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量子情報勉強会,量子ゲートについて
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Hata Ryosuke
October 07, 2019
Science
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量子情報勉強会,量子ゲートについて
代表的な論理ゲートと量子ゲートをまとめて,
足し算を行ってみました。
1量子ビットに対して量子ゲートを作用させたときの,
ブロッホ球の描写を行いました。
Hata Ryosuke
October 07, 2019
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Transcript
ྔࢠใษڧձ ྔࢠήʔτͷछྨ ാɹྒྷհ
ࢀߟจݙ • ͍ͪΜ͍͞͠ྔࢠίϯϐϡʔλʔͷ ڭຊ ਓؾߨࢣ͕ڭ͑Δੈք͕͢Δ࠷ ৽ςΫϊϩδʔ (ʮ͍ͪΜ͍͞͠ڭ ຊʯγϦʔζ) ື༤Ұ (ஶ)
• Quantum Optics, Miguel Orzag, Chap 21 • Blochٿͷඳࣸ, QuTiP, ίʔυ https://github.com/ryosukehata/qutip/blob/ master/03_QuTiP_Bolch_sphere_gate.ipynb
ྔࢠήʔτͬͯʁ • ݹయίϯϐϡʔλཧήʔτͷྔࢠ൛ • ݹయίϯϐϡʔλͷཧήʔτͱʁʁ → ೖྗʹରԠͨ͠ग़ྗ͕ಘΒΕΔճ࿏ – ྫ.ANDճ࿏ – ྫ.XORճ࿏ ը૾wikipediaΑΓ
https://ja.wikipedia.org/wiki / %E8%AB%96%E7%90%86%E5% ೖྗA ೖྗB ग़ྗ ̌ ̌ ̌ ̌ ̍ ̌ ̍ ̌ ̌ ̍ ̍ ̍ ೖྗA ೖྗB ग़ྗ ̌ ̌ ̌ ̌ ̍ ̍ ̍ ̌ ̍ ̍ ̍ ̌ ೖྗΛసͤ͞Δ NOTήʔτ͋Δ
ཧήʔτʹ͍ͭͯ
ཧήʔτʹΑΔԋࢉ̍ ؆୯ͷͨΊʹ͠ࢉʹ͍ͭͯߟ͑Δ ίϯϐϡʔλ̎ਐͳͷͰɼ ̍ʴ̍ʹ̍̌ ͱͳΔɻ ͜ͷԋࢉΛཧήʔτʹΑͬͯͲͷΑ͏ʹ ߦ͏ͷ͔ߟ͑ͯΈΔ
ཧήʔτʹΑΔԋࢉ̎ XORճ࿏ ANDճ࿏ ೖྗA ೖྗB ग़ྗ ̌ ̌ ̌ ̌
̍ ̌ ̍ ̌ ̌ ̍ ̍ ̍ ೖྗA ೖྗB ग़ྗ ̌ ̌ ̌ ̌ ̍ ̍ ̍ ̌ ̍ ̍ ̍ ̌ ANDճ࿏ ೖग़ྗ XORճ࿏ ೖग़ྗ ্ܻΓ ࠓͷܻ A B
ཧήʔτʹΑΔԋࢉ̏ XORճ࿏ ANDճ࿏ ೖྗA ೖྗB ग़ྗ ̌ ̌ ̌ ̌
̍ ̌ ̍ ̌ ̌ ̍ ̍ ̍ ೖྗA ೖྗB ग़ྗ ̌ ̌ ̌ ̌ ̍ ̍ ̍ ̌ ̍ ̍ ̍ ̌ ANDճ࿏ ೖग़ྗ XORճ࿏ ೖग़ྗ ̌ ̍ ̍ ̌
ཧήʔτʹΑΔԋࢉ̐ XORճ࿏ ANDճ࿏ ೖྗA ೖྗB ग़ྗ ̌ ̌ ̌ ̌
̍ ̌ ̍ ̌ ̌ ̍ ̍ ̍ ೖྗA ೖྗB ग़ྗ ̌ ̌ ̌ ̌ ̍ ̍ ̍ ̌ ̍ ̍ ̍ ̌ ANDճ࿏ ೖग़ྗ XORճ࿏ ೖग़ྗ ̍ ̍ ̍ ̍
ྔࢠήʔτʹ͍ͭͯ
ྔࢠήʔτͷछྨ දతͳྔࢠήʔτ ɾ̍ྔࢠϏοτʹ࡞༻͢Δήʔτ - ΞμϚʔϧήʔτ - Ґ૬ήʔτ - XήʔτɼYήʔτɼZήʔτ ɾෳྔࢠϏοτʹ࡞༻͢Δήʔτ
- CNOTήʔτʢ̎ྔࢠϏοτʣ - τϑΥϦήʔτʢ̏ྔࢠϏο τʣ
ྔࢠήʔτͷઆ໌ͷલʹ දهʹ͍ͭͯઆ໌͢Δɻ ೖྗA ೖྗB ̌ ̌ ̌ ̍ ̍ ̌
̍ ̍ ཧήʔτ ྔࢠήʔτ ݹయίϯϐϡʔλͱྔࢠίϯϐϡʔλͰ ҟͳΔ෦ಛʹͳ͍ɻ Ϗοτͷ̌̍Ͱهࡌ͢Δɻ ※ࠓ͍ͬͯΔ̒̐bitPC Ұͷೖྗ͕̒̐ݸ ೖྗA ೖྗB |̌> |̌> |̌> |̍> |̍> |̌> |̍> |̍> ೖྗ |̌̌> |̌̍> |̍̌> |̍̍> ʹ
̍ྔࢠϏοτʹ࡞༻͢Δήʔτ Blochٿ ̍ྔࢠϏοτͷ࡞༻ ࠨਤͷΑ͏ͳBlochٿ໘্ Ͱߟ͑Δɻ ݹయίϯϐϡʔλࠨਤ Z࣠ํͷ̌,͔̍͠ଘࡏ ͠ͳ͍͕ɼྔࢠίϯϐϡ ʔλͰؒͷঢ়ଶΛར༻ ՄೳͰ͋Δɻ
ྫ͑ɼ+xํͱ-xํͰঢ়ଶ͕ҟͳΔɻ
̍ྔࢠϏοτͷදه ೖྗೋछྨ ೖྗ |̌> |̍> ࣜͰͷهड़ ঢ়ଶΛ࡞Δͱ͖ʹߦྻͰ࡞༻ͤ͞Δɻ ࠷ޙʹଌఆΛߦ͍ɼ͔͔̌̍ΛಘΔɻ
ΞμϚʔϧήʔτ H ঢ়ଶΛࠞͥΔ |̌ʼʹ࡞༻ |̍ʼʹ࡞༻
ΞμϚʔϧήʔτ H ঢ়ଶΛࠞͥΔ |̌ʼʹ࡞༻ |̍ʼʹ࡞༻
ΞμϚʔϧήʔτ ଌఆ H ঢ়ଶΛࠞͥΔ ଌఆʢ̍ʹͳΔظʣͷܭࢉΛߦ͏ ̑̌ˋͷ֬Ͱ̍ʹͳΓɼ ঢ়ଶ͕͍ࠞͬͯ͟ΔͱΘ͔Δɻ
Ґ૬ήʔτ φ ̌ɼ̍ͷ࣠ʢ͉࣠ʣΛج४ʹճసͤ͞Δ |̍>ʹ࡞༻ ΞμϚʔϧήʔτͷޙ ͷঢ়ଶʹ࡞༻(φ=-π/4)
ΞμϚʔϧήʔτͱҐ૬ήʔτΛΈ߹Θ ͤΔ͜ͱͰ̍ྔࢠϏοτʹҙͷঢ়ଶΛ࡞ Ͱ͖Δɻ ҙͷ̍ྔࢠϏοτͷঢ়ଶ̍ H θ H π/2+ φ ճ࿏ͪ͜Β͔Β
ܭࢉͪ͜Β͔Β
ঢ়ଶ|̌ʼʹ࡞༻ͤͯ͞ΈΔ ҙͷ̍ྔࢠϏοτͷঢ়ଶ̎ θ=̌, πͷͱ͖ɼ|̌ʼͱͳΓɼ θ=π/2, 3π/2ͷͱ͖ɼ|̍ʼͱͳΔɻ θ্͕هҎ֎ͷͱ͖ʹɼφΛૢ࡞ͯ͠ҙͷ ঢ়ଶΛ࡞ΕΔɻ
ෳྔࢠϏοτʹ࡞༻͢Δήʔτ ̎ྔࢠϏοτ CNOTήʔτ ೖྗA ೖྗB |̌> |̌> |̌> |̍> |̍>
|̌> |̍> |̍> ೖྗ |̌̌> |̌̍> |̍̌> |̍̍> ʹ ೖྗA͕̍ͷͱ͖͚ͩɼ ೖྗBͷΛస͢Δɻ ॳظঢ়ଶ͕C=̍ɼA=B=D=̌ͷͱ͖ɼCNOT ήʔτΛ࡞༻ͤ͞Δͱɼ|̍̍ʼ͚͕ͩΔ
ෳྔࢠϏοτʹ࡞༻͢Δήʔτ ̏ྔࢠϏοτ τϑΥϦήʔτ CNOTͷ̏ྔࢠϏοτ൛ ̎ͭͷྔࢠϏοτ͕̍ͳΒɼ࠷ޙͷϏοτΛ సͤ͞Δɻ
ྔࢠήʔτΛͬͯ͠ࢉΛߦ͏ ೖྗA ೖྗB CNOT ήʔτ ग़ྗ ࠓͷܻ ग़ྗ ্ܻΓ |̌ʼ
|̌ʼ τϑΥϦή ʔτ ͜ͷߏͰ ཧήʔτΛ ͬͨ͠ࢉ ͱಉ͡ܭࢉ͕ Ͱ͖Δɻ
ྔࢠήʔτΛͬͨԋࢉ·ͱΊ ͨͩͷ͠ࢉΛߦ͏͚ͩͰ͋Εɼ ࠓͷίϯϐϡʔλΛͬͨ΄͏͕ ࠷దԽ͞Ε͍ͯΔͨΊߴ ͨͩɼ్தͰग़͖ͯͨΑ͏ͳ̌ͱ̍ͷؒͷঢ় ଶΛ͏ͷͰ͋ΕߴԽͰ͖Δɻ ྔࢠίϯϐϡʔλ༻ͷΞϧΰϦζϜ͕ඞཁʂ