From ed25e1e84ab5a7ab6c35a18514b90294d6ea08b2 Mon Sep 17 00:00:00 2001 From: Philippe Faist Date: Fri, 20 Dec 2024 21:14:49 +0100 Subject: [PATCH 1/3] Correct Typo - Update qudit_3_6_2.yml (#356) --- codes/quantum/qudits/nonstabilizer/qudit_3_6_2.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/codes/quantum/qudits/nonstabilizer/qudit_3_6_2.yml b/codes/quantum/qudits/nonstabilizer/qudit_3_6_2.yml index 1ab94cb9d..bc0d92cc7 100644 --- a/codes/quantum/qudits/nonstabilizer/qudit_3_6_2.yml +++ b/codes/quantum/qudits/nonstabilizer/qudit_3_6_2.yml @@ -12,7 +12,7 @@ introduced: '\cite{arxiv:2104.05122}' # Qudits labeled by Z_6 description: | - Six-qudit error-detecting code with logical dimension \(K=6\) that is obtained from a particular \hyperref[topic:ame]{AME state} that serves as a solution to the 36 officers of Euler problem. + Three-qudit error-detecting code with logical dimension \(K=6\) that is obtained from a particular \hyperref[topic:ame]{AME state} that serves as a solution to the 36 officers of Euler problem. The code is obtained from a \(((4,1,3))_{\mathbb{Z}_6}\) code. relations: From 1ff22ba470fe4944a2867c48112a2264e0a4ebc7 Mon Sep 17 00:00:00 2001 From: VVA2024 Date: Fri, 20 Dec 2024 15:35:45 -0500 Subject: [PATCH 2/3] refs expt --- codes/classical/analog/lattice/root/hexagonal.yml | 4 ++-- codes/quantum/categories/string_net/fibonacci.yml | 2 +- .../oscillators/fock_state/rotation/number_phase.yml | 2 +- codes/quantum/qubits/dynamic/floquet/honeycomb.yml | 2 +- .../stabilizer/topological/color/2d_color/2d_color.yml | 2 +- .../topological/color/2d_color/488_color/488_color.yml | 5 ++++- .../color/2d_color/triangular_color/triangular_color.yml | 8 +++++++- .../quantum/qubits/stabilizer/topological/color/color.yml | 2 +- .../stabilizer/topological/surface/2d_surface/surface.yml | 5 +++-- .../qubits/subsystem/qldpc/bbs/bacon_shor/bacon_shor.yml | 1 + .../qubits/subsystem/topological/kitaev_honeycomb.yml | 2 +- 11 files changed, 23 insertions(+), 12 deletions(-) diff --git a/codes/classical/analog/lattice/root/hexagonal.yml b/codes/classical/analog/lattice/root/hexagonal.yml index 3667eb7f1..59b6b0590 100644 --- a/codes/classical/analog/lattice/root/hexagonal.yml +++ b/codes/classical/analog/lattice/root/hexagonal.yml @@ -11,8 +11,8 @@ name: '\(A_2\) hexagonal lattice' description: | Two-dimensional lattice that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. - Its dual is the \textit{honeycomb lattice}. - The \textit{ruby lattice} is a fattened honeycomb lattice interpolating between the honeycomb and hexagonal lattices. + Its dual is the \textit{honeycomb tiling}, which is not a lattice (since the points do not form a group under addition) but which consists of two hexagonal lattices. + The \textit{ruby lattice} is a fattened honeycomb tiling interpolating between the honeycomb tiling and hexagonal lattice. It's generator matrix is \begin{align} diff --git a/codes/quantum/categories/string_net/fibonacci.yml b/codes/quantum/categories/string_net/fibonacci.yml index 90be23d25..d70d3105c 100644 --- a/codes/quantum/categories/string_net/fibonacci.yml +++ b/codes/quantum/categories/string_net/fibonacci.yml @@ -18,7 +18,7 @@ description: | The second type of encoding is into the degenerate fusion space of a number of anyonic quasiparticle excitations of the Levin-Wen model. This can equivalently constructed by braiding holes in a spherical geometry \cite[Sec. 5]{arxiv:1002.2816}. -protection: When defined on a \(L \times L\) tailed honeycomb lattice on a torus, the code distance for ground-state encoding is \(L\). +protection: When defined on a \(L \times L\) tailed honeycomb tiling on a torus, the code distance for ground-state encoding is \(L\). features: # rate: Rate of \(n = k d^2\) for the ground-state encoding. diff --git a/codes/quantum/oscillators/fock_state/rotation/number_phase.yml b/codes/quantum/oscillators/fock_state/rotation/number_phase.yml index 4d79fea5c..415d24c68 100644 --- a/codes/quantum/oscillators/fock_state/rotation/number_phase.yml +++ b/codes/quantum/oscillators/fock_state/rotation/number_phase.yml @@ -33,7 +33,7 @@ protection: | features: decoders: - - 'Modular phase measurement done in the logical \(X\), or dual, basis has zero uncertainty in the case of ideal number phase codes. This is equivalent to a quantum measurement of the spectrum of the Susskind–Glogower phase operator. Approximate number-phase codes are characterized by vanishing phase uncertainty. Such measurements can be utilized for Knill error correction (a.k.a. telecorrection \cite{arxiv:quant-ph/0601066}), which is based on teleportation \cite{arxiv:quant-ph/0410199,arxiv:quant-ph/0312190}. This type of error correction avoids the complicated correction procedures typical in Fock-state codes, but requires a supply of clean codewords \cite{arxiv:1901.08071}. Performance of this method was analyzed in Ref. \cite{arxiv:2108.01009}.' + - 'Modular phase measurement done in the logical \(X\), or dual, basis has zero uncertainty in the case of ideal number phase codes. This is equivalent to a quantum measurement of the spectrum of the Susskind–Glogower phase operator. Approximate number-phase codes are characterized by vanishing phase uncertainty. Such measurements can be utilized for Knill error correction (a.k.a. telecorrection \cite{arxiv:quant-ph/0601066}), which is based on teleportation \cite{arxiv:quant-ph/0410199,arxiv:quant-ph/0312190}. This type of error correction avoids the complicated correction procedures typical in Fock-state codes, but requires a supply of clean codewords \cite{arxiv:1901.08071}. Performance of this method was analyzed in Ref. \cite{arxiv:2108.01009}, and it was extended in Ref. \cite{arxiv:2412.15134}.' - 'Number measurement can be done by extracting modular number information using a CROT gate \(\mathrm{e}^{(2\pi \mathrm{i} / NM) \hat n \otimes \hat n}\) and performing phase measurements \cite{preset:Helstrom,doi:10.1007/978-88-7642-378-9} on an ancillary mode. See Section 4.B.1 of Ref. \cite{arxiv:1901.08071}.' fault_tolerance: diff --git a/codes/quantum/qubits/dynamic/floquet/honeycomb.yml b/codes/quantum/qubits/dynamic/floquet/honeycomb.yml index 2c8b5d984..7accf1548 100644 --- a/codes/quantum/qubits/dynamic/floquet/honeycomb.yml +++ b/codes/quantum/qubits/dynamic/floquet/honeycomb.yml @@ -73,7 +73,7 @@ relations: - code_id: kitaev_honeycomb detail: 'The Kitaev honeycomb model Hamiltonian is a sum of checks of the honeycomb Floquet code \cite{arxiv:2107.02194}.' - code_id: hexagonal - detail: 'The honeycomb Floquet code is defined on the honeycomb lattice.' + detail: 'The honeycomb Floquet code is defined on the honeycomb tiling.' # Begin Entry Meta Information diff --git a/codes/quantum/qubits/stabilizer/topological/color/2d_color/2d_color.yml b/codes/quantum/qubits/stabilizer/topological/color/2d_color/2d_color.yml index 1022d7e08..96449946c 100644 --- a/codes/quantum/qubits/stabilizer/topological/color/2d_color/2d_color.yml +++ b/codes/quantum/qubits/stabilizer/topological/color/2d_color/2d_color.yml @@ -33,7 +33,7 @@ description: | # They anti-commute when they cross an odd number of times and have a different color and type. # \begin{figure} # \includegraphics{colorCodeHoneycombHighlightedChecksAdjColor.svg} - # \caption{Stabilizer generators and string operators of a 2D color code defined on a honeycomb lattice on a torus. + # \caption{Stabilizer generators and string operators of a 2D color code defined on a honeycomb tiling on a torus. # The plaquette operators generate the stabilizer group of the toric code where each face corresponds to an X or Z plaquette operator. The string operators # are pairs of X and Z logical operators that wrap around the torus. There are only four independent string operators, so there are two independent colors for the string operators \cite{arxiv:1311.0277}.} # \label{figure:toric-code-operators} diff --git a/codes/quantum/qubits/stabilizer/topological/color/2d_color/488_color/488_color.yml b/codes/quantum/qubits/stabilizer/topological/color/2d_color/488_color/488_color.yml index 4b927c0af..2bf8da7b5 100644 --- a/codes/quantum/qubits/stabilizer/topological/color/2d_color/488_color/488_color.yml +++ b/codes/quantum/qubits/stabilizer/topological/color/2d_color/488_color/488_color.yml @@ -60,6 +60,9 @@ features: - 'Phenomenological noise: \(3.05(4)\%\) under IP decoder \cite[Table I]{arxiv:1108.5738} and \(2.08(1)\%\) under projection decoder \cite{arxiv:1402.3037}.' - 'Circuit-level noise: \(0.082(3)\%\) under IP decoder, \(0.143(1)\%\) under projection decoder \cite{arxiv:1402.3037}, \(0.143\%\) under matching decoder \cite{arxiv:1407.5103}, and an analytic lower bound of \(\approx 0.1\%\) \cite{arxiv:0907.1708} (see \cite[Table I]{arxiv:1108.5738}).' +realizations: + - 'Rydberg atomic devices: logical magic-state distillation using distance-three and five 4.8.8 color codes, observing an improvement in logical fidelity on a device by Quera \cite{arxiv:2412.15165}.' + relations: parents: @@ -68,7 +71,7 @@ relations: - code_id: triangular_color detail: 'Lattice surgery scheme for a hybrid 6.6.6-4.8.8 layout yields lower resource overhead when compared to analogous surface code scheme \cite{arxiv:2201.07806}.' - code_id: hypercubic - detail: 'The 4.8.8 (square-octagon) tiling is obtained by applying a fattening procedure to the honeycomb lattice \cite{arxiv:cond-mat/0607736}.' + detail: 'The 4.8.8 (square-octagon) tiling is obtained by applying a fattening procedure to the honeycomb tiling \cite{arxiv:cond-mat/0607736}.' # Begin Entry Meta Information diff --git a/codes/quantum/qubits/stabilizer/topological/color/2d_color/triangular_color/triangular_color.yml b/codes/quantum/qubits/stabilizer/topological/color/2d_color/triangular_color/triangular_color.yml index 490c31448..a61b4c7be 100644 --- a/codes/quantum/qubits/stabilizer/topological/color/2d_color/triangular_color/triangular_color.yml +++ b/codes/quantum/qubits/stabilizer/topological/color/2d_color/triangular_color/triangular_color.yml @@ -50,6 +50,9 @@ features: - 'Möbius matching decoder gives low logical failure rate \cite{arxiv:2108.11395} and has an open-source implementation called Chromöbius \cite{arxiv:2312.08813}.' - 'AMBP4, a quaternary version \cite{arxiv:2202.06612} of the MBP decoder \cite{arxiv:2104.13659}.' - 'MaxSAT-based decoder \cite{arxiv:2303.14237}.' + - 'Most likely error (MLE) decoder \cite{arxiv:2412.14256}.' + - 'Neural network decoder \cite{arxiv:2412.14256}.' + fault_tolerance: - 'Fault-tolerant syndrome extraction circuits using flag qubits \cite{arxiv:1708.02246,arxiv:1911.00355}.' @@ -68,12 +71,15 @@ features: - 'A \hyperref[topic:measurement-threshold]{measurement threshold} of one \cite{arxiv:2402.00145}.' +realizations: + - 'Superconducting qubits: transversal Clifford gates, randomized logical benchmarking, and magic-state injection demonstrated on distance-three and five triangular color codes on the Willow device by Google Quantum AI \cite{arxiv:2412.14256}. Logical state teleportation using lattice surgery performed between two distance-three color codes.' + relations: parents: - code_id: 2d_color cousins: - code_id: hexagonal - detail: 'The triangular color code is defined on a trivalent lattice such as the honeycomb lattice.' + detail: 'The triangular color code is defined on the honeycomb tiling.' # Begin Entry Meta Information diff --git a/codes/quantum/qubits/stabilizer/topological/color/color.yml b/codes/quantum/qubits/stabilizer/topological/color/color.yml index 82b95b82d..cacc93d88 100644 --- a/codes/quantum/qubits/stabilizer/topological/color/color.yml +++ b/codes/quantum/qubits/stabilizer/topological/color/color.yml @@ -20,7 +20,7 @@ description: | See also a construction based on the more general quantum pin codes \cite{arxiv:1906.11394}. -# For 2-dimensional color code, the lattice must be such that it is 3-valent and has 3-colorable faces, such as a honeycomb lattice. +# For 2-dimensional color code, the lattice must be such that it is 3-valent and has 3-colorable faces, such as a honeycomb tiling. # The qubits are placed on the vertices and two stabilizer generators are placed on each face \cite{arxiv:1311.0277}. 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 54e682c26..2d15aeec2 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/surface.yml @@ -88,13 +88,13 @@ features: - 'Magic-state distillation protocols \cite{arxiv:1208.0928,arxiv:1209.0510,arxiv:2212.00813,arxiv:2403.03991} leading up to magic-state cultivation \cite{arxiv:2409.17595}.' - 'Framework of fault tolerance utilizing ZX calculus \cite{doi:10.1007/978-3-540-70583-3_25,arxiv:0906.4725} that is applicable to MBQC, FBQC, and conventional computation versions of the surface code \cite{arxiv:2303.08829}.' - 'Single-shot state preparation \cite{arxiv:1904.01502} and MWPM decoding \cite{arxiv:2209.09774}.' - - 'Syndrome extraction circuits consisting of CNOT gates and ancillary measurements \cite{arxiv:1208.0928}. Measurement schedules can be optimized using spacetime circuit codes to yield what is know as the \textit{3CX surface code} \cite{arxiv:2302.02192}. Schedules can also be optimized via ZX calculus \cite{doi:10.1007/978-3-540-70583-3_25,arxiv:0906.4725}. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed \cite{arxiv:2007.00307,arxiv:2206.12780,arxiv:2310.12981}, with the scheme in Ref. \cite{arxiv:2206.12780} being a special case of DWR.' + - 'Syndrome extraction circuits consisting of CNOT gates and ancillary measurements \cite{arxiv:1208.0928}. Measurement schedules can be optimized using spacetime circuit codes to yield what is known as the \textit{3CX surface code} \cite{arxiv:2302.02192}. Schedules can also be optimized via ZX calculus \cite{doi:10.1007/978-3-540-70583-3_25,arxiv:0906.4725}. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed \cite{arxiv:2007.00307,arxiv:2206.12780,arxiv:2310.12981}, with the scheme in Ref. \cite{arxiv:2206.12780} being a special case of DWR.' decoders: - 'Using data from multiple syndrome measurements prior to decoding allows for correcting syndrome measurement errors. The surface code requires \hyperref[topic:asymptotics]{order} \(O(d)\) extraction rounds in order to gain a reliable estimate. Syndrome measurements are \hyperref[topic:effective-distance]{distance-preserving} because syndrome extraction circuits can be designed to avoid \hyperref[topic:effective-distance]{hook errors} \cite{arxiv:quant-ph/0110143}.' - - 'Syndrome extraction circuits consist of CNOT gates and ancillary measurements since this is a stabilizer code \cite{arxiv:1208.0928}. Measurement schedules can be optimized using spacetime circuit codes to yield the \textit{3CX surface code} \cite{arxiv:2302.02192}. Schedules can also be optimized via ZX calculus \cite{doi:10.1007/978-3-540-70583-3_25,arxiv:0906.4725}. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed \cite{arxiv:2007.00307,arxiv:2206.12780,arxiv:2310.12981}, with the scheme in Ref. \cite{arxiv:2206.12780} being a special case of DWR.' + - 'Syndrome extraction circuits consist of CNOT gates and ancillary measurements since this is a stabilizer code \cite{arxiv:1208.0928}. Measurement schedules can be optimized using spacetime circuit codes to yield what is known as the \textit{3CX surface code} \cite{arxiv:2302.02192}. Schedules can also be optimized via ZX calculus \cite{doi:10.1007/978-3-540-70583-3_25,arxiv:0906.4725}. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed \cite{arxiv:2007.00307,arxiv:2206.12780,arxiv:2310.12981}, with the scheme in Ref. \cite{arxiv:2206.12780} being a special case of DWR.' - 'Expanding diamonds decoder correcting errors of some maximum fractal dimension \cite{manual:{Andrew Landahl, private communication, 2023}}. The sub-threshold failure probability scales as \((p/p_{\text{th}})^{d^\beta}\), where \(p_{\text{th}}\) is the threshold and \(\beta = \log_3 2\).' - 'Minimum weight perfect-matching (MWPM) \cite{arxiv:quant-ph/0110143,arxiv:1307.1740} (based on work by Edmonds on finding a matching in a graph \cite{doi:10.4153/CJM-1965-045-4,doi:10.6028/jres.069B.013}), which takes time up to polynomial in \(n\) for the surface code. @@ -141,6 +141,7 @@ features: realizations: - | Signatures of corresponding topological phase of matter detected in superconducting circuits \cite{arxiv:2104.01180} and two-dimensional Rydberg atomic arrays \cite{arxiv:2104.04119}. + - 'Measurement schedules associated with the 3CX surface code realized in superconducting qubits on the Willow device by Google Quantum AI \cite{arxiv:2412.14360}.' notes: - 'Introduction to computation with the surface code \cite{doi:10.21468/SciPostPhysLectNotes.49,arxiv:1504.01444}.' diff --git a/codes/quantum/qubits/subsystem/qldpc/bbs/bacon_shor/bacon_shor.yml b/codes/quantum/qubits/subsystem/qldpc/bbs/bacon_shor/bacon_shor.yml index 626ffce76..504194c58 100644 --- a/codes/quantum/qubits/subsystem/qldpc/bbs/bacon_shor/bacon_shor.yml +++ b/codes/quantum/qubits/subsystem/qldpc/bbs/bacon_shor/bacon_shor.yml @@ -41,6 +41,7 @@ features: general_gates: - 'Piecably fault-tolerant circuits can be employed to construct non-transversal gates effectively \cite{manual:{Yoder, Theodore., \emph{DSpace@MIT} Practical Fault-Tolerant Quantum Computation (2018)}}.' - 'Subsystem lattice surgery \cite{arxiv:1609.08062}.' + - 'Measurement-free deformation protocol realizing the \(CCZ\) gate \cite{arxiv:2412.15187}.' fault_tolerance: - 'Fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes that is effective against highly biased noise \cite{arxiv:1211.1400}.' - 'Pieceably fault-tolerant circuits can be employed to construct non-transversal gates effectively \cite{manual:{Yoder, Theodore., \emph{DSpace@MIT} Practical Fault-Tolerant Quantum Computation (2018)}}.' diff --git a/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml b/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml index 3bb4c1640..c1ebaad6e 100644 --- a/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml +++ b/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml @@ -47,7 +47,7 @@ relations: This code can be obtained from the square-lattice surface code by \hyperref[topic:gauging-out]{gauging out} the anyon \(em\) \cite[Sec. 7.3]{arxiv:2211.03798}. During this process, the square lattice is effectively expanded to a hexagonal lattice \cite[Fig. 12]{arxiv:2211.03798}.' - code_id: hexagonal - detail: 'The Kitaev honeycomb model is defined on the honeycomb lattice.' + detail: 'The Kitaev honeycomb model is defined on the honeycomb tiling.' - code_id: topological detail: 'The Kitaev honeycomb model realizes all anyon theories of the 16-fold way, i.e., all minimal modular extensions of the \(\mathbb{Z}_2^{(1)}\) anyon theory \cite{arxiv:cond-mat/0506438}\cite[Footnote 25]{arxiv:2211.03798}. This includes the (non-Abelian) Ising-anyon topological order \cite{arxiv:cond-mat/0506438} (a.k.a. \(p+ip\) superconducting phase \cite{arxiv:1104.5485}) as well as Abelian \(\mathbb{Z}_2\) topological order.' From 6d472a6adf94337d09b424623b211df9e089dd32 Mon Sep 17 00:00:00 2001 From: VVA2024 Date: Fri, 20 Dec 2024 16:15:55 -0500 Subject: [PATCH 3/3] ref tqsg --- .../qldpc/homological/balanced_product/quantum_expander.yml | 2 ++ codes/quantum/qubits/stabilizer/qldpc/homological/dlv.yml | 2 ++ .../balanced_product/lp/matrix/expander_lifted_product.yml | 2 ++ 3 files changed, 6 insertions(+) diff --git a/codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/quantum_expander.yml b/codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/quantum_expander.yml index 7685b2821..6d9f2df92 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/quantum_expander.yml +++ b/codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/quantum_expander.yml @@ -41,6 +41,8 @@ relations: detail: 'Quantum expander codes are single-shot \cite{arxiv:1808.03821}.' cousins: - code_id: expander + - code_id: topological + detail: 'Quantum expander codes realize topological quantum spin glass order \cite{arxiv:2412.13248}.' # detail: 'Quantum expander codes are constructed from classical expander codes.' diff --git a/codes/quantum/qubits/stabilizer/qldpc/homological/dlv.yml b/codes/quantum/qubits/stabilizer/qldpc/homological/dlv.yml index c45bfed15..eb58959d3 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/homological/dlv.yml +++ b/codes/quantum/qubits/stabilizer/qldpc/homological/dlv.yml @@ -26,6 +26,8 @@ relations: - code_id: qltc detail: 'DLV codes have linear dimension and inverse poly-logarithmic relative distance and soundness, assuming a conjecture about random linear maps \cite{arxiv:2402.07476}. Applying distance amplification and soundness amplification yields asymptotically constant soundness, \hyperref[topic:asymptotics]{order} \(\Theta(n)\) distance, \hyperref[topic:asymptotics]{order} \(\Theta(n)\) dimension, but poly-logarithmic locality \cite[Table 4]{arxiv:2309.05541}.' + - code_id: topological + detail: 'DLV codes are expected to realize topological quantum spin glass order \cite{arxiv:2412.13248}.' # Begin Entry Meta Information diff --git a/codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/matrix/expander_lifted_product.yml b/codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/matrix/expander_lifted_product.yml index a9378410a..e9b51813f 100644 --- a/codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/matrix/expander_lifted_product.yml +++ b/codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/matrix/expander_lifted_product.yml @@ -45,6 +45,8 @@ relations: detail: 'Expander lifted-product codes are products of regular \(q\)-ary Tanner codes defined on expander graphs \cite{doi:10.1090/S0273-0979-06-01126-8}.' - code_id: random detail: 'Expander lifted-product codes are quantum CSS codes that utilize short classical codes in their construction which need to satisfy some properties (Ref. \cite{arxiv:2111.03654}, Lemma 10). It is shown that such codes exist, but they are not explicitly constructed. Such codes can be obtained by repeated random sampling or by performing a search of all codes of desired length. Nevertheless, since the length of the desired short codes does not scale with \(n\), this construction is effectively explicit.' + - code_id: topological + detail: 'Expander lifted-product codes are expected to realize topological quantum spin glass order \cite{arxiv:2412.13248}.' # Begin Entry Meta Information