『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。
৽य़ɾߴઐΧϯϑΝin େࡕ2019/01/12 kosenconf-123shinshun ͷΉͷΉʢ@nomunomu0504ʣ
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ڈʹҾ͖ଓ͖…• ڈͷ৽य़ΧϯϑΝʹࢀՃͯ͠·ͨ͠• @John_bardera ͔Β࣮ߦҕһΛ͍·ͨ͠ ʢϞϯΤφϑϧηοτʣ
πΠʔτ͢Δͱ͖ʹ…#ͷΉͷΉ
ࠓͷൃද༰• ྔࢠίϯϐϡʔλʹ͍ͭͯ - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍ - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔
ࠓͷൃද༰ྔࢠίϯϐϡʔλΛઐͱ͞Ε͍ͯΔํ• ྔࢠίϯϐϡʔλʹ͍ͭͯ - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍ - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔
ࠓͷൃද༰ྔࢠɾྔࢠྗֶઐ߈ͷํ• ྔࢠίϯϐϡʔλʹ͍ͭͯ - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍ - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔ྔࢠίϯϐϡʔλΛઐͱ͞Ε͍ͯΔํ
ࠓͷൃද༰• ྔࢠίϯϐϡʔλʹ͍ͭͯ - ྔࢠίϯϐϡʔλͱݹయతίϯϐϡʔλͱͷҧ͍ - ͲͷΑ͏ʹԋࢉΛߦͳ͍ͬͯΔͷ͔ྔࢠɾྔࢠྗֶઐ߈ͷํྔࢠίϯϐϡʔλΛઐͱ͞Ε͍ͯΔํझຯൣғͰֶश͍ͯ͠Δ༰Ͱ͢ ʢߨٛҰऔ͍ͬͯ·ͤΜɻऔΓͨͯ͘ߴઐʹ͋Γ·ͤΜʣؒҧ͍͕͋Γ·ͨ͠Β%.ૹ͍͚ͬͯͨͩΔͱ ϓϨθϯλʔتͼ·͢
ྔࢠίϯϐϡʔλͱ• ྔࢠྗֶͷੑ࣭ΛͬͯߴʹܭࢉͰ͖Δίϯϐϡʔλ• ͋ΔఔͷαΠζͷྔࢠίϯϐϡʔλ͕͋Ε ʹΑͬͯεύίϯΑΓߴʹܭࢉͰ͖Δ• ྔࢠίϯϐϡʔλͷ࣮ݱํ๏̎छྨ͋Δ• ྔࢠήʔτํࣜ• ྔࢠΞχʔϦϯάํࣜ> ࠓճઆ໌͢Δͷʰྔࢠήʔτํࣜʱ
ྔࢠίϯϐϡʔλͰߴܭࢉͰ͖Δ͜ͱ1. ܥ• Ҽղ• ࢄର• ϕϧํఔࣜ• Ψε• ߹ಉθʔλؔ2. زԿܥ• ݁ͼෆมྔ• Persistent Homology3. ઢܗܥ• ߦྻͷྦྷ• ߦྻͷ֊ͳͲ
1. ܥ• Ҽղ• ࢄର• ϕϧํఔࣜ• Ψε• ߹ಉθʔλؔޙͰ࣮ࡍʹಋग़ͬͯΈ·͢ʂྔࢠίϯϐϡʔλͰߴܭࢉͰ͖Δ͜ͱ2. زԿܥ• ݁ͼෆมྔ• Persistent Homology3. ઢܗܥ• ߦྻͷྦྷ• ߦྻͷ֊ͳͲ
Ͳͷ͙Β͍ૣ͘ܭࢉͰ͖Δͷ͔• nϏοτͷҼղ• ݹయతίϯϐϡʔλ(Ұൠମ;Δ͍๏)(e1.9(ln n)13(ln ln n)23)• ྔࢠίϯϐϡʔλ(ShorΞϧΰϦζϜ)((log n)2(log log n)(log log log n))• 1024ϏοτͷҼղʹ͔͔Δܭࢉྔ• ݹయతίϯϐϡʔλɿ• ྔࢠίϯϐϡʔλɹɿ(10278)(1061)
• 1024ϏοτͷҼղʹ͔͔Δܭࢉྔ• ݹయతίϯϐϡʔλɿ• ྔࢠίϯϐϡʔλɹɿͲͷ͙Β͍ૣ͘ܭࢉͰ͖Δͷ͔• nϏοτͷҼղ• ݹయతίϯϐϡʔλ(Ұൠମ;Δ͍๏)(e1.9(ln n)13(ln ln n)23)• ྔࢠίϯϐϡʔλ(ShorΞϧΰϦζϜ)((log n)2(log log n)(log log log n))(10278)(1061)
ͲͷΑ͏ͳܭࢉΛ͍ͯ͠Δͷ͔• ݹయతίϯϐϡʔλͳΒɺ͋Δॲཧܥ͓͍ͯ ೖྗ͕ಉ͡ͳΒৗʹಉ݁͡Ռ͕ಘΒΕΔɻ• ྔࢠίϯϐϡʔλͰɺ͋Δԋࢉࢠܥʹ͓͍ͯ ೖྗ͕ಉ͡Ͱʰԋࢉ݁ՌʱҟͳΔՄೳੑ͕͋Δɻྔࢠίϯϐϡʔλܭࢉͷਖ਼֬͞ʢ֬ʣΛग़ྗ͢Δ
֤ίϯϐϡʔλͷجૅԋࢉʹ͍ͭͯ
ݹయతίϯϐϡʔλͷجૅ• Β͕͍ͭར༻͍ͯ͠ΔίϯϐϡʔλΛ ʮݹయతίϯϐϡʔλʯͱݺͿ• ݹయతίϯϐϡʔλͷܭࢉ୯Ґʮbitʯ 0 ͔ 1 ͷͲͪΒ͔ͷঢ়ଶΛऔΓ͏Δɻ2ਐͰද͢• bitͷঢ়ଶిѹͷ on/off Ͱอ࣋͞Ε͍ͯΔɻ
ݹయతίϯϐϡʔλͷԋࢉ• ཧήʔτʢAND, OR, NOTʣͰԋࢉճ࿏ΛΉ• AND, NOTήʔτͷ2छྨ (͘͠NANDήʔτͳΒ1छྨ) ͕͋Ε ҙͷཧճ࿏Λ࣮ݱͰ͖Δ
ྔࢠίϯϐϡʔλͷجૅ• ྔࢠྗֶతͳঢ়ଶͷॏͶ߹ΘͤͰฒྻੑΛ࣮ݱ͢Δ• ܭࢉ୯ҐʮQubit (Quantum bit) ʯ ˠ 0 ͔ 1 ʹͳΔʮ֬Λอ࣋ʯͯ͠ԋࢉΛߦ͏• ྔࢠϏοτʮϒϥɾέοτه๏ʯͰදݱ͞ΕΔجຊతʹྔࢠྗֶʮγϡϨσΟϯΨʔܗࣜʢඍੵʣʯ͕ͩίϯϐϡʔλͰ ѻ͏ͨΊʮσΟϥοΫܗࣜʢߦྻʣʯΛ༻͍ΔԋࢉͰʮςϯιϧੵʯΛଟ༻͢Δ
ྔࢠίϯϐϡʔλͷԋࢉʢςϯιϧੵʣ• ߦྻੵ(0 00 1) ⋅ (1 00 1) = (0 00 1)֤ߦྻͷཁૉಉ࢜ͷԋࢉʢੵɾʣͦΕͧΕͷߦྻͷαΠζ n×m, m×p Ͱͳ͚ΕͳΒͳ͍ԋࢉޙͷߦྻͷαΠζ n×p ͱͳΔ
• ςϯιϧੵཁૉͱߦྻͷԋࢉͦΕͧΕͷߦྻͷαΠζΛ n×m, p×q ͱ͢Ε ԋࢉޙͷߦྻαΠζnp × mq ͱͳΔςϯιϧੵߦྻͷ֦ு(0 00 1) ⊗ (1 00 1) =0 0 0 00 0 0 00 0 1 00 0 0 1ྔࢠίϯϐϡʔλͷԋࢉʢςϯιϧੵʣ
ཁૉͱߦྻͷԋࢉͦΕͧΕͷߦྻͷαΠζΛ n×m, p×q ͱ͢Ε ԋࢉޙͷߦྻαΠζnp × mq ͱͳΔςϯιϧੵߦྻͷ֦ுྔࢠίϯϐϡʔλͷԋࢉʢςϯιϧੵʣ• ςϯιϧੵ(0 00 1) ⊗ (1 00 1) =0 0 0 00 0 0 00 0 1 00 0 0 10 ⋅ (1 00 1)0 ⋅ (1 00 1) 1 ⋅ (1 00 1)0 ⋅ (1 00 1)
ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ʣ• ϒϥɾέοτه๏α, β Λෳૉͱ͢Ε|⟩ɿϒϥ⟨||A⟩ = (αβ) ⟨A| = (α* β*) ɹɹɿɹͷෳૉڞX* Xγ, δ Λෳૉͱ͢Ε|B⟩ = (γδ) ⟨B| = (γ* δ*)ɿέοτ
ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ͷܭࢉྫʣ|A⟩ = (αβ) ⟨A| = (α* β*) ⟨B| = (γ* δ*)|B⟩ = (γδ)
ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ͷܭࢉྫʣ⟨A|B⟩ = (α* β*) (γδ) = α*γ + β*δ ← ߦྻA, Bͷੵ|A⟩ = (αβ) ⟨A| = (α* β*) ⟨B| = (γ* δ*)|B⟩ = (γδ)
ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ͷܭࢉྫʣ|A⟩ ⊗ |B⟩ = (αβ) ⊗ (γδ) =αγαδβγβδ|A⟩ ⊗ ⟨B| = (αβ) ⊗ (γ* δ*) = (αγ* αδ*βγ* βδ*)⟨A|B⟩ = (α* β*) (γδ) = α*γ + β*δ ← ߦྻA, Bͷੵ|A⟩ = (αβ) ⟨A| = (α* β*) ⟨B| = (γ* δ*)|B⟩ = (γδ)
ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ͷܭࢉྫʣ|A⟩ ⊗ |B⟩ = (αβ) ⊗ (γδ) =αγαδβγβδ|A⟩ ⊗ ⟨B| = (αβ) ⊗ (γ* δ*) = (αγ* αδ*βγ* βδ*)⟨A|B⟩ = (α* β*) (γδ) = α*γ + β*δ ← ߦྻA, Bͷੵ|A⟩ = (αβ) ⟨A| = (α* β*) ⟨B| = (γ* δ*)|B⟩ = (γδ)ߦྻA, Bͷςϯιϧੵ ˠ ԋࢉࢠʢߦྻʣ
ྔࢠίϯϐϡʔλͷԋࢉʢϒϥɾέοτه๏ͷܭࢉྫʣ|B⟩⟨B| ͷΤϧϛʔτڞͱ͍͏ߦྻA, Bͷςϯιϧੵ ˠ ԋࢉࢠʢߦྻʣ|A⟩ ⊗ |B⟩ = (αβ) ⊗ (γδ) =αγαδβγβδ|A⟩ ⊗ ⟨B| = (αβ) ⊗ (γ* δ*) = (αγ* αδ*βγ* βδ*)⟨A|B⟩ = (α* β*) (γδ) = α*γ + β*δ ← ߦྻA, Bͷੵ|A⟩ = (αβ) ⟨A| = (α* β*) ⟨B| = (γ* δ*)|B⟩ = (γδ)
• ςϯιϧੵͷলུෳͷςϯιϧੵͰද͞Ε͍ͯΔϕΫτϧ·ͱΊΔ͜ͱ͕Ͱ͖Δɻ έοτɺϒϥಉ࢜লུͰ͖Δ͕ɺࠞࡏ͍ͯ͠Δͱ͖ҙ͕ඞཁྔࢠίϯϐϡʔλͷԋࢉʢςϯιϧੵʣ|0⟩ ⊗ |1⟩ ⊗ |1⟩ ⊗ |0⟩ ⊗ |1⟩ = |0⟩|1⟩|1⟩|0⟩|1⟩= |01101⟩(⟨A| ⊗ ⟨B|) (|X⟩ ⊗ |Z⟩) = ⟨A|X⟩⟨B|Z⟩߹ʹΑͬͯςϯιϧੵΛؚΜͰͯܭࢉ݁ՌੵʹͳΔͱ͔1 12 2 11 22
• ԋࢉࢠUΛఆٛ͢ΔྔࢠίϯϐϡʔλͷԋࢉʢΤϧϛʔτڞʣ→ సஔͷෳૉڞU = (αβ)ԋࢉࢠUͷసஔtU = (α β)ԋࢉࢠUͷసஔͷෳૉڞtU* = (α* β*)ຖճɹ Λॻ͘ͷ໘ → লུه߸͋Γ·͢tU*U† = tU* ɿUͷΤϧϛʔτڞ( ɿ μΨʔ )U† †
• nྔࢠϏοτͷঢ়ଶɹɹ͕͋Δͱ͖ྔࢠίϯϐϡʔλͷԋࢉʢྔࢠࢄతϑʔϦΤมʣ|j⟩|j⟩ =12n2n−1∑k=0ei 2πkj2n|k⟩|j⟩ =12n2n−1∑k=0e−i 2πkj2n|k⟩• ٯྔࢠࢄతϑʔϦΤม
• nྔࢠϏοτͷঢ়ଶɹɹ͕͋Δͱ͖|j⟩|j⟩ =12n2n−1∑k=0ei 2πkj2n|k⟩|j⟩ =12n2n−1∑k=0e−i 2πkj2n|k⟩• ٯྔࢠࢄతϑʔϦΤมྔࢠίϯϐϡʔλͷԋࢉʢྔࢠࢄతϑʔϦΤมʣ
• nྔࢠϏοτͷঢ়ଶɹɹ͕͋Δͱ͖|j⟩|j⟩ =12n2n−1∑k=0ei 2πkj2n|k⟩|j⟩ =12n2n−1∑k=0e−i 2πkj2n|k⟩• ٯྔࢠࢄతϑʔϦΤมྔࢠίϯϐϡʔλͷԋࢉʢྔࢠࢄతϑʔϦΤมʣʂपظղੳʹ༻͍Δʂ
ྔࢠͷੑ࣭ʹ͍ͭͯ
• ࢄੑ• ෆ֬ఆੑ − ෆ֬ఆੑݪཧ( ϋΠθϯϕϧάͷݪཧ )• ೋॏੑྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ
ྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ• ࢄੑ ྔࢠ࿈ଓతͳΤωϧΪʔͰͳ͘ࢄతͳΤωϧΪʔΛ࣋ͭ ϚΫϩʢڊࢹతʣͰ࿈ଓతͰ͋Δ͕ɺϛΫϩʢඍࢹతʣͰಛఆͷ߹ʹࢄతͳΤωϧΪʔ४Ґ͔࣋ͯ͠ͳ͘ͳΔ • ཻࢠ̍ݸ͕࣋ͭΤωϧΪʔℏω ( ∵ ℏ =h2π ) hɿϓϥϯΫఆ
ྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ• ෆ֬ఆੑ − ෆ֬ఆੑݪཧʢϋΠθϯϕϧάͷݪཧʣΔx ⋅ Δpx≥ℏ2ΔxɿҐஔͷෆ֬ఆੑ Δpx ɿӡಈྔͷෆ֬ఆੑిࢠͷӡಈྔʢʣͱҐஔΛಉ࣌ʹਖ਼֬ʹΔ͜ͱͰ͖ͳ͍ ɾӡಈྔ͕͔Εʢɹɹ = 0 ʣɺҐஔ͕ෆ໌ʢɹɹ= ∞ ʣ ɾҐஔ͕͔Εʢɹɹ = 0 ʣɺӡಈྔ͕ෆ໌ʢɹɹ = ∞ ʣΔpxΔxΔx ΔpxυΠπͷཧֶऀ ϋΠθϯϕϧάʹΑͬͯఏҊ͞Εͨ
ྔࢠίϯϐϡʔλͷجૅʲྔࢠͷੑ࣭ʳ• ೋॏੑ − ೋॏεϦοτͷ࣮ݧ• ཻࢠͳΒεϦοτΛ௨ͬͯ ਅ͙ͬεΫϦʔϯʹͿ͔ͭΔ ͞Βཻࢠ̍ͭ̍ͭͷ͕Δ• ͳΒεϦοτΛ௨ΔͷͰ ׯবࣶ͕εΫϦʔϯʹͰ͖Δ
ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ• ిࢠΛೋॏεϦοτʹ௨͢ͱʁཻࢠͷੑ࣭͕εΫϦʔϯʹΈΒΕΔʢཻࢠͷʣͷੑ࣭͕εΫϦʔϯʹΈΒΕΔʢׯবࣶʣͲͪΒͩͱࢥ͍·͔͢ʁ
ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ• ిࢠΛೋॏεϦοτʹ௨͢ͱʁ• ʮཻࢠͷੑ࣭ʹݟΒΕΔిࢠͷিಥͷʯ ʮͷੑ࣭ʹݟΒΕΔׯবࣶʯͷ྆ํ͕εΫϦʔϯʹΈΒΕΔͭ·Γʮిࢠʯ ˠʮʯͷΑ͏ʹׯব͠߹͍ ʮཻࢠʯͷΑ͏ʹεΫϦʔϯʹিಥͨ͠ͱ͍͏͜ͱʹͳΔʮʯͱʮཻࢠʯͷೋॏੑ
ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ• ిࢠΛೋॏεϦοτͷલͰ؍ଌͨ͠Βʁʮʯͷੑ࣭͚ͩΛ͍࣋ͬͯΔͷ͔ʮཻࢠʯͷੑ࣭͚ͩΛ͍࣋ͬͯΔͷ͔ిࢠׂ͕͞Ε͍ͯΔͷͰͳ͍͔
ྔࢠίϯϐϡʔλͷجૅʲೋॏੑʳ• ిࢠΛೋॏεϦοτͷલͰ؍ଌͨ͠Βʁ• ʮ؍ଌʯͱ͍͏ߦҝΛߦ͏ͱʮཻࢠʯͷੑ࣭͔͠ΈΒΕͳ͔ͬͨ ˠʮ؍ଌʯΛߦ͏ͱঢ়ଶ่͕Εͯ͠·͏ޙड़ɿʮྔࢠॏͶ߹Θͤͷঢ়ଶʯʹؔ࿈͍ͯ͠Δ
ྔࢠίϯϐϡʔλͷجૅʲྔࢠঢ়ଶʹ͍ͭͯʳ• ྔࢠॏͶ߹Θͤঢ়ଶ• ྔࢠׯবޮՌ• ྔࢠͭΕঢ়ଶʢΤϯλϯάϧϝϯτʣ
ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ• ྔࢠॏͶ߹Θͤঢ়ଶ ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ
ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳ|α|2 %|β|2 %|0⟩|1⟩α|0⟩ + β|1⟩• ྔࢠॏͶ߹Θͤঢ়ଶ ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ
ྔࢠίϯϐϡʔλͷجૅʲྔࢠॏͶ߹Θͤঢ়ଶʳϧϏϯͷᆵ• ྔࢠॏͶ߹Θͤঢ়ଶ ࢄతͳঢ়ଶ͕ࠞ͟Γ߹ͬͨঢ়ଶɻ ؍ଌʹΑͬͯͲͪΒ͔ͷঢ়ଶʹऩॖ͠ɺॏͶ߹Θͤঢ়ଶ่͕ΕΔ
ྔࢠίϯϐϡʔλͷجૅʲྔࢠׯবޮՌʳ• ྔࢠׯবޮՌ ৭ʑͳঢ়ଶ͕ڧΊ߹ͬͨΓऑΊ߹ͬͨΓ͢Δ͜ͱ ͷׯবͱࣅͨΑ͏ͳݱ
ྔࢠίϯϐϡʔλͷجૅʲྔࢠͭΕঢ়ଶʳ• ྔࢠͭΕঢ়ଶʢΤϯλϯάϧϝϯτʣ ৭ʑͳঢ়ଶؒͰ૬ޓ͕ؔ͋ΓͰ͖ͳ͍ ˠ ʰγϡϨσΟϯΨʔͷೣͷύϥυοΫεʱ͕༗໊|0⟩ |1⟩ෳ߹ܥͷঢ়ଶΛςϯιϧੵΛ ༻͍ͯද͢͜ͱ͕Ͱ͖ͳ͍࣌ ྔࢠͭΕঢ়ଶͱ͍͏ |0⟩ ⊗ |1⟩ ≠ |0⟩ + |1⟩
ྔࢠίϯϐϡʔλͷجૅʲྔࢠͭΕঢ়ଶʳ• ྔࢠͭΕঢ়ଶʢΤϯλϯάϧϝϯτʣ ৭ʑͳঢ়ଶؒͰ૬ޓ͕ؔ͋ΓͰ͖ͳ͍ ˠ ʰγϡϨσΟϯΨʔͷೣͷύϥυοΫεʱ͕༗໊|0⟩ |1⟩|0⟩ ⊗ |1⟩ ≠ |0⟩ + |1⟩ͲΏ͜ͱʁෳ߹ܥͷঢ়ଶΛςϯιϧੵΛ ༻͍ͯද͢͜ͱ͕Ͱ͖ͳ͍࣌ ྔࢠͭΕঢ়ଶͱ͍͏
ྔࢠίϯϐϡʔλͷجૅʲྔࢠͭΕঢ়ଶʳ• Τϯλϯάϧϝϯτঢ়ଶͱඇΤϯλϯάϧϝϯτঢ়ଶ࣍ͷ̎ͭͷঢ়ଶA, Bʢ ɹɹɹɹ ɹʣΛߟ͑Δ|ψA⟩, |ψB⟩|ψA⟩ =12(|00⟩ + |01⟩) |ψB⟩ =12(|00⟩ + |11⟩)
ྔࢠίϯϐϡʔλͷجૅʲྔࢠͭΕঢ়ଶʳ→ ঢ়ଶAςϯιϧੵͰද͢͜ͱ͕Ͱ͖Δ͕ঢ়ଶBͰ͖ͳ͍• Τϯλϯάϧϝϯτঢ়ଶͱඇΤϯλϯάϧϝϯτঢ়ଶ࣍ͷ̎ͭͷঢ়ଶA, Bʢ ɹɹɹɹ ɹʣΛߟ͑Δ|ψA⟩, |ψB⟩|ψA⟩ =12(|00⟩ + |01⟩) |ψB⟩ =12(|00⟩ + |11⟩)
ྔࢠίϯϐϡʔλͷجૅʲྔࢠͭΕঢ়ଶʳ|ψA⟩ = |0⟩ ⊗ 12(|0⟩ + |1⟩)• Τϯλϯάϧϝϯτঢ়ଶͱඇΤϯλϯάϧϝϯτঢ়ଶ→ ঢ়ଶAςϯιϧੵͰද͢͜ͱ͕Ͱ͖Δ͕ঢ়ଶBͰ͖ͳ͍࣍ͷ̎ͭͷঢ়ଶA, Bʢ ɹɹɹɹ ɹʣΛߟ͑Δ|ψA⟩, |ψB⟩|ψA⟩ =12(|00⟩ + |01⟩) |ψB⟩ =12(|00⟩ + |11⟩)
ྔࢠίϯϐϡʔλͷجૅʲྔࢠͭΕঢ়ଶʳঢ়ଶAɿඇΤϯλϯάϧϝϯτঢ়ଶʢͭΕঢ়ଶʹͳ͍ʣ ঢ়ଶBɿɹΤϯλϯάϧϝϯτঢ়ଶʢͭΕঢ়ଶʹ͋Δʣ• Τϯλϯάϧϝϯτঢ়ଶͱඇΤϯλϯάϧϝϯτঢ়ଶ|ψA⟩ = |0⟩ ⊗ 12(|0⟩ + |1⟩)→ ঢ়ଶAςϯιϧੵͰද͢͜ͱ͕Ͱ͖Δ͕ঢ়ଶBͰ͖ͳ͍࣍ͷ̎ͭͷঢ়ଶA, Bʢ ɹɹɹɹ ɹʣΛߟ͑Δ|ψA⟩, |ψB⟩|ψA⟩ =12(|00⟩ + |01⟩) |ψB⟩ =12(|00⟩ + |11⟩)
͔͜͜Βຊ൪ౖ౭ͷֶ & ྔࢠྗֶ
࣌ؒͷ߹্༷ʑͳ݅ͷղઆলུ͠·͢ ✨ ϒϩάͰߦؒຒΊ͠·͢ ✨
1. ܥ• Ҽղ• ࢄର• ϕϧํఔࣜ• Ψε• ߹ಉθʔλؔ͜ΕΓ·͢ྔࢠίϯϐϡʔλͰߴܭࢉͰ͖Δ͜ͱ2. زԿܥ• ݁ͼෆมྔ• Persistent Homology3. ઢܗܥ• ߦྻͷྦྷ• ߦྻͷ֊ͳͲ
• ֬తΞϧΰϦζϜʢෳճ࣮ߦ͢Εɺߴ֬ͰͨΔʣ1. NΛҼղ͢Δͱ͖ɺࣗવ p < N ΛϥϯμϜʹܾΊΔ2. gcd(p, N)Λܭࢉ͢Δ → ݁Ռ͕ > 1 ͳΒɺඇࣗ໌ͳNͷҼ3. ɹɹɹɹɹɹɹɹɹͷपظTΛݟ͚ͭΔʢྔࢠΞϧΰϦζϜʣ1. T͕حͳΒɺ1.͔ΒΓ͢2. ɹɹɹɹɹɹɹɹɹ ͳΒɺ1.͔ΒΓ͢4. ɹɹɹɹɹɹɹ͕ඇࣗ໌ͳNͷҼ →ShorͷΞϧΰϦζϜʹΑΔҼղfN(x) = px mod NpT2 + 1 ≡ 0 mod Ngcd(pT2± 1,N)pT ≡ 1 mod N ⇔ (pT2 + 1)(pT2 − 1) ≡ 0 mod N≠ 0 ≠ 0p, Nͷ࠷େެҐൃݟΞϧΰϦζϜɾҐਪఆ ɾҐൃݟ ɾؔपظൃݟ ݁Ռ͕ͳΒ ʰޓ͍ʹૉʱͰ͋Δ
• N = 57ͱ͢Δɻp = 5ͩͬͨͱ͖1. ɹɹɹɹɹɹɹͱͳΔΑ͏ͳɹɹɹɹΛબͿ ɹɹɹɹɹɹɹɹ ΑΓ2. ྔࢠϏοτΛ࡞͢Δɿ 3. ྔࢠϑʔϦΤมΛ࡞༻ͤ͞ΔɿShorͷΞϧΰϦζϜʹΑΔҼղN2 ≤ q < 2N2 q = 2k572 ≤ q < 2 ⋅ 572 q = 212 = 40961qq−1∑x=0|x⟩ ⊗ |f(x)⟩(1q )2 q−1∑y=0q−1∑x=0ei 2πxyq|y⟩ ⊗ |f(x)⟩4. ɹͷӈϏοτΛଌఆ͠ɺ ࠨϏοτΛଌఆ͢Δͱ࣍ͷ֬Ͱ yΛಘΒΕΔ⊗1q⌊qr⌋⌊ qr⌋−1∑x=0,f(x)=zei 2πrxyq2f57(x) = 5x mod 57
2. ྔࢠϏοτͷ࡞ShorͷΞϧΰϦζϜʹΑΔҼղ1qq−1∑x=0|x⟩ ⊗ |f(x)⟩ =1q(|0⟩ ⊗ |50⟩ + |1⟩ ⊗ |51⟩ + |2⟩ ⊗ |52⟩ + . . .+|18⟩ ⊗ |50⟩ + |19⟩ ⊗ |51⟩ + |20⟩ ⊗ |52⟩ + . . .+|4086⟩ ⊗ |50⟩ + |4087⟩ ⊗ |51⟩ + |4088⟩ ⊗ |52⟩+ . . . + |4095⟩ ⊗ |59⟩)
ShorͷΞϧΰϦζϜʹΑΔҼղ1qq−1∑x=0|x⟩ ⊗ |f(x)⟩ =1q(|0⟩ ⊗ |50⟩ + |1⟩ ⊗ |51⟩ + |2⟩ ⊗ |52⟩ + . . .+|18⟩ ⊗ |50⟩ + |19⟩ ⊗ |51⟩ + |20⟩ ⊗ |52⟩ + . . .+|4086⟩ ⊗ |50⟩ + |4087⟩ ⊗ |51⟩ + |4088⟩ ⊗ |52⟩+ . . . + |4095⟩ ⊗ |59⟩)2. ྔࢠϏοτͷ࡞
2. ྔࢠϏοτͷ࡞ShorͷΞϧΰϦζϜʹΑΔҼղ1qq−1∑x=0|x⟩ ⊗ |f(x)⟩ =1q(|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…
ShorͷΞϧΰϦζϜʹΑΔҼղ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩+(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩+ . . .4. पظTͷಋग़ͱҼղ
Λࣔ͢ɻ͜ͷͱ͖ࠨϏοτͷ0Λআ͘࠷খۮ͕ʰҐʱʹͳΔɻ Αͬͯ T=18 ͱͳΔShorͷΞϧΰϦζϜʹΑΔҼղ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩+(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩+ . . .|50⟩ = f57(x) = 5x mod 57 = 14. पظTͷಋग़ͱҼղ
ShorͷΞϧΰϦζϜʹΑΔҼղ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩+(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩+ . . .|50⟩ = f57(x) = 5x mod 57 = 1gcd(pT2± 1,N) ͕ඇࣗ໌ͳҼͳͷͰ…Λࣔ͢ɻ͜ͷͱ͖ࠨϏοτͷ0Λআ͘࠷খۮ͕ʰҐʱʹͳΔɻ Αͬͯ T=18 ͱͳΔ4. पظTͷಋग़ͱҼղ
4. पظTͷಋग़ͱҼղShorͷΞϧΰϦζϜʹΑΔҼղ((|0⟩ + |18⟩ + . . . + |4086⟩)|50⟩+(|1⟩ + |19⟩ + |37⟩ + . . . + |4087⟩)|51⟩+ . . .|50⟩ = f57(x) = 5x mod 57 = 1gcd(5182 + 1,57) = gcd(1953126,57) = 3gcd(5182 − 1,57) = gcd(1953124,57) = 19 ҼղͰ͖ͨʂgcd(pT2± 1,N) ͕ඇࣗ໌ͳҼͳͷͰ…Λࣔ͢ɻ͜ͷͱ͖ࠨϏοτͷ0Λআ͘࠷খۮ͕ʰҐʱʹͳΔɻ Αͬͯ T=18 ͱͳΔ
·ͱΊ• ྔࢠίϯϐϡʔλ͢ΕͱΜͰͳ͍Խ͚ ˠ ݱࡏͷPCੑೳΛӽ͑ΔͨΊʹ·ͩ·͕͔͔ͩ࣌ؒΔ• ܭࢉ݁Ռ͋͘·Ͱʰ֬ʱͰ͋Γɺ෮ܭࢉΛߦ͏ඞཁ͋Γ ˠ ܭࢉʹલճͷ֬Λ༻͍ͯܭࢉ͞ΕΔͨΊ• Q#͍ͬͯ͏ྔࢠϓϩάϥϛϯάݴޠ͋ΔͷͰݕࡧͯ͠ΈͯͶʂ ˠ ϓϩάϥϜͰ͜ͷΞϧΰϦζϜ͕؆୯ʹ͔͚Δʂ
͝੩ௌ ͋Γ͕ͱ͏͍͟͝·ͨ͠ εϥΠυɾղઆ ϒϩάʹUp͠·͢ͷͰੋඇΈ͍ͯͩ͘͞ ✨
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