford gates only can be efficiently simulated on a classical computer, according to the \textit{Gottesman-Knill theorem} \cite{arxiv:quant-ph/9807006,manual:{E. Knill, private communication, ca. 1998.}}. Universal quantum computing can be achieved using Clifford gates and a single type of non-Clifford gate, such as the \(T\) gate \cite{arxiv:quant-ph/9503016}. More generally, the \textit{Solovay-Kitaev} theorem \cite{doi:10.1070/rm1997v052n06abeh002155,doi:10.1090/gsm/047} states that any subset of gates the generates a dense subgroup of the full \(n\)-qubit gate group can be used to construct any gate to arbitrary accuracy (see \cite{arxiv:quant-ph/0505030}\cite[Appx. 3]{doi:10.1017/cbo9780511976667.019}). The task of approximating a desired gate by Clifford gates and a fixed set of non-Clifford gates is called \textit{gate compilation} or \textit{circuit synthesis}.' @@ -106,7 +114,7 @@ features: \begin{align} C_k = \{ U | U P U^{\dagger} \in C_{k-1} \}~, \end{align} - where \(P\) is any Pauli matrix, and \(C_1\) is the \hyperref[topic:complementary-channel]{Pauli group}. + where \(P\) is any Pauli matrix, and \(C_1\) is the \hyperref[topic:pauli]{Pauli group}. \end{defterm}' - 'Arbitrary \(n\)-qubit circuits can be implemented fault-tolerantly in a 3D architecture using \(O(n^{3/2}\log^3 n)\) qubits, and in a 2D architecture using only \(O(n^2 \log^3 n)\) qubits \cite{arxiv:2402.13863}.' decoders: diff --git a/codes/quantum/qubits/small_distance/iceberg.yml b/codes/quantum/qubits/small_distance/iceberg.yml index dc7509cb8..95a70f86e 100644 --- a/codes/quantum/qubits/small_distance/iceberg.yml +++ b/codes/quantum/qubits/small_distance/iceberg.yml @@ -22,7 +22,7 @@ description: | Admits a basis such that each codeword is a superposition of a computational basis state labeled by an even-weight bitstring \(b\) and a state labeled by the negation of \(b\). Its all-zero logical state is a conventional GHz state. - All of its automorphisms lie in the Clifford group \cite[Thm. 13]{arxiv:quant-ph/9704043}. + All of its automorphisms lie in the \hyperref[topic:clifford]{Clifford group} \cite[Thm. 13]{arxiv:quant-ph/9704043}. protection: 'Detects a single-qubit error.' diff --git a/codes/quantum/qubits/small_distance/small/stab_5_1_2.yml b/codes/quantum/qubits/small_distance/small/stab_5_1_2.yml index 1df5e1588..c581c9617 100644 --- a/codes/quantum/qubits/small_distance/small/stab_5_1_2.yml +++ b/codes/quantum/qubits/small_distance/small/stab_5_1_2.yml @@ -17,9 +17,9 @@ description: | features: general_gates: - - 'Fault-tolerant implementation of the Clifford group based on transversal gates and SWAPs \cite{arxiv:2112.01446}.' + - 'Fault-tolerant implementation of the \hyperref[topic:clifford]{Clifford group} based on transversal gates and SWAPs \cite{arxiv:2112.01446}.' fault_tolerance: - - 'Fault-tolerant implementation of the Clifford group based on transversal gates and SWAPs \cite{arxiv:2112.01446}.' + - 'Fault-tolerant implementation of the \hyperref[topic:clifford]{Clifford group} based on transversal gates and SWAPs \cite{arxiv:2112.01446}.' relations: diff --git a/codes/quantum/qubits/stabilizer/hermitian/stabilizer_over_gf4.yml b/codes/quantum/qubits/stabilizer/hermitian/stabilizer_over_gf4.yml index a20ef7434..b15701bc9 100644 --- a/codes/quantum/qubits/stabilizer/hermitian/stabilizer_over_gf4.yml +++ b/codes/quantum/qubits/stabilizer/hermitian/stabilizer_over_gf4.yml @@ -22,7 +22,7 @@ description: | Quaternary linear codes are Hermitian self-orthogonal (self-dual) iff they are trace-Hermitian self-orthogonal (self-dual) additive \cite[Thm. 27.4.1]{preset:HKSquantum} (\cite[Thm. 9.10.3]{doi:10.1017/CBO9780511807077}). In other words, if the underlying quaternary code is linear, then the \hyperref[topic:finite-fields]{field trace} can be removed from the definition of inner product. - All of its automorphisms lie in the Clifford group \cite[Corr. 16]{arxiv:quant-ph/9704043}. + All of its automorphisms lie in the \hyperref[topic:clifford]{Clifford group} \cite[Corr. 16]{arxiv:quant-ph/9704043}. protection: | A Hermitian self-orthogonal linear \([n,k,d]_{4}\) code yields an \([[n,n-2k]]\) qubit stabilizer code with distance no less than \(d\); this is the \textit{qubit Hermitian construction}. diff --git a/codes/quantum/qubits/stabilizer/holographic/happy.yml b/codes/quantum/qubits/stabilizer/holographic/happy.yml index c9bdc4275..2d005cd7f 100644 --- a/codes/quantum/qubits/stabilizer/holographic/happy.yml +++ b/codes/quantum/qubits/stabilizer/holographic/happy.yml @@ -45,7 +45,7 @@ features: - 'Heisenberg-picture encoding is done through \textit{tensor pushing}. Each bulk operator (logical) is pushed to an operator supported on a portion of the boundary region (physical). Pushing all the bulk operators through results in reconstruction of the boundary.' transversal_gates: - - 'For locality-preserving physical gates on the boundary, the set of transversally implementable logical operations in the bulk is strictly contained in the Clifford group \cite{arxiv:2103.13404}.' + - 'For locality-preserving physical gates on the boundary, the set of transversally implementable logical operations in the bulk is strictly contained in the \hyperref[topic:clifford]{Clifford group} \cite{arxiv:2103.13404}.' decoders: - 'Hierarchical recovery model \cite{arxiv:1503.06237}.' diff --git a/codes/quantum/qubits/stabilizer/magic/k-divisible/quantum_divisible.yml b/codes/quantum/qubits/stabilizer/magic/k-divisible/quantum_divisible.yml index bb29ab534..70587d08f 100644 --- a/codes/quantum/qubits/stabilizer/magic/k-divisible/quantum_divisible.yml +++ b/codes/quantum/qubits/stabilizer/magic/k-divisible/quantum_divisible.yml @@ -43,7 +43,7 @@ relations: - code_id: generalized_quantum_divisible detail: 'Generalized level-\(\nu\) quantum divisible codes reduce to quantum level-\(\nu\) divisible codes when \(t\) is a vector with \(\pm 1\) entries. The classical code formed by their \(X\)-type stabilizer generator matrix is \(\nu\)-even \cite[pg. 10]{arxiv:1709.08658}. - Both types of codes realize transversal gates outside of the Clifford group.' + Both types of codes realize transversal gates outside of the \hyperref[topic:clifford]{Clifford group}.' cousins: - code_id: divisible detail: 'The \(X\)-type stabilizers of a level-\(\nu\) quantum divisible code form a \(\nu\)-even linear binary code.' diff --git a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml index 8bbc37ec3..f5e3042fa 100644 --- a/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml +++ b/codes/quantum/qubits/stabilizer/qubit_stabilizer.yml @@ -190,7 +190,7 @@ relations: - code_id: single_shot 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}.' + detail: 'Stabilizer states on \(n\) qubits form complex projective 3-designs \cite{arxiv:1510.02767}, while the \hyperref[topic:clifford]{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 \(d-1\) \hyperref[topic:ad]{AD} errors \cite{arxiv:1001.2356}.' - code_id: ampdamp diff --git a/codes/quantum/qubits/stabilizer/topological/color/3d_color/stab_15_1_3.yml b/codes/quantum/qubits/stabilizer/topological/color/3d_color/stab_15_1_3.yml index 8b97b0fc0..64682907c 100644 --- a/codes/quantum/qubits/stabilizer/topological/color/3d_color/stab_15_1_3.yml +++ b/codes/quantum/qubits/stabilizer/topological/color/3d_color/stab_15_1_3.yml @@ -29,7 +29,7 @@ features: magic_scaling_exponent: 'Magic-state yield parameter \( \gamma= \log_d (n/k)\approx 2.47\) \cite[Box 2]{arxiv:1612.07330}\cite{arxiv:1703.07847}.' transversal_gates: - - 'This is the smallest qubit stabilizer code with a (strongly) transversal gate outside of the Clifford group \cite{arxiv:2210.14066}.' + - 'This is the smallest qubit stabilizer code with a (strongly) transversal gate outside of the \hyperref[topic:clifford]{Clifford group} \cite{arxiv:2210.14066}.' - 'A transversal logical \(T^\dagger\) is implemented by applying a \(T\) gate on every qubit \cite{arxiv:quant-ph/9610011,arxiv:1403.2734,arxiv:1612.07330}.' - 'A subsystem version yields a transversal \(CCZ\) gate \cite{arxiv:1304.3709}.' diff --git a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/stellated_dodecahedron_css.yml b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/stellated_dodecahedron_css.yml index 12a0bde24..cc1252a7f 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/stellated_dodecahedron_css.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/stellated_dodecahedron_css.yml @@ -19,7 +19,7 @@ description: | features: transversal_gates: - - 'Clifford group of four of the eight logical qubits can be done by transversal gates combined with qubit permutations \cite{arxiv:2202.06647}.' + - '\hyperref[topic:clifford]{Clifford group} of four of the eight logical qubits can be done by transversal gates combined with qubit permutations \cite{arxiv:2202.06647}.' relations: parents: diff --git a/codes/quantum/spins/single_spin/j_gross.yml b/codes/quantum/spins/single_spin/j_gross.yml index c90a1a905..4a0f742f5 100644 --- a/codes/quantum/spins/single_spin/j_gross.yml +++ b/codes/quantum/spins/single_spin/j_gross.yml @@ -14,7 +14,7 @@ description: | A single-spin code designed to realize a discrete group of gates using \(SU(2)\) rotations. Codewords are subspaces of a spin's Hilbert space that house irreducible representations (irreps) of a discrete subgroup of \(SU(2)\). - The first realization \cite{arxiv:2005.10910} used the single-qubit Clifford group (i.e., the binary octahedral (\(2O\)) subgroup of \(SU(2)\)). + The first realization \cite{arxiv:2005.10910} used the single-qubit \hyperref[topic:clifford]{Clifford group} (i.e., the binary octahedral (\(2O\)) subgroup of \(SU(2)\)). Code construction is done by restricting the \(SU(2)\) irrep to \(2O\), and determining the carrier spaces of any nontrivial irreps of \(2O\). Since irreps of \(2O\) do not appear in integer spins, half-integer spins are used. A simple example of a codespace is a projection onto an instance of a particular irrep of \(2O\), referred to as either \( \varrho_4 \) or \( \varrho_5 \).