From 40bba9bbb9f23badd8f995acd21582e17ddbbbbd Mon Sep 17 00:00:00 2001 From: VVA2024 Date: Mon, 12 Aug 2024 18:07:19 -0400 Subject: [PATCH] ~ --- codes/quantum/properties/asymmetric_qecc.yml | 2 +- codes/quantum/qubits/qubit_concatenated.yml | 5 ++++- codes/quantum/qubits/qubits_into_qubits.yml | 9 +++++---- codes/quantum/qubits/small_distance/small/stab_5_1_3.yml | 2 +- .../qldpc/concatenated/concatenated_steane.yml | 6 +++--- codes/quantum/qubits/stabilizer/quantum_bch.yml | 2 +- .../quantum/qubits/stabilizer/rm/quantum_hamming_css.yml | 2 +- .../topological/color/3d_color/stab_15_1_3.yml | 2 +- .../topological/surface/2d_surface/surface.yml | 2 +- .../qubits/subsystem/qldpc/bacon_shor/bacon_shor.yml | 8 ++++---- 10 files changed, 22 insertions(+), 18 deletions(-) diff --git a/codes/quantum/properties/asymmetric_qecc.yml b/codes/quantum/properties/asymmetric_qecc.yml index 9039b0646..876acb9ac 100644 --- a/codes/quantum/properties/asymmetric_qecc.yml +++ b/codes/quantum/properties/asymmetric_qecc.yml @@ -31,7 +31,7 @@ features: - 'A CNOT gate continuously connected to the identity cannot be noise-bias-preserving in finite dimensions \cite{arxiv:0806.0383}\cite[Appx. A]{arxiv:1904.09474}.' threshold: - - 'A lower bound on concatenated thresholds with CSS codes under biased noise \cite{arxiv:0710.1301}.' + - 'A lower bound on \hyperref[topic:computational-threshold]{concatenated thresholds} with CSS codes under biased noise \cite{arxiv:0710.1301}.' fault_tolerance: - 'Fault-tolerant noise-bias-preserving computation scheme \cite{arxiv:0806.0383}.' diff --git a/codes/quantum/qubits/qubit_concatenated.yml b/codes/quantum/qubits/qubit_concatenated.yml index 48e82fa15..de574724f 100644 --- a/codes/quantum/qubits/qubit_concatenated.yml +++ b/codes/quantum/qubits/qubit_concatenated.yml @@ -24,7 +24,10 @@ features: - 'The effective channel for a concatenation of codes is the composition of the codes'' effective channels \cite{arxiv:quant-ph/0206061}.' - 'Message passing algorithm for concatenated codes can be equivalent to ML decoding \cite{arxiv:quant-ph/0606126}.' threshold: - - 'The first methods to achieve a fault-tolerant \hyperref[topic:computational-threshold]{computational threshold} use concatenated qubit stabilizer codes \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}. Such methods require constant-space and polylogarithmic time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time \cite{arxiv:2207.08826}.' + - 'The first methods to achieve a fault-tolerant \hyperref[topic:computational-threshold]{computational threshold} use concatenated qubit stabilizer codes \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}. + Such thresholds are called \hyperref[topic:computational-threshold]{concatenated thresholds}, + These methods require constant-space and polylogarithmic time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time \cite{arxiv:2207.08826}.' + relations: diff --git a/codes/quantum/qubits/qubits_into_qubits.yml b/codes/quantum/qubits/qubits_into_qubits.yml index 015330a23..b1bebb97f 100644 --- a/codes/quantum/qubits/qubits_into_qubits.yml +++ b/codes/quantum/qubits/qubits_into_qubits.yml @@ -120,10 +120,11 @@ features: threshold: - '\begin{defterm}{Computational threshold} \label{topic:computational-threshold} - A fault-tolerant computational threshold is the maximum noise rate in a noise model below which any logical computation of size \(M\) can be executed on a physical-qubit architecture to arbitrary accuracy and with an overhead of \hyperref[topic:asymptotics]{order} \(O(M\text{polylog}M)\). - The first methods to achieve a computational threshold use concatenated stabilizer codes \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}. - Such methods require constant-space and polylogarithmic-time overhead, but concatentions using quantum Hamming codes improve this to quasi-polylogarithmic time \cite{arxiv:2207.08826}. - Fault-tolerant computations with no notion of locality can be made local on a 2D or 3D geometry with minimial overhead \cite{arxiv:2402.13863}. + A fault-tolerant computational threshold is the maximum noise rate in a particular single-parameter noise model below which any logical computation of size \(M\) can be executed on a physical-qubit architecture to arbitrary accuracy and with an overhead of \hyperref[topic:asymptotics]{order} \(O(M\text{polylog}M)\). + The first methods to achieve a computational threshold use recursively concatenated stabilizer code families \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; such a threshold is called a \textit{concatenated threshold}. + Such methods require constant-space and polylogarithmic-time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time \cite{arxiv:2207.08826}. + Subsequently, thresholds were determined for infinite families of lattice stabilizer codes, starting with the toric code \cite{arxiv:quant-ph/0110143}; such a threshold is colloquially called a \textit{topological threshold}. + Fault-tolerant computations with no notion of locality can be made local on a 2D or 3D geometry with minimal overhead \cite{arxiv:2402.13863}. \end{defterm}' - '\begin{defterm}{Measurement threshold} \label{topic:measurement-threshold} diff --git a/codes/quantum/qubits/small_distance/small/stab_5_1_3.yml b/codes/quantum/qubits/small_distance/small/stab_5_1_3.yml index 11796c9aa..82df108bf 100644 --- a/codes/quantum/qubits/small_distance/small/stab_5_1_3.yml +++ b/codes/quantum/qubits/small_distance/small/stab_5_1_3.yml @@ -55,7 +55,7 @@ features: - 'Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived \cite{arxiv:2203.01706}.' threshold: - - 'Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.' + - 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.' fault_tolerance: - 'Pieceable fault-tolerant CZ, CNOT, and CCZ gates \cite{arxiv:1603.03948}.' diff --git a/codes/quantum/qubits/stabilizer/qldpc/concatenated/concatenated_steane.yml b/codes/quantum/qubits/stabilizer/qldpc/concatenated/concatenated_steane.yml index 5f818cfe9..73f283684 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/concatenated/concatenated_steane.yml +++ b/codes/quantum/qubits/stabilizer/qldpc/concatenated/concatenated_steane.yml @@ -12,7 +12,7 @@ introduced: '\cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054}' description: | A member of the family of \([[7^m,1,3^m]]\) CSS codes, each of which is a recursive level-\(m\) concatenatenation of the Steane code. - This family is one of the first to admit a concatenated threshold \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054,arxiv:quant-ph/0207119,arxiv:quant-ph/0410047,arxiv:quant-ph/0604090}. + This family is one of the first to admit a \hyperref[topic:computational-threshold]{concatenated threshold} \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054,arxiv:quant-ph/0207119,arxiv:quant-ph/0410047,arxiv:quant-ph/0604090}. protection: 'Code performance against general Pauli channels has been worked out \cite{arxiv:quant-ph/0111003,arxiv:quant-ph/0206061}.' @@ -23,9 +23,9 @@ features: fault_tolerance: - 'There exist fault-tolerant syndrome extraction protocols for the concatenated Steane code \cite{arxiv:2403.09978}.' code_capacity_threshold: - - 'This family is one of the first to admit a concatenated threshold \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054,arxiv:quant-ph/0207119,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}.' + - 'This family is one of the first to admit a \hyperref[topic:computational-threshold]{concatenated threshold} \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054,arxiv:quant-ph/0207119,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}.' threshold: - - 'Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}; see also \cite{arxiv:quant-ph/0406025}.' + - 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}; see also \cite{arxiv:quant-ph/0406025}.' - 'A \hyperref[topic:measurement-threshold]{measurement threshold} of one \cite{arxiv:2402.00145}.' relations: diff --git a/codes/quantum/qubits/stabilizer/quantum_bch.yml b/codes/quantum/qubits/stabilizer/quantum_bch.yml index 0a264d4f4..8137e3d08 100644 --- a/codes/quantum/qubits/stabilizer/quantum_bch.yml +++ b/codes/quantum/qubits/stabilizer/quantum_bch.yml @@ -18,7 +18,7 @@ features: general_gates: - 'Magic-state distillation protocols \cite{arxiv:1709.02789}.' threshold: - - 'Semi-analytical estimates of concatenated thresholds \cite{arxiv:quant-ph/0207119}. + - 'Semi-analytical estimates of \hyperref[topic:computational-threshold]{concatenated thresholds} \cite{arxiv:quant-ph/0207119}. Qubit BCH codes are difficult to study numerically \cite{arxiv:0711.1556}.' notes: diff --git a/codes/quantum/qubits/stabilizer/rm/quantum_hamming_css.yml b/codes/quantum/qubits/stabilizer/rm/quantum_hamming_css.yml index 6becac36d..12c9dbe53 100644 --- a/codes/quantum/qubits/stabilizer/rm/quantum_hamming_css.yml +++ b/codes/quantum/qubits/stabilizer/rm/quantum_hamming_css.yml @@ -28,7 +28,7 @@ features: - 'Syndrome measurement can be done with two ancillary flag qubits \cite{arxiv:1705.02329}.' - 'Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads \cite{arxiv:2207.08826}.' threshold: - - 'Concatenated thresholds requiring constant-space and quasi-polylogarithmic time overhead \cite{arxiv:2207.08826}.' + - '\hyperref[topic:measurement-threshold]{Concatenated threshold} requiring constant-space and quasi-polylogarithmic time overhead \cite{arxiv:2207.08826}.' relations: parents: 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 2e4c8a8fa..8b97b0fc0 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 @@ -37,7 +37,7 @@ features: - 'Code is often used in magic-state distillation protocols because of its transversal \(T\) gate \cite{arxiv:quant-ph/0403025}.' threshold: - - 'Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.' + - 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.' fault_tolerance: - 'A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code \cite{arxiv:1304.3709,arxiv:1403.2734,arxiv:1703.03860,arxiv:2210.14074}.' diff --git a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml index 257c52650..663c96583 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml @@ -120,7 +120,7 @@ features: A threshold of \(0.41\%\) when concatenated with the \([[4,2,2]]\) code \cite{arxiv:1604.04062}.' - 'Phenomenological noise: \(3.3\%\) for square tiling \cite{arxiv:quant-ph/0401101}, and \(2.93(2)\%\) using several rounds of syndrome measurement \cite{arxiv:quant-ph/0207088}.' - 'Quasistatic phase damping and readout noise: \(2.85\%\) \cite{arxiv:2401.04530}.' - - 'Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.' + - 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.' realizations: diff --git a/codes/quantum/qubits/subsystem/qldpc/bacon_shor/bacon_shor.yml b/codes/quantum/qubits/subsystem/qldpc/bacon_shor/bacon_shor.yml index 908ccd769..a37026fc5 100644 --- a/codes/quantum/qubits/subsystem/qldpc/bacon_shor/bacon_shor.yml +++ b/codes/quantum/qubits/subsystem/qldpc/bacon_shor/bacon_shor.yml @@ -70,11 +70,11 @@ features: code_capacity_threshold: - 'The number of check operators scales sublinearly with system size, so the Bacon-Shor codes alone do not exhibit a topological threshold in the \(m_1,m_2 \to \infty\) limit \cite{arxiv:1903.03937}. However, a threshold can be obtained from concatenated Bacon-Shor codes that are further restricted to planar geometries, whose recovery circuit is a subset of a circuit used by a larger bona-fide Bacon-Shor code \cite{arxiv:2305.12046}. - This threshold differs from a concatenated threshold in that there are no long-range connectivity requirements.' - - 'Lower bounds for the concatenated threshold of various small Bacon-Shor codes are tabulated in \cite[Table I]{arxiv:quant-ph/0610063}.' - - '\(2.02 \times 10^{-5}\) concatenated threshold for the concatenated \([[9,1,3,3]]\) Bacon-Shor code \cite{arxiv:0805.4213}.' + This threshold differs from a \hyperref[topic:computational-threshold]{concatenated threshold} in that there are no long-range connectivity requirements.' + - 'Lower bounds for the \hyperref[topic:computational-threshold]{concatenated threshold} of various small Bacon-Shor codes are tabulated in \cite[Table I]{arxiv:quant-ph/0610063}.' + - '\(2.02 \times 10^{-5}\) \hyperref[topic:computational-threshold]{concatenated threshold} for the concatenated \([[9,1,3,3]]\) Bacon-Shor code \cite{arxiv:0805.4213}.' threshold: - - 'Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.' + - 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}.' - 'The Bacon-Shor code has a \hyperref[topic:measurement-threshold]{measurement threshold} of zero \cite{arxiv:2402.00145}.' decoders: - 'Message passing for \([[9,1,3,3]]\) Bacon-Shor code \cite{arxiv:0806.2188}.'