高専カンファレンス新春 in 大阪

by Hiroki Nomura

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৽य़ɾߴઐΧϯϑΝ in େࡕ 2019/01/12 kosenconf-123shinshun  ͷΉͷΉʢ@nomunomu0504ʣ

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ڈ೥ʹҾ͖ଓ͖… • ڈ೥ͷ৽य़ΧϯϑΝʹ΋ࢀՃͯ͠·ͨ͠ • @John_bardera ͔Β࣮ߦҕһ௕৆Λ໯͍·ͨ͠  ʢϞϯΤφϑϧηοτʣ

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πΠʔτ͢Δͱ͖ʹ͸… #ͷΉͷΉ

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ࠓ೔ͷൃද಺༰ • ྔࢠίϯϐϡʔλʹ͍ͭͯ  - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍  - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔

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ࠓ೔ͷൃද಺༰ ྔࢠίϯϐϡʔλΛઐ໳ͱ͞Ε͍ͯΔํ • ྔࢠίϯϐϡʔλʹ͍ͭͯ  - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍  - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔

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ࠓ೔ͷൃද಺༰ ྔࢠ࿦ɾྔࢠྗֶઐ߈ͷํ • ྔࢠίϯϐϡʔλʹ͍ͭͯ  - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍  - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔ ྔࢠίϯϐϡʔλΛઐ໳ͱ͞Ε͍ͯΔํ

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ࠓ೔ͷൃද಺༰ • ྔࢠίϯϐϡʔλʹ͍ͭͯ  - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍  - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔ ྔࢠ࿦ɾྔࢠྗֶઐ߈ͷํ ྔࢠίϯϐϡʔλΛઐ໳ͱ͞Ε͍ͯΔํ झຯൣғͰֶश͍ͯ͠Δ಺༰Ͱ͢  ʢߨٛ౳Ұ੾औ͍ͬͯ·ͤΜɻऔΓͨͯ͘΋ߴઐʹ͋Γ·ͤΜʣ ؒҧ͍͕͋Γ·ͨ͠Β%.౳ૹ͍͚ͬͯͨͩΔͱ  ϓϨθϯλʔ͸تͼ·͢

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ྔࢠίϯϐϡʔλͱ͸ • ྔࢠྗֶͷੑ࣭Λ࢖ͬͯߴ଎ʹܭࢉͰ͖Δίϯϐϡʔλ • ͋Δఔ౓ͷαΠζͷྔࢠίϯϐϡʔλ͕͋Ε͹  ໰୊ʹΑͬͯ͸εύίϯΑΓ΋ߴ଎ʹܭࢉͰ͖Δ • ྔࢠίϯϐϡʔλͷ࣮ݱํ๏͸̎छྨ͋Δ • ྔࢠήʔτํࣜ • ྔࢠΞχʔϦϯάํࣜ > ࠓճઆ໌͢Δͷ͸ʰྔࢠήʔτํࣜʱ

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ྔࢠίϯϐϡʔλͰߴ଎ܭࢉͰ͖Δ͜ͱ 1. ਺࿦ܥ • Ҽ਺෼ղ • ཭ࢄର਺໰୊ • ϕϧํఔࣜ • Ψ΢ε࿨ • ߹ಉθʔλؔ਺ 2. زԿܥ • ݁ͼ໨ෆมྔ • Persistent Homology 3. ઢܗ୅਺ܥ • ߦྻͷྦྷ৐ • ߦྻͷ֊৐ ͳͲ

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1. ਺࿦ܥ • Ҽ਺෼ղ • ཭ࢄର਺໰୊ • ϕϧํఔࣜ • Ψ΢ε࿨ • ߹ಉθʔλؔ਺ ޙ൒Ͱ࣮ࡍʹ ಋग़΍ͬͯΈ·͢ʂ ྔࢠίϯϐϡʔλͰߴ଎ܭࢉͰ͖Δ͜ͱ 2. زԿܥ • ݁ͼ໨ෆมྔ • Persistent Homology 3. ઢܗ୅਺ܥ • ߦྻͷྦྷ৐ • ߦྻͷ֊৐ ͳͲ

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Ͳͷ͙Β͍ૣ͘ܭࢉͰ͖Δͷ͔ • nϏοτ੔਺ͷҼ਺෼ղ • ݹయతίϯϐϡʔλ(Ұൠ਺ମ;Δ͍๏) (e1.9(ln n) 1 3(ln ln n) 2 3) • ྔࢠίϯϐϡʔλ(ShorΞϧΰϦζϜ) ((log n)2(log log n)(log log log n)) • 1024Ϗοτ੔਺ͷҼ਺෼ղʹ͔͔Δܭࢉྔ • ݹయతίϯϐϡʔλɿ • ྔࢠίϯϐϡʔλɹɿ (10278) (1061)

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• 1024Ϗοτ੔਺ͷҼ਺෼ղʹ͔͔Δܭࢉྔ • ݹయతίϯϐϡʔλɿ • ྔࢠίϯϐϡʔλɹɿ Ͳͷ͙Β͍ૣ͘ܭࢉͰ͖Δͷ͔ • nϏοτ੔਺ͷҼ਺෼ղ • ݹయతίϯϐϡʔλ(Ұൠ਺ମ;Δ͍๏) (e1.9(ln n) 1 3(ln ln n) 2 3) • ྔࢠίϯϐϡʔλ(ShorΞϧΰϦζϜ) ((log n)2(log log n)(log log log n)) (10278) (1061)

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ͲͷΑ͏ͳܭࢉΛ͍ͯ͠Δͷ͔ • ݹయతίϯϐϡʔλͳΒɺ͋Δॲཧܥ͓͍ͯ  ೖྗ஋͕ಉ͡ͳΒৗʹಉ݁͡Ռ͕ಘΒΕΔɻ • ྔࢠίϯϐϡʔλͰ͸ɺ͋Δԋࢉࢠܥʹ͓͍ͯ  ೖྗ஋͕ಉ͡Ͱ΋ʰԋࢉ݁Ռʱ͸ҟͳΔՄೳੑ͕͋Δɻ ྔࢠίϯϐϡʔλ͸ܭࢉͷਖ਼֬͞ʢ֬཰ʣΛग़ྗ͢Δ

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֤ίϯϐϡʔλͷ جૅԋࢉʹ͍ͭͯ

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ݹయతίϯϐϡʔλͷجૅ • ๻Β͕͍ͭ΋ར༻͍ͯ͠ΔίϯϐϡʔλΛ  ʮݹయతίϯϐϡʔλʯͱݺͿ • ݹయతίϯϐϡʔλͷܭࢉ୯Ґ͸ʮbitʯ  0 ͔ 1 ͷͲͪΒ͔ͷঢ়ଶΛऔΓ͏Δɻ2ਐ਺Ͱද͢ • bitͷঢ়ଶ͸ిѹͷ on/off Ͱอ࣋͞Ε͍ͯΔɻ

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ݹయతίϯϐϡʔλͷԋࢉ • ࿦ཧήʔτʢAND, OR, NOTʣͰԋࢉճ࿏Λ૊Ή • AND, NOTήʔτͷ2छྨ (΋͘͠͸NANDήʔτͳΒ1छྨ) ͕͋Ε͹  ೚ҙͷ࿦ཧճ࿏Λ࣮ݱͰ͖Δ

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ྔࢠίϯϐϡʔλͷجૅ • ྔࢠྗֶతͳঢ়ଶͷॏͶ߹ΘͤͰฒྻੑΛ࣮ݱ͢Δ • ܭࢉ୯Ґ͸ʮQubit (Quantum bit) ʯ  ˠ 0 ͔ 1 ʹͳΔʮ֬཰Λอ࣋ʯͯ͠ԋࢉΛߦ͏ • ྔࢠϏοτ͸ʮϒϥɾέοτه๏ʯͰදݱ͞ΕΔ جຊతʹྔࢠྗֶ͸ʮγϡϨσΟϯΨʔܗࣜʢඍੵ෼ʣʯ͕ͩίϯϐϡʔλͰ  ѻ͏ͨΊʮσΟϥοΫܗࣜʢߦྻʣʯΛ༻͍Δ ԋࢉͰ͸ʮςϯιϧੵʯΛଟ༻͢Δ

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ྔࢠίϯϐϡʔλͷԋࢉʢςϯιϧੵʣ • ߦྻੵ ( 0 0 0 1) ⋅ ( 1 0 0 1) = ( 0 0 0 1) ֤ߦྻͷཁૉಉ࢜ͷԋࢉʢੵɾ࿨ʣ ͦΕͧΕͷߦྻͷαΠζ͸ n×m, m×p Ͱͳ͚Ε͹ͳΒͳ͍ ԋࢉޙͷߦྻͷαΠζ͸ n×p ͱͳΔ

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• ςϯιϧੵ ཁૉͱߦྻͷԋࢉ ͦΕͧΕͷߦྻͷαΠζΛ n×m, p×q ͱ͢Ε͹  ԋࢉޙͷߦྻαΠζ͸np × mq ͱͳΔ ςϯιϧੵ͸ߦྻͷ֦ு ( 0 0 0 1) ⊗ ( 1 0 0 1) = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 ྔࢠίϯϐϡʔλͷԋࢉʢςϯιϧੵʣ

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ཁૉͱߦྻͷԋࢉ ͦΕͧΕͷߦྻͷαΠζΛ n×m, p×q ͱ͢Ε͹  ԋࢉޙͷߦྻαΠζ͸np × mq ͱͳΔ ςϯιϧੵ͸ߦྻͷ֦ு ྔࢠίϯϐϡʔλͷԋࢉʢςϯιϧੵʣ • ςϯιϧੵ ( 0 0 0 1) ⊗ ( 1 0 0 1) = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 ⋅ ( 1 0 0 1) 0 ⋅ ( 1 0 0 1) 1 ⋅ ( 1 0 0 1) 0 ⋅ ( 1 0 0 1)

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ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ʣ • ϒϥɾέοτه๏ α, β Λෳૉ਺ͱ͢Ε͹ |⟩ ɿϒϥ ⟨| |A⟩ = ( α β) ⟨A| = (α* β*) ɹɹɿɹͷෳૉڞ໾ X* X γ, δ Λෳૉ਺ͱ͢Ε͹ |B⟩ = ( γ δ) ⟨B| = (γ* δ*) ɿέοτ

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ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ͷܭࢉྫʣ |A⟩ = ( α β) ⟨A| = (α* β*) ⟨B| = (γ* δ*) |B⟩ = ( γ δ)

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ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ͷܭࢉྫʣ ⟨A|B⟩ = (α* β*) ( γ δ) = α*γ + β*δ ← ߦྻA, Bͷ಺ੵ |A⟩ = ( α β) ⟨A| = (α* β*) ⟨B| = (γ* δ*) |B⟩ = ( γ δ)

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ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ͷܭࢉྫʣ |A⟩ ⊗ |B⟩ = ( α β) ⊗ ( γ δ) = αγ αδ βγ βδ |A⟩ ⊗ ⟨B| = ( α β) ⊗ (γ* δ*) = ( αγ* αδ* βγ* βδ*) ⟨A|B⟩ = (α* β*) ( γ δ) = α*γ + β*δ ← ߦྻA, Bͷ಺ੵ |A⟩ = ( α β) ⟨A| = (α* β*) ⟨B| = (γ* δ*) |B⟩ = ( γ δ)

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ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ͷܭࢉྫʣ |B⟩ ⟨B| ͸ ͷ Τϧϛʔτڞ໾ͱ͍͏ ߦྻA, Bͷςϯιϧੵ  ˠ ԋࢉࢠʢߦྻʣ |A⟩ ⊗ |B⟩ = ( α β) ⊗ ( γ δ) = αγ αδ βγ βδ |A⟩ ⊗ ⟨B| = ( α β) ⊗ (γ* δ*) = ( αγ* αδ* βγ* βδ*) ⟨A|B⟩ = (α* β*) ( γ δ) = α*γ + β*δ ← ߦྻA, Bͷ಺ੵ |A⟩ = ( α β) ⟨A| = (α* β*) ⟨B| = (γ* δ*) |B⟩ = ( γ δ)

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• ςϯιϧੵͷলུ ෳ਺ͷςϯιϧੵͰද͞Ε͍ͯΔϕΫτϧ͸·ͱΊΔ͜ͱ͕Ͱ͖Δɻ  έοτɺϒϥಉ࢜͸লུͰ͖Δ͕ɺࠞࡏ͍ͯ͠Δͱ͖͸஫ҙ͕ඞཁ ྔࢠίϯϐϡʔλͷԋࢉʢςϯιϧੵʣ |0⟩ ⊗ |1⟩ ⊗ |1⟩ ⊗ |0⟩ ⊗ |1⟩ = |0⟩|1⟩|1⟩|0⟩|1⟩ = |01101⟩ (⟨A| ⊗ ⟨B|) (|X⟩ ⊗ |Z⟩) = ⟨A|X⟩⟨B|Z⟩ ৔߹ʹΑͬͯ͸ςϯιϧੵΛؚΜͰͯ΋ܭࢉ݁Ռ͸಺ੵʹͳΔͱ͔ 1 1 2 2 1 1 2 2

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• ԋࢉࢠUΛఆٛ͢Δ ྔࢠίϯϐϡʔλͷԋࢉʢΤϧϛʔτڞ໾ʣ → సஔͷෳૉڞ໾ U = ( α β) ԋࢉࢠUͷసஔ͸ tU = (α β) ԋࢉࢠUͷసஔͷෳૉڞ໾͸ tU* = (α* β*) ຖճɹ Λॻ͘ͷ͸໘౗ → লུه߸͋Γ·͢ tU* U† = tU* ɿUͷΤϧϛʔτڞ໾( ɿ μΨʔ ) U† †

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ྔࢠͷੑ࣭ʹ͍ͭͯ

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• ཭ࢄੑ • ෆ֬ఆੑ − ෆ֬ఆੑݪཧ( ϋΠθϯϕϧάͷݪཧ ) • ೋॏੑ ྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ • ཭ࢄੑ  ྔࢠ͸࿈ଓతͳΤωϧΪʔͰ͸ͳ͘཭ࢄతͳΤωϧΪʔΛ࣋ͭ  ϚΫϩʢڊࢹతʣͰ͸࿈ଓతͰ͋Δ͕ɺϛΫϩʢඍࢹతʣͰ͸ಛఆͷ৔߹ʹ཭ࢄ తͳΤωϧΪʔ४Ґ͔࣋ͯ͠ͳ͘ͳΔ • ཻࢠ̍ݸ͕࣋ͭΤωϧΪʔ ℏω ( ∵ ℏ = h 2π ) hɿϓϥϯΫఆ਺

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ • ෆ֬ఆੑ − ෆ֬ఆੑݪཧʢϋΠθϯϕϧάͷݪཧʣ Δx ⋅ Δpx ≥ ℏ 2 ΔxɿҐஔͷෆ֬ఆੑ Δpx ɿӡಈྔͷෆ֬ఆੑ ిࢠͷӡಈྔʢ଎౓ʣͱҐஔΛಉ࣌ʹਖ਼֬ʹ஌Δ͜ͱ͸Ͱ͖ͳ͍  ɾӡಈྔ͕෼͔Ε͹ʢɹɹ = 0 ʣɺҐஔ͕ෆ໌ʢɹɹ= ∞ ʣ  ɾҐஔ͕෼͔Ε͹ʢɹɹ = 0 ʣɺӡಈྔ͕ෆ໌ʢɹɹ = ∞ ʣ Δpx Δx Δx Δpx υΠπͷ෺ཧֶऀ ϋΠθϯϕϧάʹΑͬͯఏҊ͞Εͨ

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ • ೋॏੑ − ೋॏεϦοτͷ࣮ݧ • ཻࢠͳΒ͹εϦοτΛ௨ͬͯ  ਅͬ௚͙εΫϦʔϯʹͿ͔ͭΔ  ͞Βཻࢠ̍ͭ̍ͭͷ੻͕࢒Δ • ೾ͳΒ͹εϦοτΛ௨ΔͷͰ  ׯবࣶ͕εΫϦʔϯʹͰ͖Δ

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ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ • ిࢠΛೋॏεϦοτʹ௨͢ͱʁ ཻࢠͷੑ࣭͕εΫϦʔϯʹΈΒΕΔʢཻࢠͷ੻ʣ ೾ͷੑ࣭͕εΫϦʔϯʹΈΒΕΔʢׯবࣶʣ ͲͪΒͩͱࢥ͍·͔͢ʁ

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ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ • ిࢠΛೋॏεϦοτʹ௨͢ͱʁ • ʮཻࢠͷੑ࣭ʹݟΒΕΔిࢠͷিಥͷ੻ʯ  ʮ೾ͷੑ࣭ʹݟΒΕΔׯবࣶʯͷ྆ํ͕εΫϦʔϯʹΈΒΕΔ ͭ·Γʮిࢠʯ͸  ˠʮ೾ʯͷΑ͏ʹׯব͠߹͍  ʮཻࢠʯͷΑ͏ʹεΫϦʔϯʹিಥͨ͠ͱ͍͏͜ͱʹͳΔ ʮ೾ʯͱʮཻࢠʯͷೋॏੑ

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ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ • ిࢠΛೋॏεϦοτͷલͰ؍ଌͨ͠Βʁ ʮ೾ʯͷੑ࣭͚ͩΛ͍࣋ͬͯΔͷ͔ ʮཻࢠʯͷੑ࣭͚ͩΛ͍࣋ͬͯΔͷ͔ ిࢠ͕෼ׂ͞Ε͍ͯΔͷͰ͸ͳ͍͔

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ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ • ిࢠΛೋॏεϦοτͷલͰ؍ଌͨ͠Βʁ • ʮ؍ଌʯͱ͍͏ߦҝΛߦ͏ͱʮཻࢠʯͷੑ࣭͔͠ΈΒΕͳ͔ͬͨ  ˠʮ؍ଌʯΛߦ͏ͱঢ়ଶ่͕Εͯ͠·͏ ޙड़ɿʮྔࢠॏͶ߹Θͤͷঢ়ଶʯʹؔ࿈͍ͯ͠Δ

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠঢ়ଶʹ͍ͭͯʳ • ྔࢠॏͶ߹Θͤঢ়ଶ • ྔࢠׯবޮՌ • ྔࢠ΋ͭΕঢ়ଶʢΤϯλϯάϧϝϯτʣ

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ • ྔࢠॏͶ߹Θͤঢ়ଶ  ཭ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ  ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ • ྔࢠॏͶ߹Θͤঢ়ଶ  ཭ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ  ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ |α|2 % |β|2 % |0⟩ |1⟩ α|0⟩ + β|1⟩ • ྔࢠॏͶ߹Θͤঢ়ଶ  ཭ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ  ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ ϧϏϯͷᆵ • ྔࢠॏͶ߹Θͤঢ়ଶ  ཭ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ  ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠׯবޮՌʳ • ྔࢠׯবޮՌ  ৭ʑͳঢ়ଶ͕ڧΊ߹ͬͨΓऑΊ߹ͬͨΓ͢Δ͜ͱ  ೾ͷׯবͱࣅͨΑ͏ͳݱ৅

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠ΋ͭΕঢ়ଶʳ • ྔࢠ΋ͭΕঢ়ଶʢΤϯλϯάϧϝϯτʣ  ৭ʑͳঢ়ଶؒͰ૬ޓؔ܎͕͋Γ෼཭Ͱ͖ͳ͍  ˠ ʰγϡϨσΟϯΨʔͷೣͷύϥυοΫεʱ͕༗໊ |0⟩ |1⟩ ෳ߹ܥͷঢ়ଶΛςϯιϧੵΛ  ༻͍ͯද͢͜ͱ͕Ͱ͖ͳ͍࣌  ྔࢠ΋ͭΕঢ়ଶͱ͍͏ |0⟩ ⊗ |1⟩ ≠ |0⟩ + |1⟩

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠ΋ͭΕঢ়ଶʳ • ྔࢠ΋ͭΕঢ়ଶʢΤϯλϯάϧϝϯτʣ  ৭ʑͳঢ়ଶؒͰ૬ޓؔ܎͕͋Γ෼཭Ͱ͖ͳ͍  ˠ ʰγϡϨσΟϯΨʔͷೣͷύϥυοΫεʱ͕༗໊ |0⟩ |1⟩ |0⟩ ⊗ |1⟩ ≠ |0⟩ + |1⟩ ͲΏ͜ͱʁ ෳ߹ܥͷঢ়ଶΛςϯιϧੵΛ  ༻͍ͯද͢͜ͱ͕Ͱ͖ͳ͍࣌  ྔࢠ΋ͭΕঢ়ଶͱ͍͏

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠ΋ͭΕঢ়ଶʳ • Τϯλϯάϧϝϯτঢ়ଶͱඇΤϯλϯάϧϝϯτঢ়ଶ ࣍ͷ̎ͭͷঢ়ଶA, Bʢ ɹɹɹɹ ɹʣΛߟ͑Δ |ψA ⟩, |ψB ⟩ |ψA ⟩ = 1 2 (|00⟩ + |01⟩) |ψB ⟩ = 1 2 (|00⟩ + |11⟩)

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠ΋ͭΕঢ়ଶʳ → ঢ়ଶA͸ςϯιϧੵͰද͢͜ͱ͕Ͱ͖Δ͕ঢ়ଶB͸Ͱ͖ͳ͍ • Τϯλϯάϧϝϯτঢ়ଶͱඇΤϯλϯάϧϝϯτঢ়ଶ ࣍ͷ̎ͭͷঢ়ଶA, Bʢ ɹɹɹɹ ɹʣΛߟ͑Δ |ψA ⟩, |ψB ⟩ |ψA ⟩ = 1 2 (|00⟩ + |01⟩) |ψB ⟩ = 1 2 (|00⟩ + |11⟩)

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠ΋ͭΕঢ়ଶʳ |ψA ⟩ = |0⟩ ⊗ 1 2 (|0⟩ + |1⟩) • Τϯλϯάϧϝϯτঢ়ଶͱඇΤϯλϯάϧϝϯτঢ়ଶ → ঢ়ଶA͸ςϯιϧੵͰද͢͜ͱ͕Ͱ͖Δ͕ঢ়ଶB͸Ͱ͖ͳ͍ ࣍ͷ̎ͭͷঢ়ଶA, Bʢ ɹɹɹɹ ɹʣΛߟ͑Δ |ψA ⟩, |ψB ⟩ |ψA ⟩ = 1 2 (|00⟩ + |01⟩) |ψB ⟩ = 1 2 (|00⟩ + |11⟩)

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ྔࢠίϯϐϡʔλͷجૅʲྔࢠ΋ͭΕঢ়ଶʳ ঢ়ଶAɿඇΤϯλϯάϧϝϯτঢ়ଶʢ΋ͭΕঢ়ଶʹͳ͍ʣ  ঢ়ଶBɿɹΤϯλϯάϧϝϯτঢ়ଶʢ΋ͭΕঢ়ଶʹ͋Δʣ • Τϯλϯάϧϝϯτঢ়ଶͱඇΤϯλϯάϧϝϯτঢ়ଶ |ψA ⟩ = |0⟩ ⊗ 1 2 (|0⟩ + |1⟩) → ঢ়ଶA͸ςϯιϧੵͰද͢͜ͱ͕Ͱ͖Δ͕ঢ়ଶB͸Ͱ͖ͳ͍ ࣍ͷ̎ͭͷঢ়ଶA, Bʢ ɹɹɹɹ ɹʣΛߟ͑Δ |ψA ⟩, |ψB ⟩ |ψA ⟩ = 1 2 (|00⟩ + |01⟩) |ψB ⟩ = 1 2 (|00⟩ + |11⟩)

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͔͜͜Βຊ൪ ౖ౭ͷ਺ֶ & ྔࢠྗֶ

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࣌ؒͷ౎߹্ ༷ʑͳ৚݅ͷղઆ͸লུ͠·͢    ✨ ϒϩάͰߦؒຒΊ͠·͢ ✨

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1. ਺࿦ܥ • Ҽ਺෼ղ • ཭ࢄର਺໰୊ • ϕϧํఔࣜ • Ψ΢ε࿨ • ߹ಉθʔλؔ਺ ͜Ε΍Γ·͢ ྔࢠίϯϐϡʔλͰߴ଎ܭࢉͰ͖Δ͜ͱ 2. زԿܥ • ݁ͼ໨ෆมྔ • Persistent Homology 3. ઢܗ୅਺ܥ • ߦྻͷྦྷ৐ • ߦྻͷ֊৐ ͳͲ

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• ֬཰తΞϧΰϦζϜʢෳ਺ճ࣮ߦ͢Ε͹ɺߴ֬཰Ͱ౰ͨΔʣ 1. NΛҼ਺෼ղ͢Δͱ͖ɺࣗવ਺ p < N ΛϥϯμϜʹܾΊΔ 2. gcd(p, N)Λܭࢉ͢Δ → ݁Ռ͕ > 1 ͳΒ͹ɺඇࣗ໌ͳNͷҼ਺ 3. ɹɹɹɹɹɹɹɹɹͷपظTΛݟ͚ͭΔʢྔࢠΞϧΰϦζϜʣ 1. T͕ح਺ͳΒɺ1.͔Β΍Γ௚͢ 2. ɹɹɹɹɹɹɹɹɹ ͳΒɺ1.͔Β΍Γ௚͢ 4. ɹɹɹɹɹɹɹ͕ඇࣗ໌ͳNͷҼ਺      → ShorͷΞϧΰϦζϜʹΑΔҼ਺෼ղ fN (x) = px mod N pT 2 + 1 ≡ 0 mod N gcd(pT 2 ± 1,N) pT ≡ 1 mod N ⇔ (pT 2 + 1)(pT 2 − 1) ≡ 0 mod N ≠ 0 ≠ 0 p, Nͷ࠷େެ໿਺ Ґ਺ൃݟΞϧΰϦζϜ ɾҐ਺ਪఆ໰୊  ɾҐ਺ൃݟ໰୊  ɾؔ਺पظൃݟ໰୊  ݁Ռ͕ͳΒ͹  ʰޓ͍ʹૉʱͰ͋Δ

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• N = 57ͱ͢Δɻp = 5ͩͬͨͱ͖ 1. ɹɹɹɹɹɹɹͱͳΔΑ͏ͳɹɹɹɹΛબͿ  ɹɹɹɹɹɹɹɹ ΑΓ 2. ྔࢠϏοτΛ࡞੒͢Δɿ  3. ྔࢠϑʔϦΤม׵Λ࡞༻ͤ͞Δɿ ShorͷΞϧΰϦζϜʹΑΔҼ਺෼ղ N2 ≤ q < 2N2 q = 2k 572 ≤ q < 2 ⋅ 572 q = 212 = 4096 1 q q−1 ∑ x=0 |x⟩ ⊗ |f(x)⟩ ( 1 q ) 2 q−1 ∑ y=0 q−1 ∑ x=0 ei 2πxy q |y⟩ ⊗ |f(x)⟩ 4. ɹͷӈϏοτΛଌఆ͠ɺ  ࠨϏοτΛଌఆ͢Δͱ࣍ͷ֬཰Ͱ  yΛಘΒΕΔ ⊗ 1 q⌊q r ⌋ ⌊ q r ⌋−1 ∑ x=0,f(x)=z ei 2πrxy q 2 f57 (x) = 5x mod 57

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ShorͷΞϧΰϦζϜʹΑΔҼ਺෼ղ 1 q q−1 ∑ x=0 |x⟩ ⊗ |f(x)⟩ = 1 q (|0⟩ ⊗ |50⟩ + |1⟩ ⊗ |51⟩ + |2⟩ ⊗ |52⟩ + . . . +|18⟩ ⊗ |50⟩ + |19⟩ ⊗ |51⟩ + |20⟩ ⊗ |52⟩ + . . . +|4086⟩ ⊗ |50⟩ + |4087⟩ ⊗ |51⟩ + |4088⟩ ⊗ |52⟩ + . . . + |4095⟩ ⊗ |59⟩) 2. ྔࢠϏοτͷ࡞੒

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2. ྔࢠϏοτͷ࡞੒ ShorͷΞϧΰϦζϜʹΑΔҼ਺෼ղ 1 q q−1 ∑ x=0 |x⟩ ⊗ |f(x)⟩ = 1 q (|0⟩ ⊗ |50⟩ + |1⟩ ⊗ |51⟩ + |2⟩ ⊗ |52⟩ + . . . +|18⟩ ⊗ |50⟩ + |19⟩ ⊗ |51⟩ + |20⟩ ⊗ |52⟩ + . . . +|4086⟩ ⊗ |50⟩ + |4087⟩ ⊗ |51⟩ + |4088⟩ ⊗ |52⟩ + . . . + |4095⟩ ⊗ |59⟩) ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩ +(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩ + . . . +(|17⟩ + |35⟩ + |53⟩ + . . . + |4085⟩)|517⟩) ੔ཧ͢Δͱ…

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ShorͷΞϧΰϦζϜʹΑΔҼ਺෼ղ ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩ +(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩ + . . . 4. पظTͷಋग़ͱҼ਺෼ղ

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Λࣔ͢ɻ͜ͷͱ͖ࠨϏοτͷ0Λআ͘ ࠷খۮ਺͕ʰҐ਺ʱʹͳΔɻ  Αͬͯ T=18 ͱͳΔ ShorͷΞϧΰϦζϜʹΑΔҼ਺෼ղ ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩ +(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩ + . . . |50⟩ = f57 (x) = 5x mod 57 = 1 4. पظTͷಋग़ͱҼ਺෼ղ

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ShorͷΞϧΰϦζϜʹΑΔҼ਺෼ղ ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩ +(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩ + . . . |50⟩ = f57 (x) = 5x mod 57 = 1 gcd(pT 2 ± 1,N) ͕ඇࣗ໌ͳҼ਺ͳͷͰ… Λࣔ͢ɻ͜ͷͱ͖ࠨϏοτͷ0Λআ͘ ࠷খۮ਺͕ʰҐ਺ʱʹͳΔɻ  Αͬͯ T=18 ͱͳΔ 4. पظTͷಋग़ͱҼ਺෼ղ

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4. पظTͷಋग़ͱҼ਺෼ղ ShorͷΞϧΰϦζϜʹΑΔҼ਺෼ղ ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩ +(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩ + . . . |50⟩ = f57 (x) = 5x mod 57 = 1 gcd(518 2 + 1,57) = gcd(1953126,57) = 3 gcd(518 2 − 1,57) = gcd(1953124,57) = 19   Ҽ਺෼ղͰ͖ͨʂ gcd(pT 2 ± 1,N) ͕ඇࣗ໌ͳҼ਺ͳͷͰ… Λࣔ͢ɻ͜ͷͱ͖ࠨϏοτͷ0Λআ͘ ࠷খۮ਺͕ʰҐ਺ʱʹͳΔɻ  Αͬͯ T=18 ͱͳΔ

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·ͱΊ • ྔࢠίϯϐϡʔλ͸׬੒͢Ε͹ͱΜͰ΋ͳ͍Խ͚෺  ˠ ݱࡏͷPCੑೳΛӽ͑ΔͨΊʹ͸·ͩ·͕͔͔ͩ࣌ؒΔ • ܭࢉ݁Ռ͸͋͘·Ͱʰ֬཰஋ʱͰ͋Γɺ൓෮ܭࢉΛߦ͏ඞཁ͋Γ  ˠ ౎౓ܭࢉʹલճͷ֬཰஋Λ༻͍ͯܭࢉ͞ΕΔͨΊ • Q#͍ͬͯ͏ྔࢠϓϩάϥϛϯάݴޠ΋͋ΔͷͰݕࡧͯ͠ΈͯͶʂ  ˠ ϓϩάϥϜͰ͜ͷΞϧΰϦζϜ͕؆୯ʹ͔͚Δʂ

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͝੩ௌ   ͋Γ͕ͱ͏͍͟͝·ͨ͠ εϥΠυɾղઆ౳͸  ϒϩάʹUp͠·͢ͷͰੋඇΈ͍ͯͩ͘͞ ✨