高専カンファレンス新春 in 大阪

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。

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Hiroki Nomura

January 12, 2019
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  1. 9.

    ྔࢠίϯϐϡʔλͰߴ଎ܭࢉͰ͖Δ͜ͱ 1. ਺࿦ܥ • Ҽ਺෼ղ • ཭ࢄର਺໰୊ • ϕϧํఔࣜ •

    Ψ΢ε࿨ • ߹ಉθʔλؔ਺ 2. زԿܥ • ݁ͼ໨ෆมྔ • Persistent Homology 3. ઢܗ୅਺ܥ • ߦྻͷྦྷ৐ • ߦྻͷ֊৐ ͳͲ
  2. 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
  3. 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)
  4. 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)
  5. 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
  6. 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
  7. 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
  8. 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)
  9. 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
  10. 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)
  11. 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)
  12. 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
  13. 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)
  14. 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
  15. 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† †
  16. 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׵
  17. 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
  18. 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
  19. 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
  20. 44.

    ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ |α|2 % |β|2 % |0⟩ |1⟩ α|0⟩ + β|1⟩

    • ྔࢠॏͶ߹Θͤঢ়ଶ
 ཭ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ
 ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ
  21. 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⟩)
  22. 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⟩)
  23. 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
  24. 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໰୊
 ݁Ռ͕ͳΒ͹
 ʰޓ͍ʹૉʱͰ͋Δ
  25. 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
  26. 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
  27. 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⟩)
  28. 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࡞੒
  29. 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…
  30. 62.

    ShorͷΞϧΰϦζϜʹΑΔҼ਺෼ղ ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩

    +(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩ + . . . 4. पظTͷಋग़ͱҼ਺෼ղ
  31. 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
  32. 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
  33. 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