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20180723 量子コンピュータの量子化学への応用; Bravyi-Kitaev基底の実装
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量子コンピュータで量子化学の問題を解く; Bravyi-Kitaev基底で回路数を削減できる
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20180723 量子コンピュータの量子化学への応用; Bravyi-Kitaev基底の実装
1.
Bravyi-Kitaev変換と量子化学へ の応用 (review) 2018/7/23 中田真秀 “The Bravyi-Kitaev transforma>on for quantum computa>on of electronic structure” Jacob T. Seeley, Mar>n J. Richard, Peter J. Love [arXiv:1208.5986v1]のご紹介
2.
量子コンピュータと量子化学 への応用の問題点 • ファインマンの提唱した量子コンピュータは量子系の シミュレーションを高速にする。 • 例えば量子コンピュータを使うと量子化学計算におけ るfullCIが超高速になる(場合が多い)。 –
O(n!) から O(n4)程度(qubitの数) – 演算回数も多分polynomialでスケールするだろう • 量子コンピュータでの回路設計では、いわゆるAND/ OR/NOTに相当するものはユニタリ回路である。 • 問題点量子化学計算を行うとこのユニタリ回路がたく さん必要となる。 – 回路は少なければ少ないほど計算が速くなり、誤差も少 なくなる
3.
Qubitに対するrequirement The Physical Implementa1on of Quantum Computa1on David P. DiVincenzo arXiv:quant-ph/0002077v3 1. A scalable physical system with well characterized qubits 2. The ability to ini>alize the state of the qubits to a simple fiducial state, such as |000…> 3. Long relevant decoherence >mes, much longer than the gate opera>on >me 4. A “universal” set of quantum gate. 5. A qubit-specific measurement capability これは多体電子系ではspin系など、量子の統計性が入ってこない。「1qubit目はずっと1qubit目」と固定 されている… というふうにある量子状態たちがqubitであるにはいくつかの要請がある(ようです)
4.
多体波動関数とqubitの違い • 多体の波動関数は粒子の入れ替えについて反対称 or 対称 • Qubitはbit入れ替えについて対称でも反対称でもない。 –
場所が特定できる=n bit目が明確にわかる そこで湧き上がる疑問 粒子の統計性は実装できるか? – 少し考えたらわかるが、できる。 • 古典コンピュータでもboson/fermionは扱える。 • 古典コンピュータでも量子コンピュータでもTuring machineとしては同じ問題が解 けるのみである。 – 演算子を作用させるときに粒子数を数えさせて、explictに計算。 – NaïveにはO(n)の余計な計算が必要 (nは粒子数) • Jordan-Wigner変換 (x1, x2, · · · , xi, · · · , xj, · · · , xN ) = (x1, x2, · · · , xj, · · · , xi, · · · , xN )<latexit sha1_base64="9ZMBeMY9LibbHwHz9Zts3zDo8EY=">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</latexit><latexit 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5.
Bravyi-Kitaev変換 • 粒子の統計性による符号をO(log n)程度で計算する (nは粒子数)[arXiv:quant-ph/0003137] – O(n) -> O(log n)とものすごい高速化 – 回路の数を減らせる • 量子化学への応用[arXiv:1208.5986v1] 今回はここらへんの紹介
6.
Fermionの第二量子化について(Jordan-Wigner) • Occupa>on number vectorを考える。これは線形空間をなす (ni=0 or 1) • このベクトルに作用する反交換関係を満たす生成・消滅演算子を定義す る これらから、 1.
軌道に2つ同じ粒子は入らない 2. 粒子数演算子の固有値は0 or 1 (辻褄が合っている) 3. が出てくる a† |0i = |1i<latexit sha1_base64="kfXXhSQFCcGVr/tTQIdmvHedzgs=">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</latexit><latexit 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sha1_base64="kfXXhSQFCcGVr/tTQIdmvHedzgs=">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</latexit> a† |1i = 0<latexit 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sha1_base64="OwIeVHohm1gubn5Pe/gfhXNzdRI=">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</latexit><latexit sha1_base64="OwIeVHohm1gubn5Pe/gfhXNzdRI=">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</latexit><latexit sha1_base64="OwIeVHohm1gubn5Pe/gfhXNzdRI=">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</latexit> a|1i = |0i<latexit sha1_base64="BK5ZASRI4UcCNNqnvWNplyfxXT0=">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</latexit><latexit sha1_base64="BK5ZASRI4UcCNNqnvWNplyfxXT0=">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</latexit><latexit sha1_base64="BK5ZASRI4UcCNNqnvWNplyfxXT0=">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</latexit><latexit sha1_base64="BK5ZASRI4UcCNNqnvWNplyfxXT0=">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</latexit> a|0i = 0<latexit 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sha1_base64="BA2RnaMnSUJ2V2Yyb5TUvesoVbs=">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</latexit><latexit sha1_base64="BA2RnaMnSUJ2V2Yyb5TUvesoVbs=">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</latexit> {ar, a† s} = rs<latexit 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7.
Fermionの第二量子化について 電子の詰め方を以下のようにすると 最終的に必要な表式は Hamiltonian Jordan-Wignerの方法 S = n1
+ n2 + · · · + ns 1<latexit sha1_base64="EyhjEQU/6UtrXVC+pPAbbTJP2rM=">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</latexit><latexit 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8.
量子コンピュータで量子化学計算:これまで • Qubitを波動関数とみなし、初期波動関数と厳密解が近いと いう仮定のもと波動関数にe-iHtを掛け位相推定を行う ß ここ は変えない •
粒子の統計性は演算子に直接入れる(Jordan-Wigner) – 波動関数は粒子の入れ替えに対して反対称 – Qubitはビットの入れ替えに対しては対称でも反対称でもない • Jordan-Wignerの方法では Qubitに統計性を入れ込むのにかかるコストがO(n)
9.
Jordan-Wignerの実装 occpa>on number vectorをqubitと同一視する 1 qubitに作用する演算子 は、量子回路 として実装できる。 フェルミ粒子の生成・消滅演算子はこれらから、量子回路として実装できる 基本的なところはこれでok |n1n2 · ·
· ns · · · i = |q0i ⌦ |q1i ⌦ · · · ⌦ |qni<latexit sha1_base64="z7mLuk/dusmH1QC+zp2J0LkShrw=">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</latexit><latexit 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sha1_base64="z7mLuk/dusmH1QC+zp2J0LkShrw=">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</latexit> ˆQ+ |0i = |1i, ˆQ+ |1i = 0, ˆQ |1i = |0i, ˆQ |0i = 0,<latexit sha1_base64="wkQFIBAuo0NINzU6hKe1eB8zCKc=">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</latexit><latexit sha1_base64="wkQFIBAuo0NINzU6hKe1eB8zCKc=">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</latexit><latexit sha1_base64="wkQFIBAuo0NINzU6hKe1eB8zCKc=">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</latexit><latexit sha1_base64="wkQFIBAuo0NINzU6hKe1eB8zCKc=">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</latexit> ˆQ+ = |1ih0| = 1 2 ( x i y ) ˆQ = |0ih1| = 1 2 ( x + i y ) <latexit sha1_base64="9QwyjH3kbhDPdYtrPqXClvgHULk=">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</latexit><latexit sha1_base64="Bn27mZ/nVar4v3ax+M9YBwDpQfA=">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</latexit><latexit sha1_base64="Bn27mZ/nVar4v3ax+M9YBwDpQfA=">AAADBXichVG/axRBFH67aoznj1ySQsVmMEQiMcfbQDQIgRAby1ziJYFscsyOc3dDZn+wO3d4Wa8W/AcsbIwgImInthY2ljYWsbFVLCPYWPhud8EfR+Jbdt9737zv229mvEirxCDuW/ax4yeGTg6fKp0+c/bcSHl0bC0J27GQNRHqMN7weCK1CmTNKKPlRhRL7ntarns7t/rr6x0ZJyoM7phuJLd83gxUQwluCKqXn7gtblh1e5otsPvMYW7Mg6aWzNV5RkIXmNuIuUidXjrbY1Nuopo+377HZphiRdO9ytxWEnEhU6zMCb/HCt2ZTBcHdJ0jdKf/1K2XJ7CCWbDBwimKicX5zx/Of/pycTksPwcX7kIIAtrgg4QADNUaOCT0bIIDCBFhW5ASFlOlsnUJPSgRt01TkiY4oTv0bVK3WaAB9X3NJGML+oumNyYmg0n8iC/wAN/jS/yKPw/VSjONvpcuZS/nyqg+8vDC6o//snzKBlq/WUd6NtCA+cyrIu9RhvR3IXJ+Z/fRwerNlcn0Cj7Fb+R/D/fxHe0g6HwXz6py5TGU6AKcf497sFibrThYcap0E0uQxzBcgsswRed9AxbhNixDDYQ1ZF2z5qzr9gP7lf3afpOP2lbBGYe/wn77C1RGwzI=</latexit><latexit sha1_base64="omamPnx8qLL6/NzAMKLD+jenABo=">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</latexit> a† j = 1⌦n j 1 ⌦ ˆQ+ ⌦ [ z⌦j ]<latexit sha1_base64="CvXxxCk5Z9OkAUDEvIio5jGB9q4=">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</latexit><latexit 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