https:\/\/dojo.qulacs.org\/ja\/latest\/notebooks\/2.2_Hadamard_test.html<\/a>)<\/p>\n\n\n\n\u6700\u521d\u306b\u3001\u88dc\u52a9\u7cfb\u306b\u30a2\u30c0\u30de\u30fc\u30eb\u30b2\u30fc\u30c8\u3092\u4f5c\u7528\u3055\u305b\u305f\u5f8c\u3001\u5236\u5fa1U\u30b2\u30fc\u30c8\u3092\u4f5c\u7528\u3055\u305b\u307e\u3059(U\u306f\u6ce8\u76ee\u7cfb\u306b\u5bfe\u3057\u3066\u671f\u5f85\u5024\u3092\u6c42\u3081\u305f\u3044\u30e6\u30cb\u30bf\u30ea\u6f14\u7b97\u5b50)\u3002\u305d\u306e\u5f8c\u3001\u3055\u3089\u306b\u30a2\u30c0\u30de\u30fc\u30eb\u30b2\u30fc\u30c8\u3092\u4f5c\u7528\u3055\u305b\u308b\u91cf\u5b50\u56de\u8def\u3067\u3059\u3002\u3053\u306e\u6642\u70b9\u3067\u306e(\u88dc\u52a9\u7cfb+\u6ce8\u76ee\u7cfb\u306e)\u72b6\u614b\u3092 $|\\psi_H\\rangle$ \u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u6b21\u306e\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n\n\n\n
$$\\langle \\psi_H | Z_0 | \\psi_H \\rangle = \\text{Re}\\langle \\psi | U | \\psi \\rangle \\tag{8}$$<\/p>\n\n\n\n
\u5de6\u8fba\u306f\u3001\u88dc\u52a9\u7cfb+\u6ce8\u76ee\u7cfb\u306e\u72b6\u614b $|\\psi_H\\rangle$ \u306b\u5bfe\u3059\u308b\u88dc\u52a9\u7cfb\u306e $Z$ \u671f\u5f85\u5024($Z_0$ \u671f\u5f85\u5024)\u3067\u3059\u3002\u4e00\u65b9\u3001\u53f3\u8fba\u306f\u3001\u6ce8\u76ee\u7cfb\u306e\u72b6\u614b $|\\psi\\rangle$ \u306b\u5bfe\u3059\u308b $U$ \u306e\u671f\u5f85\u5024\u306e\u5b9f\u90e8\u3067\u3059\u3002\u5f93\u3063\u3066\u3001\u4e0a\u8a18\u306e\u91cf\u5b50\u56de\u8def\u3092\u901a\u3057\u305f\u5f8c\u306b\u3001\u88dc\u52a9\u7cfb\u306e $Z$ \u6e2c\u5b9a\u3092\u7e70\u308a\u8fd4\u3057\u884c\u3063\u3066\u5f97\u305f\u6e2c\u5b9a\u5024\u304b\u3089\u5e73\u5747\u5024\u3092\u4f5c\u308b\u3068\u3001\u305d\u308c\u306f\u3001\u6ce8\u76ee\u7cfb\u306b\u5bfe\u3059\u308b $U$ \u306e\u671f\u5f85\u5024\u306e\u5b9f\u90e8\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u30a2\u30c0\u30de\u30fc\u30eb\u30c6\u30b9\u30c8\u306e\u91cf\u5b50\u56de\u8def\u3067\u5bc6\u5ea6\u6f14\u7b97\u5b50\u306e\u671f\u5f85\u5024 $\\mathrm{Tr}\\left[ (O \\otimes I) U_{\\mathrm{SWAP}} (\\rho \\otimes \\rho) \\right]$ \u3092\u6c42\u3081\u308b\u969b\u306f\u3001\u6ce8\u76ee\u7cfb\u306e\u521d\u671f\u72b6\u614b\u3092 \\(\\rho\\otimes\\rho\\) \u3068\u3057\u3001\\(U=(O\\otimes I)U_{\\mathrm{SWAP}}\\) \u3068\u3057\u307e\u3059\u3002\u305f\u3060\u3057\u3001\\(O\\) \u306f\u30e6\u30cb\u30bf\u30ea\u6f14\u7b97\u5b50\u3067\u3042\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002 \u5206\u6bcd\u306f \\(O=I\\) \u3068\u3059\u308c\u3070\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u4e00\u65b9\u3001\u5206\u5b50\u306b\u3064\u3044\u3066\u306f\u3001\u4e00\u822c\u306b \\(H_p\\) \u304c\u30e6\u30cb\u30bf\u30ea\u3067\u306f\u306a\u3044\u305f\u3081\u3001\\(H_p\\) \u3092\u30d1\u30a6\u30ea\u6f14\u7b97\u5b50\u306e\u7dda\u5f62\u7d50\u5408\u306b\u5206\u89e3\u3057\u3001\u5404\u9805\u306b\u3064\u3044\u3066\u30a2\u30c0\u30de\u30fc\u30eb\u30c6\u30b9\u30c8\u3092\u884c\u3046\u3053\u3068\u3067\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n(8)\u5f0f\u306e\u8a3c\u660e<\/strong><\/summary>\n\u307e\u305a\u3001\u91cf\u5b50\u56de\u8def\u306e\u5404\u30b9\u30c6\u30c3\u30d7\u3092\u8ffd\u3063\u3066 $|\\psi_H\\rangle$ \u3092\u5177\u4f53\u7684\u306b\u6c42\u3081\u307e\u3059\u3002\u88dc\u52a9\u7cfb\u306e\u521d\u671f\u72b6\u614b\u3092 $|0\\rangle$\u3001\u6ce8\u76ee\u7cfb\u306e\u521d\u671f\u72b6\u614b\u3092 $|\\psi\\rangle$ \u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
\u30b9\u30c6\u30c3\u30d71:\u88dc\u52a9\u7cfb\u306b\u30a2\u30c0\u30de\u30fc\u30eb\u30b2\u30fc\u30c8\u3092\u4f5c\u7528<\/strong><\/p>\n\n\n\n$$|0\\rangle|\\psi\\rangle \\;\\longrightarrow\\; \\frac{1}{\\sqrt{2}}\\left(|0\\rangle+|1\\rangle\\right)|\\psi\\rangle$$<\/p>\n\n\n\n
\u30b9\u30c6\u30c3\u30d72:\u88dc\u52a9\u7cfb\u3092\u5236\u5fa1\u30d3\u30c3\u30c8\u3068\u3057\u305f\u5236\u5fa1$U$\u30b2\u30fc\u30c8\u3092\u4f5c\u7528<\/strong><\/p>\n\n\n\n\u88dc\u52a9\u7cfb\u304c $|1\\rangle$ \u306e\u3068\u304d\u3060\u3051\u6ce8\u76ee\u7cfb\u306b $U$ \u304c\u4f5c\u7528\u3059\u308b\u306e\u3067\u3001<\/p>\n\n\n\n
$$\\frac{1}{\\sqrt{2}}\\left(|0\\rangle+|1\\rangle\\right)|\\psi\\rangle \\;\\longrightarrow\\; \\frac{1}{\\sqrt{2}}\\left(|0\\rangle|\\psi\\rangle+|1\\rangle U|\\psi\\rangle\\right)$$<\/p>\n\n\n\n
\u30b9\u30c6\u30c3\u30d73:\u88dc\u52a9\u7cfb\u306b\u518d\u3073\u30a2\u30c0\u30de\u30fc\u30eb\u30b2\u30fc\u30c8\u3092\u4f5c\u7528<\/strong><\/p>\n\n\n\n$H|0\\rangle=\\frac{1}{\\sqrt{2}}(|0\\rangle+|1\\rangle)$\u3001$H|1\\rangle=\\frac{1}{\\sqrt{2}}(|0\\rangle-|1\\rangle)$ \u3092\u4f7f\u3046\u3068\u3001<\/p>\n\n\n\n
$$\\frac{1}{\\sqrt{2}}\\left(|0\\rangle|\\psi\\rangle+|1\\rangle U|\\psi\\rangle\\right) \\;\\longrightarrow\\; \\frac{1}{2}|0\\rangle\\left(|\\psi\\rangle+U|\\psi\\rangle\\right)+\\frac{1}{2}|1\\rangle\\left(|\\psi\\rangle-U|\\psi\\rangle\\right)$$<\/p>\n\n\n\n
\u3053\u308c\u304c $|\\psi_H\\rangle$ \u3067\u3059\u3002<\/p>\n\n\n\n
$$|\\psi_H\\rangle = \\frac{1}{2}|0\\rangle\\left(|\\psi\\rangle+U|\\psi\\rangle\\right)+\\frac{1}{2}|1\\rangle\\left(|\\psi\\rangle-U|\\psi\\rangle\\right)$$<\/p>\n\n\n\n
$Z_0$\u671f\u5f85\u5024\u306e\u8a08\u7b97<\/strong><\/p>\n\n\n\n$Z_0$ \u306f\u88dc\u52a9\u7cfb\u306b\u306e\u307f\u4f5c\u7528\u3059\u308b\u30d1\u30a6\u30ea$Z$\u6f14\u7b97\u5b50\u3067\u3001$Z|0\\rangle=|0\\rangle$\u3001$Z|1\\rangle=-|1\\rangle$ \u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n
$$Z_0|\\psi_H\\rangle = \\frac{1}{2}|0\\rangle\\left(|\\psi\\rangle+U|\\psi\\rangle\\right)-\\frac{1}{2}|1\\rangle\\left(|\\psi\\rangle-U|\\psi\\rangle\\right)$$<\/p>\n\n\n\n
\u3053\u308c\u3068 $\\langle \\psi_H|$ \u306e\u5185\u7a4d\u3092\u53d6\u308a\u307e\u3059\u3002\u88dc\u52a9\u7cfb\u306e $|0\\rangle,|1\\rangle$ \u306f\u76f4\u4ea4\u3059\u308b\u306e\u3067\u3001<\/p>\n\n\n\n
$$\\langle \\psi_H | Z_0 | \\psi_H \\rangle = \\frac{1}{4}\\Big(\\langle\\psi|+\\langle\\psi|U^\\dagger\\Big)\\Big(|\\psi\\rangle+U|\\psi\\rangle\\Big) – \\frac{1}{4}\\Big(\\langle\\psi|-\\langle\\psi|U^\\dagger\\Big)\\Big(|\\psi\\rangle-U|\\psi\\rangle\\Big)$$<\/p>\n\n\n\n
\u53f3\u8fba\u306e\u5404\u9805\u3092\u5c55\u958b\u3059\u308b\u3068\u3001<\/p>\n\n\n\n
$$\\frac{1}{4}\\Big(\\langle\\psi|\\psi\\rangle+\\langle\\psi|U|\\psi\\rangle+\\langle\\psi|U^\\dagger|\\psi\\rangle+\\langle\\psi|U^\\dagger U|\\psi\\rangle\\Big) – \\frac{1}{4}\\Big(\\langle\\psi|\\psi\\rangle-\\langle\\psi|U|\\psi\\rangle-\\langle\\psi|U^\\dagger|\\psi\\rangle+\\langle\\psi|U^\\dagger U|\\psi\\rangle\\Big)$$<\/p>\n\n\n\n
$\\langle\\psi|\\psi\\rangle$ \u306e\u9805\u3068 $\\langle\\psi|U^\\dagger U|\\psi\\rangle$ \u306e\u9805\u306f\u6253\u3061\u6d88\u3057\u5408\u3044\u3001\u6b8b\u308b\u306e\u306f<\/p>\n\n\n\n
$$\\langle \\psi_H | Z_0 | \\psi_H \\rangle = \\frac{1}{2}\\Big(\\langle\\psi|U|\\psi\\rangle+\\langle\\psi|U^\\dagger|\\psi\\rangle\\Big)$$<\/p>\n\n\n\n
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u3001$\\langle\\psi|U^\\dagger|\\psi\\rangle
=\\langle\\psi|U|\\psi\\rangle^*$\u306a\u306e\u3067\u3001 $\\langle\\psi|U^\\dagger|\\psi\\rangle=\\langle\\psi|U|\\psi\\rangle$ \u3068\u306a\u308a\u3001<\/p>\n\n\n\n
$$
\\begin{aligned}
\\langle\\psi_H|Z_0|\\psi_H\\rangle
&=
\\frac{1}{2}
\\left(
\\langle\\psi|U|\\psi\\rangle
+
\\langle\\psi|U|\\psi\\rangle^*
\\right)\\\\
&=
\\operatorname{Re}\\langle\\psi|U|\\psi\\rangle
\\end{aligned}
$$<\/p>\n<\/details>\n\n\n\n
\u3057\u304b\u3057\u3001\u3053\u306e\u65b9\u6cd5\u306b\u306f\u6b21\u306e\u8ab2\u984c\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n