Leverrier, Anthony

Londe, Vivien

Zémor, Gilles

INRIA a CCSD electronic archive server

Quantum local testabilityarXiv:1911.03069Anthony Leverrier, Inria(with Vivien Londe and Gilles Zémor)27 nov 2019Symmetry, Phases of Matter, and Resources in Quantum ComputingA. Leverrier quantum LTC 27 nov 2019 1 / 25We want:a local Hamiltonian such thatI with degenerate ground space (quantum code)I the energy of an error scales linearly with the size of the errorA. Leverrier quantum LTC 27 nov 2019 2 / 25The current research on quantum error correctionmostly concerned with the goal of building a (large) quantum computerdesire for realistic constructionsI LDPC codes: the generators of the stabilizer group act on a small number of qubitsI spatial/geometrical locality: qubits on a 2D/3D latticeI main contenders: surface codes, or 3D variantsA fairly reasonable and promising approachI good performance for topological codes: efficient decoders, high thresholdI overhead still quite large for fault-tolerance (magic state distillation) but the numbersare improving regularlyIs this it?A. Leverrier quantum LTC 27 nov 2019 3 / 25Better quantum LDPC codes?from a math/coding point of view, topological codes in 2D-3D are not that goodI 2D toric code Jn, k = O(1), d = O(√n)KI topological codes on 2D Euclidean manifold (Bravyi, Poulin, Terhal 2010)kd2 ≤ cnI topological codes on 2D hyperbolic manifold (Delfosse 2014)kd2 ≤ c(log k)2nI things are better in 4D hyp. space: Guth-Lubotzky 2014 (also Londe-Leverrier 2018)Jn, k = Θ(n), d = nαK, for α ∈ [0.2, 0.3]what can we get by relaxing geometric locality in 3D?I we still want an LDPC construction, but allow for non local generatorsI a nice mathematical topic with many frustrating open questions!A. Leverrier quantum LTC 27 nov 2019 4 / 25Classical LDPC codes are well understoodsparse parity-check matrix H ∈ Fm×n2 :C = kerHI good codes with k = Θ(n), d = Θ(n) can be found by picking H at randomI efficient decoding with belief propagationA. Leverrier quantum LTC 27 nov 2019 5 / 25quantum LDPC codes remain poorly understoodstabilizer group S = 〈g1, . . . , gm〉 with gi ∈ Pn (n-qubit Pauli group) such that [gi, gj] = 0LDPC:I |gi| small (constant or log)I ∀` ∈ [n], ]{i : ` ∈ supp(gi)} smallC = {|ψ〉 ∈ (C2)⊗n : gi|ψ〉 = |ψ〉, ∀i ∈ [m]}The big questions (for me!)I what kind of parameters are possible for qLDPC?I efficient decoding??I links with Hamiltonian complexityA. Leverrier quantum LTC 27 nov 2019 6 / 25quantum LDPC codes with large minimum distanceBeating the√n of the toric code is very hard!I Freedman, Meyer, Luo (2002): construction based on S1 × S2d ∝ n1/2 log1/4 nI Kaufman, Kazhdan, Lubotzky (2016): construction based on Ramanujan complexesdX ∝ n, dZ ∝ log n+ balancing technique (Hastings 2017)=⇒ d ∝ n1/2 log1/2 nI construction by Hastings (2017) he conjectures could yield d ∝ n1−εA. Leverrier quantum LTC 27 nov 2019 7 / 25quantum LDPC codes with large minimum distancebest minimum distance when asking for constant ratehypergraph product codes (Tillich, Zémor 2009) of two good classical LDPC codesJn,Θ(n),Θ(√n)KI note 1: generalization of the toric code (product of 2 repetition codes)I note 2: existence of codes with d ∝ n by relaxing the LDPC condition to√n-localgenerators (Bravyi, Hastings 2014)Do good qLDPC codes exist?A. Leverrier quantum LTC 27 nov 2019 8 / 25Decoding quantum LDPC codesI essentially solved for topological codes!What about general codes?I belief propagation: several issues (Poulin, Chung 2008)I lots of small cyclesI many symmetric patterns (half generators) where the decoder gets stuckI how to deal with degenerescence??I greedy decoding in local ballsI for 4D hyperbolic codes (Hastings 2014)I small-set-flip for quantum expander codes (Leverrier, Tillich, Zémor 2015, Fawzi,Grospellier, Leverrier 2018)A. Leverrier quantum LTC 27 nov 2019 9 / 25Small-set-flip for quantum expander codesI consider a classical expander code (Sipser, Spielman 1996), i.e. such that its factorgraph is an expanderI hypergraph product code =⇒ quantum expander code Jn,Θ(n),Θ(√n)Ksmall-set-flip decodingI for each gi: consider all patterns of errors within gi and apply the one that decreasesthe syndrome weight the most (if it exists)I repeat while possibleI correct arbitrary errors of weight O(√n)I locality of SSF =⇒ distant clusters of errors are also dealt withI cst threshold for local stochastic errors on both qubits and syndrome measurementsI reasonable performance in practice: threshold around 6-7 % with noiseless syndromemeasurement and ≈ 3% for noisy syndrome measurement for phenomenological noisemodel (Grospellier, Krishna 2018, Grospellier, Grouès, Krishna, Leverrier 2019)A. Leverrier quantum LTC 27 nov 2019 10 / 25Soundness and local testabilityThe analysis of the bit-flip decoder for classical expander codes and SSF for quantumexpander codes relies on the soundness of the codes:soundness of quantum expander codesfor any error e such that |e| := d(e, C) ≤ c√n,|s(e)| ≥ η|e|for η = cst, and s(e) the syndromeIf true for any e, then locally testable code=⇒ easy to distinguish between codewords and words far from the code, making aconstant number of queries to the word.Many applications in the classical setting, mostly in theoretical CS, e.g. for PCP theoremreview paper by Goldreich (2006)A. Leverrier quantum LTC 27 nov 2019 11 / 25Quantum locally testable codesI notion introduced by Aharonov, Eldar (2015)I applications remain a bit unclear at the moment, essentially in HamitloniancomplexityI qLTC with linear minimum distance would establish the NLTS conjecture (Eldar,Harrow 2017)I existing qLTC (this talk) allows to prove an average-case version of NLTS (Eldar 2019)I strong form of confinement of errors (Stephen’s talk yesterday)I link with single-shot decoding (Campbell 2018)I definition requires to quantize notions of distance to code and weight of the syndromeA. Leverrier quantum LTC 27 nov 2019 12 / 25qLTC with soundness ηI q-local quantum code C =⇒ Hamiltonian HC = 1qm ∑mi=112 (1− gi)I projector ΠCt on t-fattening of the codeCt := Span{(A1 ⊗ · · · ⊗An)|ψ〉 : |ψ〉 ∈ C, |{i : Ai 6= 1}| ≤ t}DC = ∑tt(ΠCt −ΠCt−1)A quantum code is locally testable with soundness η ifHC ηNDC (energy ≥ η × distance)2 known constructionsI Hastings (2017): η = 1log3 n , k = 2I this work: η = 1log2 n , k = 1, possibly also for k = ω(1)?A. Leverrier quantum LTC 27 nov 2019 13 / 25Examples of codes which are NOT locally testablesI 2D toric code: errors of weight Ω(√n) and constant energyI D-dimensional toric codeI quantum expander codes: errors of weight Ω(√n) and constant energyA. Leverrier quantum LTC 27 nov 2019 14 / 25The hemicubic code constructionalternative name: the projective code (QIP’19)A. Leverrier quantum LTC 27 nov 2019 15 / 25Main properties of the hemicubic codesalmost LDPC: log-localThe simplest version: 1 logical qubitJN, 1, d ≥√N/1.62KI locally testable with η = Ω(1log2 N), open whether η = Θ(1logN)?I efficient decoder for adversarial errors of size dpolylog(N)The general case: k = Nαexplicit parameters of the form: JN,poly(N),poly(N)KI conjectured local testabilityA. Leverrier quantum LTC 27 nov 2019 16 / 25Idea behind the construction: homological codes with large min distance?Geometric interpretation of N and d for surfaces:I N ≈ area of the surfaceI d = systole of the surface, length of theshortest loop which is not the boundaryof a 2D subregion of the surfaceI idea: minimize N at fixed dI work on surface with positive curvature=⇒ sphere (requires some identificationto get a logical qubit)A. Leverrier quantum LTC 27 nov 2019 17 / 25The real projective planeI identify antipodal points =⇒ some loops are notboundaries: homologyI 1 logical qubitI systole = πI area = 2π =⇒ systole >√areaI ?=⇒ D >√NNot an infinite family of quantum codes...Solution: increase the ambient dimension (similar to Hastings 2016)A. Leverrier quantum LTC 27 nov 2019 18 / 25A discrete real projective planeI identify pairs of antipodal faces of the cubeI N = 6 (qubits on edges)I DX = 3 (smallest non trivial cycle)I DZ = 2 (smallest non trivial cocycle)I D = min(DX,DZ) = 2I N = DXDZA. Leverrier quantum LTC 27 nov 2019 19 / 25A discrete real projective planeI identify pairs of antipodal faces of the cubeI N = 6 (qubits on edges)I DX = 3 (smallest non trivial cycle)I DZ = 2 (smallest non trivial cocycle)I D = min(DX,DZ) = 2I N = DXDZA. Leverrier quantum LTC 27 nov 2019 19 / 25A discrete real projective planeI identify pairs of antipodal faces of the cubeI N = 6 (qubits on edges)I DX = 3 (smallest non trivial cycle)I DZ = 2 (smallest non trivial cocycle)I D = min(DX,DZ) = 2I N = DXDZA. Leverrier quantum LTC 27 nov 2019 19 / 25A discrete real projective planeI identify pairs of antipodal faces of the cubeI N = 6 (qubits on edges)I DX = 3 (smallest non trivial cycle)I DZ = 2 (smallest non trivial cocycle)I D = min(DX,DZ) = 2I N = DXDZA. Leverrier quantum LTC 27 nov 2019 19 / 25The hemicubic code: a discrete real projective n-spaceI n-hemicube: antipodal quotient of the n-hypercubeI qubits on p-faces (1 ≤ p ≤ n− 1), generators on (p± 1)-facesI N = (np)2n−p−1, K = 1I DX = (np) (minimal nontrivial cycle has a p-face in every direction)I DZ = 2n−p−1 (minimal nontrivial cocycle consists of all p-faces in a given direction)I N = DXDZI DX ≈ DZ ≈√N for p = αn with α ≈ 0.227.This code has already appeared in the literature in a completely different form relying onKhovanov homology (Audoux 2013).A. Leverrier quantum LTC 27 nov 2019 20 / 25Local testabilty of the hemicubic codeRecall that we want to prove that a lower bound on the syndrome weight:1qm|s(e)| ≥ ηNd(e, C), ∀e ∈ PNHemicubic code: m = Θ(N), q = Θ(logN)We will prove |s(e)| = Ω(d(e,C)logN), which implies η = Ω(1log2 N).Geometric interpretationfor a homological code, |s(e)| is the weight of a boundary and d(e, C) is the minimalweight of its filling.We are looking for filling inequalities.A. Leverrier quantum LTC 27 nov 2019 21 / 25A filling inequality for the n-hypercubeThe syndrome is a boundary B. We are looking for a filling F of it of low weight.Filling inequality by Dotterrer (2012)I qubits on 2-faces, checks on edgesI send the syndrome to the left by fillingwith horizontal squaresI iterateI choose the order of directions carefullyDotterrer’s boundI Dotterrer’s algorithm yields: |B| ≥ cst |F| =⇒ |S(e)| ≥ cst d(e, C)I here: no homology, but a similar approach works for the hemicubeI our current analysis loses a log factor compared to DotterrerA. Leverrier quantum LTC 27 nov 2019 22 / 25A filling inequality for the n-hypercubeThe syndrome is a boundary B. We are looking for a filling F of it of low weight.Filling inequality by Dotterrer (2012)I qubits on 2-faces, checks on edgesI send the syndrome to the left by fillingwith horizontal squaresI iterateI choose the order of directions carefullyDotterrer’s boundI Dotterrer’s algorithm yields: |B| ≥ cst |F| =⇒ |S(e)| ≥ cst d(e, C)I here: no homology, but a similar approach works for the hemicubeI our current analysis loses a log factor compared to DotterrerA. Leverrier quantum LTC 27 nov 2019 22 / 25A filling inequality for the n-hypercubeThe syndrome is a boundary B. We are looking for a filling F of it of low weight.Filling inequality by Dotterrer (2012)I qubits on 2-faces, checks on edgesI send the syndrome to the left by fillingwith horizontal squaresI iterateI choose the order of directions carefullyDotterrer’s boundI Dotterrer’s algorithm yields: |B| ≥ cst |F| =⇒ |S(e)| ≥ cst d(e, C)I here: no homology, but a similar approach works for the hemicubeI our current analysis loses a log factor compared to DotterrerA. Leverrier quantum LTC 27 nov 2019 22 / 25A filling inequality for the n-hypercubeThe syndrome is a boundary B. We are looking for a filling F of it of low weight.Filling inequality by Dotterrer (2012)I qubits on 2-faces, checks on edgesI send the syndrome to the left by fillingwith horizontal squaresI iterateI choose the order of directions carefullyDotterrer’s boundI Dotterrer’s algorithm yields: |B| ≥ cst |F| =⇒ |S(e)| ≥ cst d(e, C)I here: no homology, but a similar approach works for the hemicubeI our current analysis loses a log factor compared to DotterrerA. Leverrier quantum LTC 27 nov 2019 22 / 25A filling inequality for the n-hypercubeThe syndrome is a boundary B. We are looking for a filling F of it of low weight.Filling inequality by Dotterrer (2012)I qubits on 2-faces, checks on edgesI send the syndrome to the left by fillingwith horizontal squaresI iterateI choose the order of directions carefullyDotterrer’s boundI Dotterrer’s algorithm yields: |B| ≥ cst |F| =⇒ |S(e)| ≥ cst d(e, C)I here: no homology, but a similar approach works for the hemicubeI our current analysis loses a log factor compared to DotterrerA. Leverrier quantum LTC 27 nov 2019 22 / 25A filling inequality for the n-hypercubeThe syndrome is a boundary B. We are looking for a filling F of it of low weight.Filling inequality by Dotterrer (2012)I qubits on 2-faces, checks on edgesI send the syndrome to the left by fillingwith horizontal squaresI iterateI choose the order of directions carefullyDotterrer’s boundI Dotterrer’s algorithm yields: |B| ≥ cst |F| =⇒ |S(e)| ≥ cst d(e, C)I here: no homology, but a similar approach works for the hemicubeI our current analysis loses a log factor compared to DotterrerA. Leverrier quantum LTC 27 nov 2019 22 / 25A filling inequality for the n-hypercubeThe syndrome is a boundary B. We are looking for a filling F of it of low weight.Filling inequality by Dotterrer (2012)I qubits on 2-faces, checks on edgesI send the syndrome to the left by fillingwith horizontal squaresI iterateI choose the order of directions carefullyDotterrer’s boundI Dotterrer’s algorithm yields: |B| ≥ cst |F| =⇒ |S(e)| ≥ cst d(e, C)I here: no homology, but a similar approach works for the hemicubeI our current analysis loses a log factor compared to DotterrerA. Leverrier quantum LTC 27 nov 2019 22 / 25A filling inequality for the n-hypercubeThe syndrome is a boundary B. We are looking for a filling F of it of low weight.Filling inequality by Dotterrer (2012)I qubits on 2-faces, checks on edgesI send the syndrome to the left by fillingwith horizontal squaresI iterateI choose the order of directions carefullyDotterrer’s boundI Dotterrer’s algorithm yields: |B| ≥ cst |F| =⇒ |S(e)| ≥ cst d(e, C)I here: no homology, but a similar approach works for the hemicubeI our current analysis loses a log factor compared to DotterrerA. Leverrier quantum LTC 27 nov 2019 22 / 25A filling inequality for the n-hypercubeThe syndrome is a boundary B. We are looking for a filling F of it of low weight.Filling inequality by Dotterrer (2012)I qubits on 2-faces, checks on edgesI send the syndrome to the left by fillingwith horizontal squaresI iterateI choose the order of directions carefullyDotterrer’s boundI Dotterrer’s algorithm yields: |B| ≥ cst |F| =⇒ |S(e)| ≥ cst d(e, C)I here: no homology, but a similar approach works for the hemicubeI our current analysis loses a log factor compared to DotterrerA. Leverrier quantum LTC 27 nov 2019 22 / 25antipodal map = translation by the classical repetition codecodewords of the repetition code: {000,111}τ111(001) = 001+ 111 = 110τ111(00∗) = 00 ∗+111 = 11∗τ111 is the antipodal map.GeneralizationQuotient of the n-cube by arbitrary linear codes?A. Leverrier quantum LTC 27 nov 2019 23 / 25The general constructionThe simple construction identifies some cell x of the n-cube with x+ 111 · · · 1.In other words, the faces are identified if they differ by an element of the repetition code.I choose a classical linear code C = [n, k, d]I associate qubits with p-faces of the n-cube, where we identify elements of a givencoset of C:x ∼ y ⇐⇒ x+ y ∈ CI many more logical qubits: k = (p+k−1p )I surprisingly, dimension and minimum distance only depend on the k and d from C,not on HA. Leverrier quantum LTC 27 nov 2019 24 / 25Perspectiveshemicubic codeI simplest version: n-cube with identified antipodal facesI d =√NI locally testableI general versionI n-cube with identification of cosets of a linear codeI explicit dimension and minimum distanceI conjectured to be locally testable?main open questionI what kind of length is possible for quantum LTC? exponential in k, polynomial?Thanks!A. Leverrier quantum LTC 27 nov 2019 25 / 25Perspectiveshemicubic codeI simplest version: n-cube with identified antipodal facesI d =√NI locally testableI general versionI n-cube with identification of cosets of a linear codeI explicit dimension and minimum distanceI conjectured to be locally testable?main open questionI what kind of length is possible for quantum LTC? exponential in k, polynomial?Thanks!A. Leverrier quantum LTC 27 nov 2019 25 / 25

Quantum local testability

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Quantum local testability

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