Londe, Vivien

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HAL Id: hal-01671456https://hal.inria.fr/hal-01671456Submitted on 22 Dec 2017HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.4D hyperbolic regular quantum codesVivien LondeTo cite this version:Vivien Londe. 4D hyperbolic regular quantum codes. Journées Informatique Quantique 2017, Nov2017, Bordeaux, France. ￿hal-01671456￿Quantum error correction Homological quantum codes Surface codes 4-dimensional codes4D hyperbolic regular quantum codesVivien Londe, Anthony Leverrier, Gilles ZémorInria Paris, Equipe SecretInstitut de Mathématiques de BordeauxJIQ’17November 10, 2017Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesTable of contentsQuantum error correctionHomological quantum codesSurface codes4-dimensional codesVivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesQuantum error correcting codesI aim: protect k qubits of information.I k logical qubitsbut n physical qubits (n > k)→ codespace of dimension 2k in a 2n Hilbert space.I distance d of the code proportional to the maximal number ofphysical qubits which can be corrupted without corrupting theinformation carried by the logical qubits.→ [[ n,k,d ]] quantum error correcting codeI asymptotics of k and d when n→∞Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesFamilies of codesI stabilizer codes: thecodespace is the1-eigenspace of a set ofcommuting stabilizers.I CSS codes: each stabilizer isa tensor product of X and Imatrices or a tensor productof Z and I matrices.I LDPC codes: each stabilizerhas a bounded number of Xor Z factors and each qubitis non trivially acted on by abounded number ofstabilizer.I homological codes: Thestabilizers are defined from acellulation of a manifold.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesExample of homological code: Toric code [Kitaev 2002]I cellulation of the torus bysquares.I To each edge correspondsa physical qubit.I To each face correspondsa Z-type stabilizer.I To each vertexcorresponds an X-typestabilizer.A torus is obtained by identifyingleft and right sides and identifyingup and down sides of the square.Stabilizers commute because each (face,vertex) pair shares an evennumber of edges.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesExample of homological code: Toric code [Kitaev 2002]I The Z-type stabilizercorresponding to the redface acts like Z on qubits1, 10, 4 and 11.I The X-type stabilizercorresponding to the bluevertex acts like X onqubits 8, 18, 9 and 15.Stabilizers commute because each (face,vertex) pair shares an evennumber of edges.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesDetection of errorsrepresents a codestate |ψ0 >.All stabilizers have eigenvalue 1.represents the state X17|ψ0 >.red stabilizers have eigenvalue -1.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesDetection of errorsrepresents a codestate |ψ0 >.All stabilizers have eigenvalue 1.represents the state X17|ψ0 >.red stabilizers have eigenvalue -1.Zstab.|ψ0 >= |ψ0 >Zred stab.X17|ψ0 >= −X17Zred stab.|ψ0 >= −X17|ψ0 >Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesDetection of errorsrepresents a codestate |ψ0 >.All stabilizers have eigenvalue 1.represents the state X17|ψ0 >.red stabilizers have eigenvalue -1.state X17X18|ψ0 >.red stabilizers have eigenvalue -1.state X16X17X18X19|ψ0 >:codestate different from |ψ0 >.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesGeometric interpretation of kI The code dimension is the rank ofthe first homology group H1 of themanifold.I Informally, it is the number ofdifferent loops of the manifold.k = 4 = rank(H1)k = 2 = rank(H1)k = 8 = rank(H1)Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesGeometric interpretation of n and dI number of physical qubits nproportional to the area ofthe manifold.I minimal distance dproportional to the systoleof the manifold.I the systole is the length ofthe shortest non contractibleloop of the manifold.d = systole ≈ 20n = 2× area ≈ 800d = systole = length of red circleVivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesSchläfli symbolsSchläfli symbols are defined recursively:I {4} is a square.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesSchläfli symbolsSchläfli symbols are defined recursively:I {4} is a square.I {4,3} is the regularpolyhedron such that eachvertex is incident to 3squares ({4}): the cube.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesSchläfli symbolsSchläfli symbols are defined recursively:I {4} is a square.I {4,3} is the regularpolyhedron such that eachvertex is incident to 3squares ({4}): the cube.I {4,3,3} is the regularpolytope such that eachedge is incident to 3 cubes({4,3}): the hypercube.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesRegular tessellationI {4,4} is the regular tessellation ofeuclidean plane such that eachvertex is incident to 4 squares.It is the grid of the toric code.I {5,4} is the regular tessellationsuch that each vertex is incident to5 squares. It is a tessellation of thehyperbolic plane.I The automorphism group Γ of the {5,4} tessellation admitsthe following presentation:Γ =< ρ0, ρ1, ρ2 | ρ20 = ρ21 = ρ21 = (ρ0ρ1)5 = (ρ1ρ2)4 = id >Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesRegular mapI Quotienting by a normal subgroup of the automorphism groupof the regular tessellation Γ yields a closed surface with thelocal structure of the regular tessellation.{6, 4}3 regular map topology of the {6, 4}3 regularmapI To this tessellated closed surface corresponds a quantum errorcorrecting code.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesLimits of surface codesI Tradeoffs for reliable quantum information storage in surfacecodes and color codes [Delfosse 2013]:kd2 ≤ C (ln k)2n– toric codes family: k = 2 and d = θ(√n)– hyperbolic codes family: k = θ(n) and d = θ(ln n)Codes based on cellulations of 4-manifolds can overcome thisbound.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codes4-dimensional codesI Cellulation of a 4-dimensional manifold by polytopes.– To each 2-cell corresponds a qubit.– To each 3-cell corresponds a Z-type stabilizer.– To each 1-cell (edge) corresponds an X-type stabilizer.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codes4-dimensional codesI Cellulation of a 4-dimensional manifold by polytopes.– To each 2-cell corresponds a qubit.– To each 3-cell corresponds a Z-type stabilizer.– To each 1-cell (edge) corresponds an X-type stabilizer.I Logical qubits correspond to 2-dimensional holes.→ the number k of logical qubits is the rank of the secondhomology group of the manifold.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codes4-dimensional codesI Cellulation of a 4-dimensional manifold by polytopes.– To each 2-cell corresponds a qubit.– To each 3-cell corresponds a Z-type stabilizer.– To each 1-cell (edge) corresponds an X-type stabilizer.I Logical qubits correspond to 2-dimensional holes.→ the number k of logical qubits is the rank of the secondhomology group of the manifold.I Minimal distance d is proportional to the 2-systole of themanifold.→ it is the smallest area of a surface surrounding a2-dimensional hole.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codes4-dimensional tessellationsI {4,3,3,4} is the canonical grid of euclidean 4-space.({4,3,3} is a 4-dimensional hypercube)Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codes4-dimensional tessellationsI {4,3,3,4} is the canonical grid of euclidean 4-space.({4,3,3} is a 4-dimensional hypercube)I {4,3,3,5} is a regular tessellation of hyperbolic 4-space H4.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codes4-dimensional tessellationsI {4,3,3,4} is the canonical grid of euclidean 4-space.({4,3,3} is a 4-dimensional hypercube)I {4,3,3,5} is a regular tessellation of hyperbolic 4-space H4.I The automorphism group Γ of {4,3,3,5} can be represented asa discrete subgroup of SO(1,4), the isometry group of H4.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codes4-dimensional tessellationsI {4,3,3,4} is the canonical grid of euclidean 4-space.({4,3,3} is a 4-dimensional hypercube)I {4,3,3,5} is a regular tessellation of hyperbolic 4-space H4.I The automorphism group Γ of {4,3,3,5} can be represented asa discrete subgroup of SO(1,4), the isometry group of H4.I Congruence subgroups of this representation yield normalsubgroups of Γ.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesCodes associated to these 4-manifoldsI We thus obtain closed 4-manifold with the {4,3,3,5} localstructure.I k = θ(n) and d ≥ n0.2 asymptotically.I kd2 ≥ (lnk)2n asymptotically.I regularity useful for decoding algorithms.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesAdvantages of 4-dimensional hyperbolic regular codesI Syndromes are cycles of edges.→ no need to pair vertices like with surface codesVivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesAdvantages of 4-dimensional hyperbolic regular codesI Syndromes are cycles of edges.→ no need to pair vertices like with surface codesI Hyperbolicity makes syndromes more redundant than ineuclidean space.→ since the boundary of a hyperbolic disk is linear in its area.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesAdvantages of 4-dimensional hyperbolic regular codesI Syndromes are cycles of edges.→ no need to pair vertices like with surface codesI Hyperbolicity makes syndromes more redundant than ineuclidean space.→ since the boundary of a hyperbolic disk is linear in its area.I Regularity of the tessellation allows one to design explicitefficient decoding algorithms.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesNormal subgroups of Γ{4,3,3,5}I Γ{4,3,3,5} =< Ri | i ∈ {0, ..., 4} >with PRiP−1 ∈M5,5(OQ[√5]) for i ∈ {0, ..., 4}OQ[√5] is the ring of integers of the number field Q[√5].I If I is an ideal of OQ[√5], there is a group homomorphism:φI :M5,5(OQ[√5])→M5,5(OQ[√5]/I)Ker(φI) is a normal subgroup of Γ{4,3,3,5} (called acongruence subgroup).Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesSummary and OutlooksSummary:I To a tessellated closed manifold and an integer i correspondsa quantum code where qubits are identified with i-faces of themanifold.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesSummary and OutlooksSummary:I To a tessellated closed manifold and an integer i correspondsa quantum code where qubits are identified with i-faces of themanifold.I 4-dimensional homological codes can outperform surfacecodes.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesSummary and OutlooksSummary:I To a tessellated closed manifold and an integer i correspondsa quantum code where qubits are identified with i-faces of themanifold.I 4-dimensional homological codes can outperform surfacecodes.I The regularity of the {4,3,3,5} construction leads to anexplicit decoding algorithm.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesSummary and OutlooksSummary:I To a tessellated closed manifold and an integer i correspondsa quantum code where qubits are identified with i-faces of themanifold.I 4-dimensional homological codes can outperform surfacecodes.I The regularity of the {4,3,3,5} construction leads to anexplicit decoding algorithm.Outlooks:I Find an explicit decoding algorithm for Z-type errors.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesSummary and OutlooksSummary:I To a tessellated closed manifold and an integer i correspondsa quantum code where qubits are identified with i-faces of themanifold.I 4-dimensional homological codes can outperform surfacecodes.I The regularity of the {4,3,3,5} construction leads to anexplicit decoding algorithm.Outlooks:I Find an explicit decoding algorithm for Z-type errors.I Investigate the performance of this family of code withrespect to a noise model.Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codesQuantum error correction Homological quantum codes Surface codes 4-dimensional codesSummary and OutlooksSummary:I To a tessellated closed manifold and an integer i correspondsa quantum code where qubits are identified with i-faces of themanifold.I 4-dimensional homological codes can outperform surfacecodes.I The regularity of the {4,3,3,5} construction leads to anexplicit decoding algorithm.Outlooks:I Find an explicit decoding algorithm for Z-type errors.I Investigate the performance of this family of code withrespect to a noise model.Thank you for your attention!Vivien Londe, Anthony Leverrier, Gilles Zémor Inria Paris, Equipe Secret Institut de Mathématiques de Bordeaux JIQ’174D hyperbolic regular quantum codes

4D hyperbolic regular quantum codes

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4D hyperbolic regular quantum codes

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