高専カンファレンス新春 in 大阪

『2019-01-12 高専カンファレンス新春 in 大阪』で発表した資料です。

Transcript

1. ৽य़ɾߴઐΧϯϑΝ in େࡕ 2019/01/12 kosenconf-123shinshun
 ͷΉͷΉʢ@nomunomu0504ʣ
2. ڈ೥ʹҾ͖ଓ͖… • ڈ೥ͷ৽य़ΧϯϑΝʹ΋ࢀՃͯ͠·ͨ͠ • @John_bardera ͔Β࣮ߦҕһ௕৆Λ໯͍·ͨ͠
 ʢϞϯΤφϑϧηοτʣ
3. πΠʔτ͢Δͱ͖ʹ͸… #ͷΉͷΉ
4. ࠓ೔ͷൃද಺༰ • ྔࢠίϯϐϡʔλʹ͍ͭͯ
 - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍
 - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔
5. ࠓ೔ͷൃද಺༰ ྔࢠίϯϐϡʔλΛઐ໳ͱ͞Ε͍ͯΔํ • ྔࢠίϯϐϡʔλʹ͍ͭͯ
 - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍
 - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔
6. ࠓ೔ͷൃද಺༰ ྔࢠ࿦ɾྔࢠྗֶઐ߈ͷํ • ྔࢠίϯϐϡʔλʹ͍ͭͯ
 - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍
 - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔ ྔࢠίϯϐϡʔλΛઐ໳ͱ͞Ε͍ͯΔํ
7. ࠓ೔ͷൃද಺༰ • ྔࢠίϯϐϡʔλʹ͍ͭͯ
 - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍
 - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔ ྔࢠ࿦ɾྔࢠྗֶઐ߈ͷํ ྔࢠίϯϐϡʔλΛઐ໳ͱ͞Ε͍ͯΔํ झຯൣғͰֶश͍ͯ͠Δ಺༰Ͱ͢
 ʢߨٛ౳Ұ੾औ͍ͬͯ·ͤΜɻऔΓͨͯ͘΋ߴઐʹ͋Γ·ͤΜʣ
   ؒҧ͍͕͋Γ·ͨ͠Β%.౳ૹ͍͚ͬͯͨͩΔͱ

   ϓϨθϯλʔ͸تͼ·͢
   
8. ྔࢠίϯϐϡʔλͱ͸ • ྔࢠྗֶͷੑ࣭Λ࢖ͬͯߴ଎ʹܭࢉͰ͖Δίϯϐϡʔλ • ͋Δఔ౓ͷαΠζͷྔࢠίϯϐϡʔλ͕͋Ε͹
 ໰୊ʹΑͬͯ͸εύίϯΑΓ΋ߴ଎ʹܭࢉͰ͖Δ • ྔࢠίϯϐϡʔλͷ࣮ݱํ๏͸̎छྨ͋Δ • ྔࢠήʔτํࣜ • ྔࢠΞχʔϦϯάํࣜ > ࠓճઆ໌͢Δͷ͸ʰྔࢠήʔτํࣜʱ
9. ྔࢠίϯϐϡʔλͰߴ଎ܭࢉͰ͖Δ͜ͱ 1. ਺࿦ܥ • Ҽ਺෼ղ • ཭ࢄର਺໰୊ • ϕϧํఔࣜ • Ψ΢ε࿨ • ߹ಉθʔλؔ਺ 2. زԿܥ • ݁ͼ໨ෆมྔ • Persistent Homology 3. ઢܗ୅਺ܥ • ߦྻͷྦྷ৐ • ߦྻͷ֊৐ ͳͲ
10. 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
11. 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)
12. • 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)
13. ͲͷΑ͏ͳܭࢉΛ͍ͯ͠Δͷ͔ • ݹయతίϯϐϡʔλͳΒɺ͋Δॲཧܥ͓͍ͯ
 ೖྗ஋͕ಉ͡ͳΒৗʹಉ݁͡Ռ͕ಘΒΕΔɻ • ྔࢠίϯϐϡʔλͰ͸ɺ͋Δԋࢉࢠܥʹ͓͍ͯ
 ೖྗ஋͕ಉ͡Ͱ΋ʰԋࢉ݁Ռʱ͸ҟͳΔՄೳੑ͕͋Δɻ ྔࢠίϯϐϡʔλ͸ܭࢉͷਖ਼֬͞ʢ֬཰ʣΛग़ྗ͢Δ
14. ֤ίϯϐϡʔλͷ جૅԋࢉʹ͍ͭͯ
15. ݹయతίϯϐϡʔλͷجૅ • ๻Β͕͍ͭ΋ར༻͍ͯ͠ΔίϯϐϡʔλΛ
 ʮݹయతίϯϐϡʔλʯͱݺͿ • ݹయతίϯϐϡʔλͷܭࢉ୯Ґ͸ʮbitʯ
 0 ͔ 1 ͷͲͪΒ͔ͷঢ়ଶΛऔΓ͏Δɻ2ਐ਺Ͱද͢ • bitͷঢ়ଶ͸ిѹͷ on/off Ͱอ࣋͞Ε͍ͯΔɻ
16. ݹయతίϯϐϡʔλͷԋࢉ • ࿦ཧήʔτʢAND, OR, NOTʣͰԋࢉճ࿏Λ૊Ή • AND, NOTήʔτͷ2छྨ (΋͘͠͸NANDήʔτͳΒ1छྨ) ͕͋Ε͹
 ೚ҙͷ࿦ཧճ࿏Λ࣮ݱͰ͖Δ
17. ྔࢠίϯϐϡʔλͷجૅ • ྔࢠྗֶతͳঢ়ଶͷॏͶ߹ΘͤͰฒྻੑΛ࣮ݱ͢Δ • ܭࢉ୯Ґ͸ʮQubit (Quantum bit) ʯ
 ˠ 0 ͔ 1 ʹͳΔʮ֬཰Λอ࣋ʯͯ͠ԋࢉΛߦ͏ • ྔࢠϏοτ͸ʮϒϥɾέοτه๏ʯͰදݱ͞ΕΔ جຊతʹྔࢠྗֶ͸ʮγϡϨσΟϯΨʔܗࣜʢඍੵ෼ʣʯ͕ͩίϯϐϡʔλͰ
 ѻ͏ͨΊʮσΟϥοΫܗࣜʢߦྻʣʯΛ༻͍Δ ԋࢉͰ͸ʮςϯιϧੵʯΛଟ༻͢Δ
18. ྔ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
19. • 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
20. 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)
21. ྔ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
22. ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ͷܭࢉྫʣ |A⟩ = ( α β) ⟨A| = (α* β*) ⟨B| = (γ* δ*) |B⟩ = ( γ δ)
23. ྔ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)
24. ྔ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)
25. ྔ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) ߦྻA, Bͷςϯιϧੵ
 ˠ ԋࢉࢠʢߦྻʣ
26. ྔ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)
27. • 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
28. • 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† †
29. • nྔࢠϏοτͷঢ়ଶɹɹ͕͋Δͱ͖ ྔࢠίϯϐϡʔλͷԋࢉʢྔࢠ཭ࢄతϑʔϦΤม׵ʣ |j⟩ |j⟩ = 1 2n 2n−1 ∑ k=0 ei 2πkj 2n |k⟩ |j⟩ = 1 2n 2n−1 ∑ k=0 e−i 2πkj 2n |k⟩ • ٯྔࢠ཭ࢄతϑʔϦΤม׵
30. • nྔࢠϏοτͷঢ়ଶɹɹ͕͋Δͱ͖ |j⟩ |j⟩ = 1 2n 2n−1 ∑ k=0 ei 2πkj 2n |k⟩ |j⟩ = 1 2n 2n−1 ∑ k=0 e−i 2πkj 2n |k⟩ • ٯྔࢠ཭ࢄతϑʔϦΤม׵ ྔࢠίϯϐϡʔλͷԋࢉʢྔࢠ཭ࢄతϑʔϦΤม׵ʣ
31. • nྔࢠϏοτͷঢ়ଶɹɹ͕͋Δͱ͖ |j⟩ |j⟩ = 1 2n 2n−1 ∑ k=0 ei 2πkj 2n |k⟩ |j⟩ = 1 2n 2n−1 ∑ k=0 e−i 2πkj 2n |k⟩ • ٯྔࢠ཭ࢄతϑʔϦΤม׵ ྔࢠίϯϐϡʔλͷԋࢉʢྔࢠ཭ࢄతϑʔϦΤม׵ʣ ʂपظղੳʹ༻͍Δʂ
32. ྔࢠͷੑ࣭ʹ͍ͭͯ
33. • ཭ࢄੑ • ෆ֬ఆੑ − ෆ֬ఆੑݪཧ( ϋΠθϯϕϧάͷݪཧ ) • ೋॏੑ ྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ
34. ྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ • ཭ࢄੑ
 ྔࢠ͸࿈ଓతͳΤωϧΪʔͰ͸ͳ͘཭ࢄతͳΤωϧΪʔΛ࣋ͭ
 ϚΫϩʢڊࢹతʣͰ͸࿈ଓతͰ͋Δ͕ɺϛΫϩʢඍࢹతʣͰ͸ಛఆͷ৔߹ʹ཭ࢄ తͳΤωϧΪʔ४Ґ͔࣋ͯ͠ͳ͘ͳΔ • ཻࢠ̍ݸ͕࣋ͭΤωϧΪʔ ℏω ( ∵ ℏ = h 2π ) hɿϓϥϯΫఆ਺
35. ྔ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
36. ྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ • ೋॏੑ − ೋॏεϦοτͷ࣮ݧ • ཻࢠͳΒ͹εϦοτΛ௨ͬͯ
 ਅͬ௚͙εΫϦʔϯʹͿ͔ͭΔ
 ͞Βཻࢠ̍ͭ̍ͭͷ੻͕࢒Δ • ೾ͳΒ͹εϦοτΛ௨ΔͷͰ
 ׯবࣶ͕εΫϦʔϯʹͰ͖Δ
37. ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ • ిࢠΛೋॏεϦοτʹ௨͢ͱʁ ཻࢠͷੑ࣭͕εΫϦʔϯʹΈΒΕΔʢཻࢠͷ੻ʣ ೾ͷੑ࣭͕εΫϦʔϯʹΈΒΕΔʢׯবࣶʣ ͲͪΒͩͱࢥ͍·͔͢ʁ
38. ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ • ిࢠΛೋॏεϦοτʹ௨͢ͱʁ • ʮཻࢠͷੑ࣭ʹݟΒΕΔిࢠͷিಥͷ੻ʯ
 ʮ೾ͷੑ࣭ʹݟΒΕΔׯবࣶʯͷ྆ํ͕εΫϦʔϯʹΈΒΕΔ ͭ·Γʮిࢠʯ͸
 ˠʮ೾ʯͷΑ͏ʹׯব͠߹͍
 ʮཻࢠʯͷΑ͏ʹεΫϦʔϯʹিಥͨ͠ͱ͍͏͜ͱʹͳΔ ʮ೾ʯͱʮཻࢠʯͷೋॏੑ
39. ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ • ిࢠΛೋॏεϦοτͷલͰ؍ଌͨ͠Βʁ ʮ೾ʯͷੑ࣭͚ͩΛ͍࣋ͬͯΔͷ͔ ʮཻࢠʯͷੑ࣭͚ͩΛ͍࣋ͬͯΔͷ͔ ిࢠ͕෼ׂ͞Ε͍ͯΔͷͰ͸ͳ͍͔
40. ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ • ిࢠΛೋॏεϦοτͷલͰ؍ଌͨ͠Βʁ • ʮ؍ଌʯͱ͍͏ߦҝΛߦ͏ͱʮཻࢠʯͷੑ࣭͔͠ΈΒΕͳ͔ͬͨ
 ˠʮ؍ଌʯΛߦ͏ͱঢ়ଶ่͕Εͯ͠·͏ ޙड़ɿʮྔࢠॏͶ߹Θͤͷঢ়ଶʯʹؔ࿈͍ͯ͠Δ
41. ྔࢠίϯϐϡʔλͷجૅʲྔࢠঢ়ଶʹ͍ͭͯʳ • ྔࢠॏͶ߹Θͤঢ়ଶ • ྔࢠׯবޮՌ • ྔࢠ΋ͭΕঢ়ଶʢΤϯλϯάϧϝϯτʣ
42. ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ • ྔࢠॏͶ߹Θͤঢ়ଶ
 ཭ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ
 ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ
43. ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ • ྔࢠॏͶ߹Θͤঢ়ଶ
 ཭ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ
 ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ
44. ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ |α|2 % |β|2 % |0⟩ |1⟩ α|0⟩ + β|1⟩ • ྔࢠॏͶ߹Θͤঢ়ଶ
 ཭ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ
 ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ
45. ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ ϧϏϯͷᆵ • ྔࢠॏͶ߹Θͤঢ়ଶ
 ཭ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ
 ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ
46. ྔࢠίϯϐϡʔλͷجૅʲྔࢠׯবޮՌʳ • ྔࢠׯবޮՌ
 ৭ʑͳঢ়ଶ͕ڧΊ߹ͬͨΓऑΊ߹ͬͨΓ͢Δ͜ͱ
 ೾ͷׯবͱࣅͨΑ͏ͳݱ৅
47. ྔࢠίϯϐϡʔλͷجૅʲྔࢠ΋ͭΕঢ়ଶʳ • ྔࢠ΋ͭΕঢ়ଶʢΤϯλϯάϧϝϯτʣ
 ৭ʑͳঢ়ଶؒͰ૬ޓؔ܎͕͋Γ෼཭Ͱ͖ͳ͍
 ˠ ʰγϡϨσΟϯΨʔͷೣͷύϥυοΫεʱ͕༗໊ |0⟩ |1⟩ ෳ߹ܥͷঢ়ଶΛςϯιϧੵΛ
 ༻͍ͯද͢͜ͱ͕Ͱ͖ͳ͍࣌
 ྔࢠ΋ͭΕঢ়ଶͱ͍͏ |0⟩ ⊗ |1⟩ ≠ |0⟩ + |1⟩
48. ྔࢠίϯϐϡʔλͷجૅʲྔࢠ΋ͭΕঢ়ଶʳ • ྔࢠ΋ͭΕঢ়ଶʢΤϯλϯάϧϝϯτʣ
 ৭ʑͳঢ়ଶؒͰ૬ޓؔ܎͕͋Γ෼཭Ͱ͖ͳ͍
 ˠ ʰγϡϨσΟϯΨʔͷೣͷύϥυοΫεʱ͕༗໊ |0⟩ |1⟩ |0⟩ ⊗ |1⟩ ≠ |0⟩ + |1⟩ ͲΏ͜ͱʁ ෳ߹ܥͷঢ়ଶΛςϯιϧੵΛ
 ༻͍ͯද͢͜ͱ͕Ͱ͖ͳ͍࣌
 ྔࢠ΋ͭΕঢ়ଶͱ͍͏
49. ྔࢠίϯϐϡʔλͷجૅʲྔࢠ΋ͭΕঢ়ଶʳ • Τϯλϯάϧϝϯτঢ়ଶͱඇΤϯλϯάϧϝϯτঢ়ଶ ࣍ͷ̎ͭͷঢ়ଶA, Bʢ ɹɹɹɹ ɹʣΛߟ͑Δ |ψA ⟩, |ψB ⟩ |ψA ⟩ = 1 2 (|00⟩ + |01⟩) |ψB ⟩ = 1 2 (|00⟩ + |11⟩)
50. ྔࢠίϯϐϡʔλͷجૅʲྔࢠ΋ͭΕঢ়ଶʳ → ঢ়ଶA͸ςϯιϧੵͰද͢͜ͱ͕Ͱ͖Δ͕ঢ়ଶB͸Ͱ͖ͳ͍ • Τϯλϯάϧϝϯτঢ়ଶͱඇΤϯλϯάϧϝϯτঢ়ଶ ࣍ͷ̎ͭͷঢ়ଶA, Bʢ ɹɹɹɹ ɹʣΛߟ͑Δ |ψA ⟩, |ψB ⟩ |ψA ⟩ = 1 2 (|00⟩ + |01⟩) |ψB ⟩ = 1 2 (|00⟩ + |11⟩)
51. ྔ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⟩)
52. ྔ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⟩)
53. ͔͜͜Βຊ൪ ౖ౭ͷ਺ֶ & ྔࢠྗֶ
54. ࣌ؒͷ౎߹্ ༷ʑͳ৚݅ͷղઆ͸লུ͠·͢
 
 ✨ ϒϩάͰߦؒຒΊ͠·͢ ✨
55. 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
56. • ֬཰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໰୊
 ݁Ռ͕
    
    ͳΒ͹
 ʰޓ͍ʹૉʱͰ͋Δ
57. • 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
58. • 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
59. 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⟩)
60. 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࡞੒
61. 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…
62. ShorͷΞϧΰϦζϜʹΑΔҼ਺෼ղ ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩ +(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩ + . . . 4. पظTͷಋग़ͱҼ਺෼ղ
63. 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
64. 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
65. 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
66. ·ͱΊ • ྔࢠίϯϐϡʔλ͸׬੒͢Ε͹ͱΜͰ΋ͳ͍Խ͚෺
 ˠ ݱࡏͷPCੑೳΛӽ͑ΔͨΊʹ͸·ͩ·͕͔͔ͩ࣌ؒΔ • ܭࢉ݁Ռ͸͋͘·Ͱʰ֬཰஋ʱͰ͋Γɺ൓෮ܭࢉΛߦ͏ඞཁ͋Γ
 ˠ ౎౓ܭࢉʹલճͷ֬཰஋Λ༻͍ͯܭࢉ͞ΕΔͨΊ • Q#͍ͬͯ͏ྔࢠϓϩάϥϛϯάݴޠ΋͋ΔͷͰݕࡧͯ͠ΈͯͶʂ
 ˠ ϓϩάϥϜͰ͜ͷΞϧΰϦζϜ͕؆୯ʹ͔͚Δʂ
67. ͝੩ௌ 
 ͋Γ͕ͱ͏͍͟͝·ͨ͠ εϥΠυɾղઆ౳͸
 ϒϩάʹUp͠·͢ͷͰੋඇΈ͍ͯͩ͘͞ ✨