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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
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
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)
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	
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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
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 sha1_base64="kfXXhSQFCcGVr/tTQIdmvHedzgs=">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</latexit><latexit sha1_base64="kfXXhSQFCcGVr/tTQIdmvHedzgs=">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</latexit><latexit sha1_base64="kfXXhSQFCcGVr/tTQIdmvHedzgs=">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</latexit>
a†
|1i = 0<latexit 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><latexit sha1_base64="OwIeVHohm1gubn5Pe/gfhXNzdRI=">AAACfXichVHLSsNAFD2N7/qquhHcBEvFhZQbERRBEN249FUVrJZJHGMwTUKSFrTqB/gDLlwpiojoT7jxB1z4CeJSwY2gt2lAVNQ7zMyZM/fcOTOje7YVhEQPCaWuvqGxqbkl2drW3tGZ6upeCtySb8ic4dquv6KLQNqWI3OhFdpyxfOlKOq2XNa3p6v7y2XpB5brLIY7nlwrCtOxNi1DhEwVUr1iPb8hTFP66p6m5n3hmLZUJ1QqpNKUpSjUn0CLQRpxzLqpC+SxARcGSihCwkHI2IZAwG0VGggec2uoMOczsqJ9iX0kWVviLMkZgtltHk1ercasw+tqzSBSG3yKzd1npYoM3dMlPdMdXdEjvf1aqxLVqHrZ4VmvaaVX6DzsXXj9V1XkOcTWp+pPzyE2MRZ5tdi7FzHVWxg1fXn36HlhfD5TGaBTemL/J/RAt3wDp/xinM/J+WMk+QO078/9EywNZzXKanMj6cmp+Cua0Yd+DPJ7j2ISM5hFjs89wBmucZN4VzLKkJKtpSqJWNODL6GMfgA3v5KL</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 sha1_base64="BA2RnaMnSUJ2V2Yyb5TUvesoVbs=">AAACdXichVHLSsNAFD2N7/qKuhFECFbFVb0RQREE0Y1LX7WFVkoSxxrMiyQtaPUH/AEXulFQET/DjT/gop8gLiu4ceFtGhAV9Q4zc+bMPXfOzOieZQYhUS0htbS2tXd0diW7e3r7+uWBwe3ALfuGyBiu5fo5XQuEZToiE5qhJXKeLzRbt0RWP1hp7Gcrwg9M19kKDz2xY2slx9wzDS1kqijLmnJMSsHXnJIllEWFinKK0hSF8hOoMUghjjVXvkUBu3BhoAwbAg5CxhY0BNzyUEHwmNtBlTmfkRntC5wgydoyZwnO0Jg94LHEq3zMOrxu1AwitcGnWNx9ViqYoCe6ozo90j090/uvtapRjYaXQ571plZ4xf7T4c23f1U2zyH2P1V/eg6xh/nIq8nevYhp3MJo6itHZ/XNhY2J6iRd0Qv7v6QaPfANnMqrcb0uNs6R5A9Qvz/3T7A9k1Ypra7PppaW46/oxAjGMMXvPYclrGINGT63ggtc4ybxJo1K49JkM1VKxJohfAlp+gODXY8W</latexit><latexit sha1_base64="BA2RnaMnSUJ2V2Yyb5TUvesoVbs=">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</latexit><latexit sha1_base64="BA2RnaMnSUJ2V2Yyb5TUvesoVbs=">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</latexit><latexit sha1_base64="BA2RnaMnSUJ2V2Yyb5TUvesoVbs=">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</latexit>
{ar, a†
s} = rs<latexit sha1_base64="dHL6CKZzDRO7tToQXkMr/mcQEig=">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</latexit><latexit sha1_base64="dHL6CKZzDRO7tToQXkMr/mcQEig=">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</latexit><latexit sha1_base64="dHL6CKZzDRO7tToQXkMr/mcQEig=">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</latexit><latexit sha1_base64="dHL6CKZzDRO7tToQXkMr/mcQEig=">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</latexit>
{ar, as} = {a†
r, a†
s} = 0<latexit sha1_base64="Pgm6ayivz2joYKt1paMV4KnoSJs=">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</latexit><latexit sha1_base64="Pgm6ayivz2joYKt1paMV4KnoSJs=">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</latexit><latexit sha1_base64="Pgm6ayivz2joYKt1paMV4KnoSJs=">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</latexit><latexit sha1_base64="Pgm6ayivz2joYKt1paMV4KnoSJs=">AAACmHichVHNLgRBEP6Mv7X+FhfhMrEhDrKpQUIkkg0H3FgWiWUzM9qamJ2ZzMxusjZewAuIOJGIiIdwcOEBHDyCOJK4OKidGREE1enuqq/qq/66W3NMw/OJHuqk+obGpuZYS7y1rb2jM9HVverZJVcXWd02bXddUz1hGpbI+oZvinXHFWpRM8Watjdby6+VhesZtrXiVxyxWVQLlrFj6KrPUD4xlqvKat4d4cWTcwfytBwCW7lttVAQYeIjCAson0hSigKTfzpK5CQR2aKduEAO27Cho4QiBCz47JtQ4fHYgAKCw9gmqoy57BlBXuAAceaWuEpwhcroHq8FjjYi1OK41tML2DqfYvJ0mSljkO7pkp7plq7okd5+7VUNetS0VHjXQq5w8p2Hvcuv/7KKvPvY/WT9qdnHDiYDrQZrdwKkdgs95Jf3j56XpzKD1SE6oyfWf0oPdMM3sMov+vmSyJwgzh+gfH/un87qaEqhlLI0nkzPRF8RQz8GMMzvPYE05rGILJ97jGvc4k7qk9LSnLQQlkp1EacHX0zKvAPPSZta</latexit>
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a2
s = a†2
s = 0<latexit sha1_base64="zq3ZI+QCAdMUxh6Y447DXzJrcYc=">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</latexit><latexit sha1_base64="zq3ZI+QCAdMUxh6Y447DXzJrcYc=">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</latexit><latexit sha1_base64="zq3ZI+QCAdMUxh6Y447DXzJrcYc=">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</latexit><latexit sha1_base64="zq3ZI+QCAdMUxh6Y447DXzJrcYc=">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</latexit>
a†
a = 1 aa†
<latexit sha1_base64="vmHpB+eMHVeXnxzBRKAeyHGWRMU=">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</latexit><latexit sha1_base64="vmHpB+eMHVeXnxzBRKAeyHGWRMU=">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</latexit><latexit sha1_base64="vmHpB+eMHVeXnxzBRKAeyHGWRMU=">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</latexit><latexit sha1_base64="vmHpB+eMHVeXnxzBRKAeyHGWRMU=">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</latexit>
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 sha1_base64="EyhjEQU/6UtrXVC+pPAbbTJP2rM=">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</latexit><latexit sha1_base64="EyhjEQU/6UtrXVC+pPAbbTJP2rM=">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</latexit><latexit sha1_base64="EyhjEQU/6UtrXVC+pPAbbTJP2rM=">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</latexit>
as|n1n2 · · · ns · · · i = 0 if ns = 0<latexit sha1_base64="G4QSNsbkJFOfMqYE7W4FSyN6tDg=">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</latexit><latexit sha1_base64="G4QSNsbkJFOfMqYE7W4FSyN6tDg=">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</latexit><latexit sha1_base64="G4QSNsbkJFOfMqYE7W4FSyN6tDg=">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</latexit><latexit sha1_base64="G4QSNsbkJFOfMqYE7W4FSyN6tDg=">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</latexit>
|n1n2 · · · n1i = (a†
1)n1
(a†
2)n2
(a†
3)n3
· · · (a†
1)n1
|0000000 · · · 0i<latexit sha1_base64="1XdddPCev8UOuEYfwEbV8vyj724=">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</latexit><latexit sha1_base64="1XdddPCev8UOuEYfwEbV8vyj724=">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</latexit><latexit sha1_base64="1XdddPCev8UOuEYfwEbV8vyj724=">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</latexit><latexit sha1_base64="1XdddPCev8UOuEYfwEbV8vyj724=">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</latexit>
a†
s|n1n2 · · · ns · · · i = ( 1)S
p
ns + 1|n1n2 · · · (ns + 1) · · · i if ns = 0<latexit sha1_base64="6aIiYUHXHuOsBmDvigSnVNzZm7s=">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</latexit><latexit sha1_base64="6aIiYUHXHuOsBmDvigSnVNzZm7s=">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</latexit><latexit sha1_base64="6aIiYUHXHuOsBmDvigSnVNzZm7s=">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</latexit><latexit sha1_base64="6aIiYUHXHuOsBmDvigSnVNzZm7s=">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</latexit>
a†
s|n1n2 · · · ns · · · i = 0 if ns = 1<latexit sha1_base64="ybTxpjFoMxL5ZnBAbkENFBox5Yk=">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</latexit><latexit sha1_base64="ybTxpjFoMxL5ZnBAbkENFBox5Yk=">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</latexit><latexit sha1_base64="ybTxpjFoMxL5ZnBAbkENFBox5Yk=">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</latexit><latexit sha1_base64="ybTxpjFoMxL5ZnBAbkENFBox5Yk=">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</latexit>
as|n1n2 · · · ns · · · i = ( 1)Sp
ns|n1n2 · · · ns 1 · · · i if ns = 1<latexit sha1_base64="MiLGaH1u14zU0r13RdE/iGbuxw4=">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</latexit><latexit 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sha1_base64="MiLGaH1u14zU0r13RdE/iGbuxw4=">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</latexit><latexit 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sha1_base64="MiLGaH1u14zU0r13RdE/iGbuxw4=">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</latexit><latexit sha1_base64="MiLGaH1u14zU0r13RdE/iGbuxw4=">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</latexit>
ˆH =
X
i,j
vi,ja†
i aj +
1
2
X
ijkl
wijkla†
i a†
jalak
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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)
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 sha1_base64="z7mLuk/dusmH1QC+zp2J0LkShrw=">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</latexit><latexit sha1_base64="z7mLuk/dusmH1QC+zp2J0LkShrw=">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</latexit><latexit 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=">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</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 sha1_base64="BqCHDf9U9S/74NIqPcPDX2lRhFY=">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</latexit><latexit sha1_base64="BqCHDf9U9S/74NIqPcPDX2lRhFY=">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</latexit><latexit sha1_base64="wH02A+j8yYi6xf5yCKKvXAWQ3tY=">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</latexit>
aj = 1⌦n j 1
⌦ ˆQ ⌦ [ z⌦j
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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

  • 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
  • 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 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  • 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=">AAACh3ichVFNL8RQFD3qe3wNNhKbxoRYjVsRRCLxsbH0NUgMk9d6qtFpm7YzCcNe/AELKxIRkdiyt/EHLPwEsSSxsXCnUxEEt+l75553z33nvad7thWERA9VSnVNbV19Q2OiqbmltS3Z3rEUuAXfkBnDtV1/RReBtC1HZkIrtOWK50uR1225rG9Pl9eXi9IPLNdZDHc8uZYXpmNtWoYImcole8R6dkOYpvTVPVKzvnBMW6rj6p6qfWS5ZIrSFIX6E2gxSCGOWTd5jiw24MJAAXlIOAgZ2xAI+FuFBoLH3BpKzPmMrGhdYh8J1ha4SnKFYHabR5Oz1Zh1OC/3DCK1wbvY/PusVNFL93RBz3RHl/RIb7/2KkU9yl52eNYrWunl2g67Fl7/VeV5DrH1qfrTc4hNjEZeLfbuRUz5FEZFX9w9el4Ym+8t9dEpPbH/E3qgWz6BU3wxzubk/DES/ADa9+v+CZYG0xqltbmh1MRU/BQN6EYP+vm+RzCBGcwiw/se4ArXuFEalQFlWBmtlCpVsaYTX0KZfAcK5pWB</latexit><latexit 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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 sha1_base64="BA2RnaMnSUJ2V2Yyb5TUvesoVbs=">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</latexit><latexit sha1_base64="BA2RnaMnSUJ2V2Yyb5TUvesoVbs=">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</latexit><latexit sha1_base64="BA2RnaMnSUJ2V2Yyb5TUvesoVbs=">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</latexit><latexit sha1_base64="BA2RnaMnSUJ2V2Yyb5TUvesoVbs=">AAACdXichVHLSsNAFD2N7/qKuhFECFbFVb0RQREE0Y1LX7WFVkoSxxrMiyQtaPUH/AEXulFQET/DjT/gop8gLiu4ceFtGhAV9Q4zc+bMPXfOzOieZQYhUS0htbS2tXd0diW7e3r7+uWBwe3ALfuGyBiu5fo5XQuEZToiE5qhJXKeLzRbt0RWP1hp7Gcrwg9M19kKDz2xY2slx9wzDS1kqijLmnJMSsHXnJIllEWFinKK0hSF8hOoMUghjjVXvkUBu3BhoAwbAg5CxhY0BNzyUEHwmNtBlTmfkRntC5wgydoyZwnO0Jg94LHEq3zMOrxu1AwitcGnWNx9ViqYoCe6ozo90j090/uvtapRjYaXQ571plZ4xf7T4c23f1U2zyH2P1V/eg6xh/nIq8nevYhp3MJo6itHZ/XNhY2J6iRd0Qv7v6QaPfANnMqrcb0uNs6R5A9Qvz/3T7A9k1Ypra7PppaW46/oxAjGMMXvPYclrGINGT63ggtc4ybxJo1K49JkM1VKxJohfAlp+gODXY8W</latexit> {ar, a† s} = rs<latexit sha1_base64="dHL6CKZzDRO7tToQXkMr/mcQEig=">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</latexit><latexit sha1_base64="dHL6CKZzDRO7tToQXkMr/mcQEig=">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</latexit><latexit sha1_base64="dHL6CKZzDRO7tToQXkMr/mcQEig=">AAACiXichVFNSxtBGH6yrZrGr7S9FHpZmioeJLxbCopQkHrpUZNGBVeX2c0YFze7y+wkkC75A/6BHjwpSCm99NpevfgHeshPKD1a6KWHvtksFBX1HWbmmWfe551nZtw48BNNNCgYDx6OjU8UH5Ump6ZnZsuPn2wmUUd5suFFQaS2XZHIwA9lQ/s6kNuxkqLtBnLLPVwb7m91pUr8KHyve7HcbYtW6O/7ntBMOeWXdmoKRy3ykOzZTdFqSWXaffONaTdloIWTqqTvlCtUpSzMm8DKQQV5rEflT7DRRAQPHbQhEUIzDiCQcNuBBULM3C5S5hQjP9uX6KPE2g5nSc4QzB7y2OLVTs6GvB7WTDK1x6cE3BUrTczRD/pMl3RBX+gn/b21VprVGHrp8eyOtDJ2Zo+e1f/cq2rzrHHwX3WnZ419LGdeffYeZ8zwFt5I3/3w8bK+UptL5+mUfrH/ExrQOd8g7P72zjZk7Rgl/gDr+nPfBJuvqhZVrY3XldW3+VcU8RwvsMDvvYRVvMM6GnzuEb7iG74bk4ZlLBsro1SjkGue4koYa/8AKyqW9A==</latexit><latexit sha1_base64="dHL6CKZzDRO7tToQXkMr/mcQEig=">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</latexit> {ar, as} = {a† r, a† s} = 0<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="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 sha1_base64="BqCHDf9U9S/74NIqPcPDX2lRhFY=">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</latexit><latexit sha1_base64="BqCHDf9U9S/74NIqPcPDX2lRhFY=">AAAC33ichVG/axRBFP6y/ornj1xiI9gMhoigHm/TGAKBoI1lLvGS4O3dMbuZbOayv9idO0iWqwULsVOxUrAQWyGtYOM/YJF/QJAUESLYWPhu76JoML5hZr755n1v5uO5SaAzQ7Q7Yp04eer0mdGzpXPnL1wcK49PLGdxJ/VUzYuDOF11ZaYCHama0SZQq0mqZOgGasXdvNu/X+mqNNNxdN9sJaoRSj/S69qThqlW+YFsOmvS91Xaaos54bhhbvdEM3dio0OViUjcEm2eTB5SzoY0otq88YuoCyfTfiib+fYh1e6JRqs8SRUqQhwF9hBMzldm3+3crD5biMs7cLCGGB46CKEQwTAOIJHxqMMGIWGugZy5lJEu7hV6KLG2w1mKMySzm7z6fKoP2YjP/ZpZofb4lYBnykqBKfpEb+iAPtJb+kI//lkrL2r0/7LFuzvQqqQ19ujy0vf/qkLeDTZ+q45RuJx9vCeDdcwUXjR7Swqm79Ib1O9uPz1Yml2cyq/RK9pjfy9plz6ww6j7zXtdVYsvUOIG2X+34yhYnq7YVLGr3Kk7GMQoruAqrnM/bmMe97CAGr/7Hp+xj6+WtB5aj60ng1RrZKi5hD/Cev4TI7q1ZA==</latexit><latexit sha1_base64="wH02A+j8yYi6xf5yCKKvXAWQ3tY=">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</latexit> aj = 1⌦n j 1 ⌦ ˆQ ⌦ [ z⌦j ]<latexit sha1_base64="LrC+WUjRX4qrdpCjcBO7GzUkE70=">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</latexit><latexit 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