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searchindex.js
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48, 50, 52, 54, 56, 58, 59, 61, 63, 65, 67, 69, 70, 72, 74], "everi": [0, 2, 4, 8, 10, 19, 27, 31, 36, 38, 40, 42, 44, 52, 61, 65], "point": [0, 8, 10, 12, 25, 27, 28, 36, 38, 40, 42, 43, 46, 48, 50, 52, 53, 56, 61, 72, 74], "dure": [0, 10, 23, 25, 37, 42, 48, 50, 65], "comput": [0, 2, 6, 10, 12, 19, 22, 23, 24, 25, 27, 28, 29, 30, 32, 34, 35, 36, 38, 40, 41, 44, 46, 48, 50, 51, 54, 55, 58, 61, 63, 65, 67, 69, 70, 72, 75, 86, 87, 91], "There": [0, 2, 4, 6, 8, 10, 19, 23, 24, 25, 28, 29, 30, 31, 34, 38, 40, 42, 46, 52, 54, 56, 58, 59, 61, 65, 74, 86, 92], "more": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 40, 42, 44, 46, 48, 49, 52, 54, 56, 59, 61, 63, 65, 67, 69, 70, 72, 74], "detail": [0, 6, 10, 12, 14, 15, 19, 28, 30, 36, 38, 40, 42, 44, 46, 52, 54, 56, 67, 74], "we": [0, 2, 4, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 35, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 59, 61, 63, 65, 67, 69, 70, 72, 87], "can": [0, 2, 4, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 34, 35, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 59, 61, 63, 65, 67, 69, 70, 72, 74, 87, 90, 92], "add": [0, 2, 4, 10, 12, 14, 15, 19, 22, 24, 28, 31, 42, 44, 46, 52, 56, 59, 63, 65, 72, 74], "than": [0, 4, 6, 8, 10, 12, 14, 15, 19, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 46, 48, 50, 52, 54, 56, 61, 67, 69, 70, 72, 74], "so": [0, 2, 4, 6, 8, 10, 12, 19, 23, 24, 25, 27, 28, 29, 30, 31, 35, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 59, 61, 63, 65, 67, 69, 70, 72, 74, 87], "write": [0, 4, 10, 12, 19, 22, 23, 24, 25, 27, 30, 31, 38, 40, 42, 46, 52, 56, 59, 61, 63, 65, 67, 69, 70, 72, 74], "down": [0, 4, 6, 8, 23, 25, 30, 31, 36, 38, 42, 46, 48, 50, 52, 56, 63, 65, 67, 69, 70, 72, 74], "c": [0, 2, 6, 8, 10, 12, 14, 15, 16, 19, 22, 23, 24, 25, 27, 28, 29, 30, 38, 40, 44, 48, 50, 52, 54, 61, 63, 67, 72, 74], "just": [0, 2, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 44, 46, 48, 52, 54, 56, 61, 63, 65, 67, 69, 70, 72, 74], "two": [0, 2, 6, 8, 10, 12, 14, 15, 19, 22, 23, 25, 27, 28, 29, 31, 32, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 55, 58, 63, 65, 67, 69, 70, 72, 74, 86], "binari": [0, 2, 12, 30, 31, 34, 38, 40, 42, 43, 46, 48, 52, 56, 58, 61, 67, 72], "valu": [0, 2, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 28, 29, 30, 31, 36, 38, 40, 43, 46, 48, 50, 52, 54, 56, 61, 65, 67, 72, 74], "For": [0, 2, 4, 6, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 46, 48, 50, 52, 54, 56, 61, 63, 65, 67, 69, 70, 72, 74], "exampl": [0, 2, 4, 6, 10, 14, 15, 19, 23, 24, 25, 27, 28, 29, 30, 31, 32, 34, 36, 40, 42, 44, 46, 48, 49, 51, 54, 56, 58, 63, 65, 67, 69, 70], "restrict": [0, 4, 8, 10, 12, 15, 23, 25, 27, 30, 40, 42, 48, 52, 61], "lift": [0, 42], "whether": [0, 6, 8, 10, 12, 25, 27, 28, 38, 42, 44, 48, 50, 52, 67, 72], "get": [0, 2, 4, 6, 8, 10, 12, 19, 22, 25, 28, 29, 30, 31, 36, 39, 42, 46, 48, 50, 52, 54, 56, 61, 63, 65, 67, 69, 70, 72, 74, 75], "from": [0, 2, 6, 8, 10, 12, 16, 17, 19, 22, 23, 24, 25, 27, 28, 29, 30, 34, 35, 36, 40, 44, 46, 48, 50, 52, 54, 55, 58, 59, 61, 63, 67, 69, 70, 72, 74, 86, 87, 90], "onli": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 23, 24, 25, 27, 28, 29, 30, 31, 34, 36, 38, 40, 42, 46, 48, 50, 52, 54, 56, 58, 59, 63, 65, 69, 70, 72, 74, 86], "need": [0, 2, 4, 8, 10, 12, 14, 19, 22, 23, 24, 27, 29, 30, 31, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 59, 61, 63, 65, 72, 74, 90], "when": [0, 2, 6, 8, 10, 12, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 46, 48, 50, 52, 54, 56, 61, 63, 65, 67, 69, 70, 72, 74], "made": [0, 19, 28, 30, 38, 40, 42, 46, 48, 52, 65, 74], "extract": [0, 10, 19, 23, 30, 31, 36, 42, 52, 54, 56, 61, 74], "output": [0, 2, 4, 6, 8, 10, 12, 16, 19, 23, 24, 25, 28, 29, 30, 31, 36, 38, 40, 42, 46, 48, 50, 52, 54, 56, 59, 61, 67, 72, 74], "At": [0, 8, 10, 28, 38, 42, 52, 54, 56, 72, 74], "must": [0, 4, 6, 10, 12, 14, 19, 23, 25, 27, 28, 29, 30, 34, 36, 38, 40, 42, 50, 52, 56, 58, 59, 63, 65, 67, 72, 74], "commit": [0, 19, 38, 42], "one": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 34, 36, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 59, 61, 63, 65, 67, 69, 70, 72, 74, 86, 90], "option": [0, 6, 8, 12, 19, 27, 37, 42, 52, 65], "other": [0, 2, 4, 8, 10, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 36, 38, 40, 41, 46, 48, 50, 52, 56, 59, 61, 63, 65, 67, 69, 70, 72, 74, 86], "time": [0, 2, 4, 6, 8, 10, 12, 14, 15, 22, 23, 24, 25, 27, 29, 30, 31, 36, 38, 40, 42, 46, 48, 49, 52, 54, 56, 59, 61, 69, 70, 72, 74], "its": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 61, 63, 65, 67, 72, 74], "someth": [0, 10, 25, 35, 36, 38, 40, 42, 44, 46, 63, 69, 70, 74, 87], "complex": [0, 2, 6, 12, 19, 22, 23, 24, 27, 30, 31, 38, 40, 43, 46, 48, 50, 52, 54, 56, 63, 65, 72, 86], "captur": [0, 14, 15, 22, 24, 28, 29, 40, 42, 44, 46, 52], "simpl": [0, 2, 6, 8, 10, 12, 19, 23, 25, 27, 28, 29, 30, 36, 38, 42, 44, 46, 47, 52, 54, 56, 59, 65, 72], "To": [0, 4, 6, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 59, 72, 74], "how": [0, 2, 4, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 28, 29, 30, 31, 32, 34, 36, 38, 40, 42, 43, 46, 48, 50, 51, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72], "describ": [0, 2, 4, 10, 12, 14, 15, 19, 23, 24, 25, 27, 28, 31, 36, 38, 40, 42, 44, 46, 48, 52, 54, 56, 63, 69, 70, 72, 74], "focu": [0, 10, 22, 29, 31, 36, 38, 40, 42, 52, 54, 67], "simplest": [0, 38, 42, 44, 46, 52, 54], "case": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 23, 24, 25, 27, 28, 29, 30, 31, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 58, 63, 65, 69, 70, 72, 74, 86], "It": [0, 2, 4, 6, 8, 10, 12, 19, 22, 23, 24, 25, 28, 29, 30, 31, 35, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 63, 65, 67, 69, 70, 72, 74, 87], "possibl": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 23, 24, 25, 27, 28, 29, 30, 36, 38, 40, 42, 44, 48, 52, 54, 56, 63, 65, 69, 70, 72, 74], "prepar": [0, 6, 10, 19, 24, 25, 27, 29, 30, 38, 39, 44, 46, 52, 54, 55, 72, 74], "which": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 34, 35, 36, 38, 40, 42, 44, 46, 48, 50, 51, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 75, 87], "definit": [0, 6, 22, 24, 25, 27, 28, 29, 30, 32, 36, 40, 42, 46, 52], "give": [0, 2, 8, 10, 19, 22, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 46, 48, 52, 61, 63, 65, 67, 69, 70, 72, 74], "outcom": [0, 2, 4, 6, 8, 10, 12, 19, 25, 28, 29, 31, 40, 42, 46, 52, 54, 56], "unimagin": [0, 42], "similarli": [0, 2, 4, 10, 14, 15, 19, 28, 38, 40, 42, 46, 50, 54, 65], "exist": [0, 8, 10, 19, 24, 25, 27, 28, 29, 30, 40, 42, 48, 56, 67, 69, 70, 72, 74], "certain": [0, 4, 6, 10, 12, 19, 31, 40, 42, 44, 46, 48, 50, 52, 56, 72, 74], "ll": [0, 6, 12, 19, 28, 30, 38, 40, 42, 46, 50, 52, 61, 65, 72, 74, 90], "These": [0, 4, 6, 8, 10, 19, 22, 23, 29, 31, 38, 40, 42, 46, 48, 52, 56, 69, 70, 74], "mutual": [0, 42, 52], "exclus": [0, 10, 42], "overlap": [0, 10, 14, 15, 22, 28, 29, 42, 48, 52, 56], "One": [0, 2, 6, 8, 19, 22, 23, 24, 25, 27, 29, 30, 31, 32, 38, 42, 44, 46, 50, 52, 56, 61, 63, 69, 70, 72], "wai": [0, 2, 4, 6, 8, 10, 14, 15, 19, 22, 25, 28, 29, 30, 31, 36, 38, 40, 42, 44, 46, 48, 52, 54, 59, 61, 63, 65, 67, 69, 70, 72, 74], "repres": [0, 4, 6, 8, 10, 12, 14, 15, 19, 22, 24, 25, 27, 28, 29, 31, 32, 36, 38, 40, 41, 43, 48, 54, 56, 61, 63, 65, 67, 69, 70, 72, 74], "mathemat": [0, 14, 15, 19, 22, 23, 24, 25, 29, 31, 38, 42, 44, 46, 48, 50, 52, 56, 74], "orthogon": [0, 4, 8, 19, 22, 24, 25, 27, 28, 29, 40, 42, 65, 69, 70, 72, 74], "vector": [0, 2, 8, 10, 14, 15, 19, 22, 23, 24, 25, 27, 29, 31, 36, 38, 40, 44, 46, 48, 50, 52, 54, 61, 69, 70], "left": [0, 2, 6, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 31, 32, 38, 40, 42, 44, 46, 50, 52, 54, 67, 72], "right": [0, 2, 4, 6, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 31, 32, 38, 40, 42, 44, 46, 48, 50, 52, 54, 61, 65, 67, 72, 90], "rangl": [0, 2, 6, 8, 12, 19, 22, 23, 24, 25, 27, 28, 29, 31, 32, 36, 40, 44, 46, 48, 50, 52, 54, 61, 63, 65, 67, 69, 70, 72, 74], "begin": [0, 2, 4, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 38, 40, 42, 44, 50, 52, 54, 61, 63, 65, 67, 72, 74], "pmatrix": [0, 10, 22, 23, 24, 27, 42, 52, 67], "end": [0, 2, 4, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 36, 38, 42, 44, 48, 50, 52, 54, 59, 61, 63, 65, 67, 72, 74], "qquad": [0, 14, 15, 22, 23, 24, 25, 28, 42, 72], "With": [0, 8, 10, 12, 19, 22, 23, 24, 25, 27, 28, 29, 31, 40, 42, 48, 50, 52, 56, 65], "consid": [0, 4, 8, 10, 12, 14, 15, 19, 22, 24, 25, 27, 28, 29, 30, 31, 34, 36, 38, 40, 42, 46, 48, 50, 52, 54, 56, 58, 59, 63, 65, 69, 70, 86], "q_0": [0, 2, 16, 17, 19, 24, 42, 61, 74], "frac": [0, 2, 4, 6, 8, 10, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 38, 40, 42, 46, 50, 52, 54, 61, 63, 67, 69, 70, 72, 74], "sqrt": [0, 2, 4, 6, 8, 10, 12, 15, 16, 17, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 42, 48, 52, 61, 63, 65, 67, 69, 70, 72, 74], "sinc": [0, 2, 4, 6, 8, 10, 12, 19, 22, 23, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 44, 46, 48, 50, 52, 61, 63, 65, 69, 70, 72, 74], "form": [0, 4, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 31, 36, 40, 42, 46, 48, 50, 52, 56, 61, 65, 67, 69, 70, 72, 74, 85, 86, 88], "orthonorm": [0, 4, 23, 24, 25, 27, 28, 32, 42, 69, 70], "basi": [0, 2, 6, 8, 10, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 34, 36, 40, 44, 46, 52, 54, 58, 61, 65, 67, 69, 70, 72, 74], "ani": [0, 2, 4, 6, 8, 10, 12, 14, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 58, 61, 63, 65, 67, 69, 70, 72, 74, 86], "2d": [0, 2, 4, 15, 30, 40, 42, 48, 53], "combin": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 24, 27, 28, 29, 34, 40, 42, 46, 50, 52, 54, 56, 58, 63, 65, 67, 69, 70, 72], "allow": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 29, 30, 31, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 59, 65, 74, 86], "u": [0, 2, 6, 10, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 36, 40, 41, 44, 46, 48, 50, 52, 54, 61, 63, 65, 67, 69, 70, 72, 74], "our": [0, 2, 4, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 40, 42, 46, 48, 50, 52, 54, 56, 61, 63, 65, 67, 69, 70, 72], "statevector": [0, 4, 12, 32, 41, 61, 63, 65], "tell": [0, 2, 4, 8, 19, 22, 23, 25, 27, 28, 40, 42, 44, 46, 52, 67, 72, 74], "everyth": [0, 19, 27, 36, 38, 42, 72, 74], "could": [0, 2, 4, 6, 8, 10, 14, 15, 19, 24, 27, 28, 29, 30, 36, 38, 40, 42, 44, 46, 48, 52, 54, 56, 63, 65, 69, 70, 72, 74], "possibli": [0, 4, 10, 27, 29, 31, 42, 48], "know": [0, 2, 8, 10, 12, 23, 24, 25, 27, 28, 29, 30, 36, 38, 40, 42, 46, 48, 52, 61, 65, 69, 70, 72, 74], "now": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 59, 61, 65, 67, 69, 70, 72, 74], "abl": [0, 6, 10, 14, 15, 19, 29, 38, 40, 42, 48, 52, 56, 59, 74, 86], "draw": [0, 2, 4, 6, 8, 10, 12, 15, 16, 17, 27, 40, 42, 48, 50, 52, 54, 61, 63, 65, 67, 72, 74], "few": [0, 4, 8, 23, 29, 30, 38, 40, 42, 44, 48, 52, 65, 74], "conclus": [0, 2, 31, 32, 41, 52, 53], "particular": [0, 6, 8, 12, 19, 27, 30, 36, 38, 40, 42, 46, 48, 50, 52, 72], "entir": [0, 10, 12, 19, 24, 27, 28, 29, 31, 37, 42, 52, 59], "instead": [0, 2, 4, 8, 10, 12, 14, 15, 19, 24, 28, 29, 30, 36, 38, 40, 42, 46, 48, 52, 56, 63, 65, 72, 74], "linear": [0, 4, 14, 15, 19, 23, 24, 25, 27, 28, 29, 30, 31, 32, 34, 36, 37, 40, 42, 46, 48, 50, 52, 54, 58, 63, 69, 70], "mechan": [0, 8, 12, 19, 22, 24, 25, 28, 30, 32, 34, 40, 41, 46, 54, 56, 58, 69, 70, 74], "typic": [0, 6, 8, 19, 27, 29, 30, 36, 38, 40, 42, 50, 52], "word": [0, 19, 23, 24, 25, 27, 28, 29, 30, 31, 34, 36, 38, 42, 58, 67, 72, 74], "superposit": [0, 2, 4, 8, 12, 23, 24, 25, 27, 30, 36, 42, 48, 63, 65, 69, 70, 72, 74], "though": [0, 14, 15, 19, 27, 28, 30, 36, 38, 40, 42, 46, 52, 67, 69, 70], "express": [0, 19, 23, 24, 25, 28, 29, 31, 32, 34, 46, 48, 50, 52, 56, 58, 72], "less": [0, 4, 6, 10, 14, 15, 19, 25, 29, 31, 38, 42, 48, 52, 54, 56, 61, 72], "manipul": [0, 42], "quantumcircuit": [0, 2, 4, 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47, 54, 61, 74], "overal": [0, 10, 14, 15, 30, 31, 38, 40, 42, 52, 54], "gamma": [0, 19, 25, 36, 40, 42, 54], "indistinguish": [0, 42], "distinct": [0, 25, 42], "term": [0, 8, 10, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 31, 32, 36, 38, 40, 46, 48, 50, 52, 54, 55, 67, 72, 88], "known": [0, 2, 4, 6, 12, 19, 23, 24, 25, 27, 29, 30, 31, 36, 40, 42, 44, 50, 52, 56, 74], "rel": [0, 2, 4, 6, 23, 24, 28, 38, 42, 52, 56, 65], "type": [0, 4, 6, 14, 15, 23, 24, 27, 32, 38, 42, 43, 48, 50, 52, 55, 59, 74], "multipl": [0, 2, 4, 8, 10, 19, 23, 24, 28, 29, 30, 32, 40, 41, 46, 47, 49, 52, 54, 56, 69, 70], "contain": [0, 4, 6, 10, 12, 19, 22, 27, 28, 30, 34, 36, 38, 42, 50, 52, 54, 58, 61, 72, 74], "q": [0, 2, 6, 8, 12, 16, 19, 27, 32, 34, 38, 40, 58, 61, 65, 72, 86], "again": [0, 4, 8, 10, 14, 15, 19, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 46, 48, 52, 63, 72, 74], "mean": [0, 2, 4, 8, 10, 12, 19, 22, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 44, 50, 52, 54, 56, 59, 67, 72, 74], "act": [0, 2, 4, 12, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 40, 42, 46, 50, 63, 65, 69, 70, 72], "chang": [0, 2, 4, 6, 8, 10, 19, 23, 24, 25, 28, 36, 38, 42, 46, 48, 52, 54, 56, 59, 63, 65, 69, 70, 72, 74], "xrightarrow": [0, 2, 42, 72, 74], "text": [0, 2, 4, 6, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 34, 38, 40, 42, 46, 48, 50, 52, 54, 58, 63, 65, 67, 69, 70, 72, 74], "sometim": [0, 4, 19, 28, 29, 42, 46, 50, 59, 65], "collaps": [0, 2, 4, 6, 19, 25, 28, 29, 42, 47, 69, 70, 74], "potent": [0, 42], "wise": [0, 14, 15, 30, 31, 42, 52], "were": [0, 4, 6, 8, 10, 19, 24, 25, 36, 40, 42, 44, 56, 59, 65, 69, 70, 72, 74], "constantli": [0, 42], "each": [0, 2, 6, 8, 10, 12, 14, 15, 19, 22, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 61, 63, 65, 67, 69, 70, 72, 74], "keep": [0, 2, 6, 10, 14, 15, 19, 25, 27, 28, 29, 36, 38, 42, 44, 46, 52, 56, 72, 74], "track": [0, 2, 14, 15, 19, 36, 38, 42, 46, 50, 74], "As": [0, 2, 4, 6, 10, 12, 14, 15, 19, 22, 25, 27, 28, 30, 31, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 59, 61, 63, 65, 67, 69, 70, 72, 74, 90], "easili": [0, 2, 8, 10, 12, 19, 22, 31, 38, 42, 44, 46, 48, 50, 52, 56, 67, 69, 70], "replac": [0, 6, 27, 36, 42, 46, 52, 56, 72, 74], "achiev": [0, 4, 10, 19, 27, 28, 30, 38, 42, 48, 52, 56, 74], "truli": [0, 27, 42], "often": [0, 2, 4, 6, 19, 24, 25, 30, 38, 40, 42, 46, 52, 56], "place": [0, 2, 4, 6, 22, 25, 38, 40, 42, 46, 52, 56, 65, 72, 74], "demonstr": [0, 8, 10, 12, 14, 15, 19, 24, 28, 36, 37, 40, 42, 47, 52, 54, 56, 74], "str": [0, 6, 10, 14, 15, 19, 38, 42, 54, 61, 72, 74], "expect": [0, 2, 6, 8, 10, 12, 14, 15, 19, 28, 29, 30, 31, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 61, 65, 72, 74], "where": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 65, 67, 69, 70, 72, 74], "re": [0, 4, 6, 8, 19, 22, 30, 36, 38, 40, 42, 46, 47, 52, 67, 72, 74], "reiniti": [0, 42], "notic": [0, 4, 10, 14, 15, 19, 36, 38, 40, 42, 50, 52, 63, 72, 74], "never": [0, 6, 38, 42, 52, 56, 59, 72], "somewhat": [0, 25, 42], "interestingli": [0, 2, 42, 46], "surviv": [0, 42], "requir": [0, 2, 4, 6, 10, 12, 14, 15, 19, 25, 27, 28, 29, 31, 32, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 61, 63, 65], "ever": [0, 19, 22, 28, 30, 42], "receiv": [0, 8, 10, 19, 22, 28, 29, 34, 40, 42, 58, 59, 74, 86, 90], "ye": [0, 28, 29, 38, 42, 44, 63, 67, 72], "answer": [0, 2, 12, 19, 25, 27, 28, 29, 30, 32, 34, 35, 40, 44, 46, 52, 56, 58, 87], "10": [0, 2, 4, 6, 8, 10, 12, 14, 15, 16, 19, 24, 25, 27, 28, 29, 31, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 61, 65, 67, 69, 70, 72, 85, 86], "like": [0, 2, 4, 6, 8, 10, 19, 22, 23, 24, 25, 27, 28, 29, 35, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 59, 61, 63, 65, 67, 72, 74, 87], "0110111110": [0, 42], "cannot": [0, 2, 4, 6, 8, 10, 19, 24, 25, 27, 28, 29, 30, 38, 40, 42, 46, 48, 52, 54, 74], "mid": [0, 6, 42], "destroi": [0, 4, 42, 69, 70, 74], "them": [0, 4, 6, 8, 10, 12, 19, 22, 28, 29, 31, 35, 36, 38, 40, 42, 46, 48, 52, 54, 61, 63, 65, 67, 72, 74, 87], "behaviour": [0, 2, 4, 19, 23, 42, 46, 52, 54, 65], "ideal": [0, 6, 14, 15, 19, 30, 40, 42, 50, 52, 56, 61], "learn": [0, 2, 4, 10, 19, 22, 25, 27, 28, 31, 35, 38, 39, 42, 46, 48, 50, 51, 53, 56, 87], "provid": [0, 4, 6, 8, 10, 12, 14, 15, 19, 25, 28, 31, 36, 38, 40, 42, 44, 48, 52, 54, 56, 61, 63, 69, 70, 72, 74], "By": [0, 6, 10, 12, 19, 24, 29, 30, 38, 42, 46, 48, 52, 59, 67], "default": [0, 6, 10, 12, 14, 19, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 59], "mimic": [0, 6, 14, 15, 30, 42], "peek": [0, 42], "includ": [0, 4, 12, 14, 15, 19, 22, 29, 31, 32, 36, 38, 40, 42, 44, 48, 50, 52, 54, 56, 61], "saw": [0, 2, 4, 27, 28, 29, 42, 44, 52, 65, 72, 74], "chapter": [0, 2, 4, 31, 42, 65, 67, 72, 74, 83], "mathbb": [0, 8, 22, 23, 24, 25, 27, 28, 29, 30, 42, 44, 50, 52, 54], "differenti": [0, 36, 42, 44, 48, 50, 52], "some": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 23, 24, 25, 27, 28, 29, 30, 31, 32, 35, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 59, 61, 63, 65, 67, 69, 70, 72, 74, 87], "descript": [0, 8, 19, 24, 25, 38, 42, 44, 54], "firstli": [0, 12, 42, 52], "confin": [0, 31, 42], "e": [0, 2, 6, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 35, 36, 38, 40, 44, 46, 48, 50, 52, 54, 61, 65, 67, 69, 70, 72, 74, 87, 91], "phi": [0, 4, 16, 19, 22, 23, 24, 25, 27, 28, 29, 30, 40, 42, 52, 54, 56, 61, 63, 65, 67, 69, 70, 74], "r": [0, 4, 6, 8, 10, 12, 14, 15, 16, 19, 22, 25, 28, 29, 30, 36, 38, 42, 44, 48, 50, 52, 54, 56, 67, 72, 74], "trigonometr": [0, 42], "sin": [0, 4, 10, 12, 16, 19, 30, 38, 42, 50, 52], "co": [0, 4, 8, 10, 16, 19, 25, 28, 30, 42, 52], "variabl": [0, 4, 8, 12, 14, 15, 19, 28, 29, 31, 36, 40, 42, 44, 47, 50, 52, 56, 59], "theta": [0, 4, 6, 8, 10, 12, 15, 17, 19, 22, 30, 36, 40, 42, 48, 50, 52, 56, 65], "interpret": [0, 27, 29, 30, 31, 34, 40, 44, 48, 58, 59], "spheric": [0, 6, 40, 42], "ordin": [0, 4, 42], "plot": [0, 2, 8, 10, 12, 14, 15, 19, 25, 28, 32, 36, 40, 41, 48, 50, 52, 56, 67, 72, 74], "singl": [0, 6, 8, 12, 14, 15, 22, 23, 24, 25, 28, 29, 30, 31, 32, 36, 40, 41, 44, 46, 48, 49, 52, 54, 56, 63, 65, 69, 70, 72, 74], "surfac": [0, 2, 36, 42], "below": [0, 2, 6, 10, 12, 15, 19, 22, 23, 24, 27, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 61, 65, 67, 72, 74], "titl": [0, 6, 12, 14, 15, 19, 34, 36, 38, 40, 42, 48, 58, 86], "coord_typ": [0, 42], "three": [0, 2, 4, 6, 10, 12, 19, 23, 24, 25, 28, 29, 31, 34, 38, 40, 42, 44, 46, 48, 52, 54, 56, 58, 63, 65, 72, 74, 86], "job": [0, 4, 6, 8, 15, 16, 17, 38, 40, 42, 46, 52, 61, 67, 72, 74, 92], "actual": [0, 4, 6, 8, 10, 12, 14, 15, 19, 22, 25, 27, 28, 31, 36, 40, 42, 48, 52, 54, 56, 63, 69, 70, 74], "notebook": [0, 2, 6, 10, 12, 14, 15, 19, 22, 36, 37, 38, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 56, 59, 65, 67, 74, 90], "evalu": [0, 10, 12, 14, 15, 19, 23, 30, 31, 36, 38, 44, 46, 48, 50, 51, 54, 56, 72], "code": [0, 14, 22, 25, 29, 32, 34, 36, 38, 44, 46, 47, 50, 52, 54, 56, 58, 61, 67, 72, 86], "under": [0, 6, 14, 15, 19, 23, 27, 29, 40, 42, 44, 52, 74], "gate": [0, 8, 12, 16, 25, 27, 29, 30, 31, 32, 34, 36, 40, 41, 44, 46, 48, 49, 52, 54, 55, 58, 67, 72, 74], "yet": [0, 8, 10, 19, 22, 23, 24, 27, 40, 42, 52], "horizont": [0, 19, 40, 42], "anoth": [0, 2, 4, 6, 8, 10, 19, 22, 27, 28, 29, 31, 36, 38, 40, 42, 46, 52, 56, 61, 65, 67, 74], "data": [0, 6, 10, 14, 15, 16, 22, 25, 29, 31, 36, 39, 42, 43, 47, 50, 52, 54, 55, 61], "function": [0, 2, 4, 6, 8, 12, 14, 15, 16, 17, 19, 22, 28, 29, 31, 32, 34, 36, 40, 42, 44, 46, 47, 50, 51, 54, 55, 58, 61, 63, 65, 67, 69, 70, 72, 74], "process": [0, 4, 6, 10, 12, 19, 22, 27, 28, 29, 31, 36, 38, 40, 42, 44, 48, 52, 54, 56, 63, 65], "why": [0, 2, 4, 10, 19, 24, 25, 27, 28, 29, 31, 36, 38, 40, 42, 44, 46, 52, 61, 63, 65], "later": [0, 6, 10, 12, 14, 15, 19, 24, 25, 29, 30, 36, 38, 42, 44, 52, 65, 74], "set": [0, 8, 10, 12, 14, 15, 22, 23, 24, 25, 27, 28, 29, 30, 31, 40, 44, 46, 48, 50, 52, 54, 55, 59, 63, 65, 67, 72, 74], "accordingli": [0, 10, 42, 52, 67], "come": [0, 4, 10, 24, 25, 38, 40, 42, 44, 52, 56, 61, 67], "ve": [0, 19, 27, 28, 38, 40, 42, 46, 52, 59, 61], "produc": [0, 2, 4, 10, 12, 14, 15, 19, 29, 30, 31, 36, 38, 42, 46, 52, 65, 67, 69, 70, 74], "human": [0, 46], "readabl": [0, 12, 42], "record": [0, 6, 19, 25, 32, 38, 42, 56, 75], "respect": [0, 4, 6, 8, 10, 12, 19, 22, 23, 24, 25, 28, 29, 30, 31, 36, 40, 42, 44, 50, 52, 54, 56, 63, 65, 74], "standard": [0, 2, 4, 6, 19, 22, 23, 25, 27, 28, 29, 30, 36, 38, 40, 42, 44, 48, 52, 56, 65], "calcul": [0, 2, 4, 10, 12, 19, 23, 24, 27, 28, 31, 36, 38, 40, 42, 44, 48, 50, 52, 54, 56, 65, 69, 70, 74], "becaus": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 23, 24, 27, 28, 29, 30, 31, 38, 42, 44, 46, 50, 52, 54, 56, 65, 72, 74], "believ": [0, 24, 30, 42, 52, 74], "difficult": [0, 2, 14, 15, 24, 27, 28, 29, 30, 40, 42, 44], "best": [0, 6, 8, 10, 36, 38, 42, 50, 51, 54, 56, 59, 67, 72], "algorithm": [0, 2, 6, 14, 15, 19, 29, 31, 32, 36, 40, 42, 44, 48, 50, 52, 56, 65], "grow": [0, 2, 19, 24, 42, 44, 48, 50, 52], "exponenti": [0, 2, 6, 19, 24, 27, 30, 34, 40, 42, 45, 48, 54, 56, 58], "small": [0, 6, 10, 12, 14, 15, 25, 36, 40, 42, 46, 48, 50, 52, 72, 74], "30": [0, 4, 14, 15, 36, 38, 40, 42, 48, 52, 56, 63, 85, 86], "trick": [0, 23, 25, 27, 32, 34, 42, 52, 54, 58, 65, 72], "speed": [0, 4, 19, 28, 38, 42, 50, 52], "design": [0, 6, 19, 25, 31, 38, 42, 48, 50, 51, 56, 61, 72, 74], "smaller": [0, 8, 10, 14, 15, 25, 28, 30, 42, 52, 56, 67, 69, 70], "make": [0, 4, 10, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 32, 36, 37, 40, 42, 44, 46, 48, 50, 52, 54, 56, 63, 72, 74], "new": [0, 4, 6, 10, 14, 15, 35, 36, 38, 40, 42, 48, 52, 54, 56, 67, 74, 87, 92], "aersimul": [0, 6, 42, 61], "experi": [0, 4, 6, 8, 19, 25, 28, 30, 31, 32, 38, 42, 48, 50, 56, 61, 65, 74], "ran": [0, 42], "importantli": [0, 6, 8, 27, 42, 50], "popular": [0, 4, 8, 14, 15, 42], "dictionari": [0, 6, 8, 10, 19, 38, 42, 56], "kei": [0, 4, 8, 10, 12, 23, 25, 29, 30, 38, 42, 50, 56, 72, 74], "string": [0, 6, 10, 12, 19, 25, 30, 31, 38, 42, 44, 61, 72, 74], "wa": [0, 2, 6, 8, 10, 12, 19, 24, 25, 27, 28, 29, 30, 36, 38, 40, 42, 44, 46, 52, 54, 56, 65, 67, 69, 70, 72, 74], "random": [0, 2, 6, 8, 12, 16, 19, 25, 27, 28, 29, 30, 31, 36, 38, 40, 44, 46, 48, 50, 52, 53, 55, 63, 72, 74], "common": [0, 4, 8, 10, 22, 23, 38, 40, 42, 44, 48, 52], "1024": [0, 10, 12, 42, 72, 74], "convent": [0, 19, 25, 29, 42], "noth": [0, 4, 8, 10, 27, 28, 31, 40, 42, 46, 52, 63, 65, 72, 74], "situat": [0, 6, 30, 40, 42, 46, 50, 52, 69, 70], "statist": [0, 2, 4, 8, 19, 28, 31, 38, 42, 46, 56, 59], "analys": [0, 19, 25, 42, 56, 61], "NOT": [0, 4, 6, 12, 22, 23, 24, 42, 61, 72], "flip": [0, 6, 8, 10, 12, 19, 29, 31, 40, 42, 46, 56, 65, 74], "dedic": [0, 31, 42], "perform": [0, 2, 4, 6, 8, 10, 12, 14, 15, 19, 23, 25, 27, 28, 30, 31, 36, 37, 40, 42, 44, 46, 48, 50, 52, 54, 56, 63, 65, 67, 69, 70, 72, 74], "ipynb": [1, 3, 5, 7, 9, 10, 11, 13, 18, 20, 21, 26, 60, 62, 64, 66, 68, 71, 73, 90], "qiskit": [2, 4, 14, 15, 16, 17, 22, 32, 34, 35, 41, 44, 58, 63, 65, 67, 75, 86, 87], "input": [2, 4, 6, 8, 10, 12, 14, 15, 16, 19, 25, 29, 30, 32, 36, 40, 41, 44, 46, 48, 50, 52, 54, 56, 61, 63, 65, 67, 72, 74], "In": [2, 4, 6, 8, 10, 12, 14, 15, 16, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 36, 38, 40, 41, 44, 46, 48, 50, 52, 54, 56, 61, 63, 65, 67, 69, 70, 72, 74, 86, 90], "interest": [2, 4, 6, 8, 10, 14, 15, 19, 22, 24, 27, 28, 31, 35, 36, 38, 40, 42, 44, 46, 50, 52, 54, 63, 65, 67, 69, 70, 87, 92], "individu": [2, 6, 10, 12, 19, 25, 27, 32, 34, 38, 40, 41, 46, 58, 74], "offer": [2, 10, 12, 22, 42, 56], "advantag": [2, 6, 19, 36, 38, 40, 42, 48, 50, 52, 56, 72, 74], "interact": [2, 4, 8, 10, 19, 24, 40, 42, 44, 46, 56], "ha": [2, 4, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 35, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 65, 69, 70, 72, 74, 86, 87], "four": [2, 8, 10, 19, 24, 25, 27, 28, 31, 32, 34, 40, 41, 52, 54, 58, 63, 74], "00": [2, 4, 6, 8, 10, 12, 15, 16, 17, 24, 25, 27, 28, 29, 31, 38, 40, 42, 44, 48, 50, 56, 59, 61, 63, 65, 67, 69, 70, 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65, 67, 69, 70, 72, 74, 75, 86, 87, 90], "note": [2, 4, 6, 8, 10, 12, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 34, 35, 38, 40, 41, 44, 48, 50, 52, 56, 58, 59, 63, 65, 67, 72, 74, 76, 77, 78, 79, 80, 81, 82, 86, 87, 88], "softwar": [2, 10, 31, 38, 42, 50, 56, 75, 86], "websit": [2, 42, 56], "order": [2, 8, 10, 12, 19, 22, 25, 28, 29, 30, 31, 34, 36, 40, 44, 46, 48, 50, 52, 53, 58, 61, 63, 65, 67, 69, 70, 72, 74, 86, 90], "consult": [2, 42], "sourc": [2, 4, 14, 29, 38, 42, 56, 59, 86], "readi": [2, 10, 36, 40, 42, 54, 75], "target": [2, 12, 16, 24, 25, 29, 38, 40, 42, 47, 52, 63, 65, 72, 74], "first": [2, 4, 6, 8, 10, 12, 14, 15, 19, 23, 24, 25, 27, 28, 29, 30, 31, 32, 34, 36, 38, 40, 41, 44, 46, 48, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 86, 90], "control": [2, 6, 10, 12, 16, 19, 24, 25, 29, 31, 40, 42, 48, 52, 63, 72, 74], "drawn": [2, 19, 42], "q0": [2, 10, 19, 42, 65, 74], "q1": [2, 19, 42, 65, 74], "cx": [2, 6, 8, 10, 12, 15, 16, 17, 42, 54, 63, 65, 72, 74], 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52, 56, 61, 74, 86], "written": [2, 4, 12, 19, 22, 23, 25, 27, 29, 36, 38, 42, 48, 50, 52, 63, 65, 69, 70, 74], "although": [2, 4, 10, 14, 15, 23, 25, 28, 38, 42, 44, 61], "top": [2, 12, 19, 25, 28, 36, 38, 42, 44, 72], "got": [2, 10, 36, 42, 72], "even": [2, 4, 8, 10, 12, 14, 15, 19, 22, 23, 25, 28, 29, 31, 36, 38, 40, 42, 46, 48, 52, 67], "light": [2, 4, 6, 8, 28, 42, 74], "year": [2, 28, 35, 38, 40, 42, 52, 56, 87], "awai": [2, 4, 19, 30, 31, 42, 46, 52, 74], "appear": [2, 4, 8, 12, 24, 25, 28, 38, 40, 42, 46, 50, 74, 90], "effect": [2, 4, 6, 8, 10, 19, 30, 38, 40, 46, 48, 52, 54, 56, 61, 65, 72, 74], "spooki": [2, 42], "action": [2, 4, 19, 22, 23, 24, 25, 29, 30, 40, 42, 61, 63, 65, 69, 70, 72], "distanc": [2, 8, 14, 15, 22, 36, 42, 52, 54, 56], "upset": [2, 42], "mani": [2, 4, 6, 10, 14, 15, 19, 22, 23, 25, 29, 30, 31, 38, 40, 42, 44, 48, 50, 52, 53, 56, 59, 61, 63, 65, 72], "physicist": [2, 22, 23, 24, 27, 42, 46, 86], "earli": [2, 23, 25, 28, 38, 42, 52, 72, 74], "20th": [2, 23, 29, 42], "centuri": [2, 23, 29, 42], "affect": [2, 6, 10, 19, 38, 42, 48, 65, 74], "share": [2, 6, 8, 19, 24, 25, 28, 29, 35, 38, 42, 44, 59, 74, 87], "commun": [2, 6, 8, 10, 19, 28, 29, 34, 35, 38, 40, 42, 54, 56, 58, 74, 87], "theorem": [2, 10, 19, 29, 32, 34, 42, 48, 58, 74], "lose": [2, 8, 28, 38, 40, 42, 56], "bloch": [2, 32, 34, 41, 44, 48, 58, 63, 65, 74, 86], "sphere": [2, 6, 32, 34, 40, 41, 44, 48, 58, 63, 65, 74, 86], "given": [2, 4, 6, 8, 10, 12, 14, 15, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 36, 38, 40, 42, 46, 48, 50, 52, 54, 56, 61, 63, 67, 69, 70, 72], "posit": [2, 14, 15, 19, 22, 23, 25, 27, 29, 36, 40, 42, 44, 46, 52, 67, 69, 70, 72], "along": [2, 4, 6, 10, 19, 25, 29, 40, 42, 46, 52, 69, 70], "axi": [2, 4, 8, 10, 12, 16, 19, 25, 36, 38, 40, 42, 44, 48, 52, 54, 65, 69, 70], "arriv": [2, 19, 27, 40, 42, 46, 72], "guarante": [2, 14, 15, 30, 31, 42, 52, 72, 74], "contrast": [2, 27, 38, 42, 46, 50, 74], "pick": [2, 14, 15, 25, 27, 29, 30, 38, 40, 42, 48, 52], "miss": [2, 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32, 34, 36, 40, 42, 44, 46, 47, 50, 52, 54, 56, 58, 61], "leftmost": [2, 42, 65, 74], "been": [2, 4, 8, 10, 12, 14, 19, 25, 27, 29, 31, 36, 38, 40, 42, 44, 50, 52, 56, 65, 74], "symmetr": [2, 14, 15, 19, 40, 42, 52, 65, 74], "fashion": [2, 25, 27, 40, 42, 52, 65], "shown": [2, 8, 10, 12, 19, 30, 31, 36, 42, 48, 52, 54, 56, 59, 65, 72, 74, 90], "sdg": [2, 4, 12, 42, 65], "applic": [2, 4, 8, 14, 15, 19, 22, 27, 29, 36, 38, 40, 42, 48, 52, 56, 59, 65], "todai": [4, 50], "session": [4, 6, 19, 30, 42, 56, 59, 65, 72, 74, 88, 90], "cheatsheet": 4, "delv": [4, 6, 27, 38], "deeper": [4, 6, 22], "cover": [4, 8, 22, 30, 38, 40, 42, 72], "due": [4, 6, 8, 10, 12, 14, 15, 19, 27, 28, 29, 30, 36, 40, 42, 48, 50, 52, 56, 64, 72, 74], "risk": [4, 42, 52, 56], "counter": [4, 25, 30, 40, 42], "idea": [4, 8, 19, 23, 27, 28, 29, 31, 32, 34, 36, 39, 42, 48, 52, 53, 56, 58, 63, 67, 74, 88], "appropri": [4, 6, 8, 19, 28, 30, 38, 42, 56, 74], "t_gate": 4, "atom": [4, 10, 19, 42, 86], "came": [4, 8, 25, 42], 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"optimis": 56, "solver": 56, "geometr": 40, "represent": 40, "categori": 40, "nought": 40, "cross": 40, "graph": 40, "recurr": 40, "qgrnn": 40}, "envversion": {"sphinx.domains.c": 2, "sphinx.domains.changeset": 1, "sphinx.domains.citation": 1, "sphinx.domains.cpp": 8, "sphinx.domains.index": 1, "sphinx.domains.javascript": 2, "sphinx.domains.math": 2, "sphinx.domains.python": 3, "sphinx.domains.rst": 2, "sphinx.domains.std": 2, "sphinx.ext.intersphinx": 1, "sphinx.ext.todo": 2, "sphinx.ext.viewcode": 1, "sphinx": 57}, "alltitles": {"Hands-on 1": [[0, "hands-on-1"]], "Preliminary": [[0, "preliminary"], [42, "preliminary"]], "Exploring Qubits with Qiskit": [[0, "exploring-qubits-with-qiskit"], [42, "exploring-qubits-with-qiskit"]], "The Rules of Measurement": [[0, "the-rules-of-measurement"], [42, "the-rules-of-measurement"]], "Normalisation": [[0, "normalisation"], [42, "normalisation"]], "In class exercise": [[0, "in-class-exercise"], [42, "in-class-exercise"]], "Alternative measurement": [[0, "alternative-measurement"], [42, "alternative-measurement"]], "Global Phase": [[0, "global-phase"], [42, "global-phase"]], "The Observer Effect": [[0, "the-observer-effect"], [42, "the-observer-effect"]], "A Note about Quantum Simulators": [[0, "a-note-about-quantum-simulators"], [42, "a-note-about-quantum-simulators"]], "The Bloch Sphere": [[0, "the-bloch-sphere"], [42, "the-bloch-sphere"]], "In Class Exercise": [[0, "id1"], [42, "id1"]], "Your first multi-qubit quantum circuit": [[0, "your-first-multi-qubit-quantum-circuit"], [42, "your-first-multi-qubit-quantum-circuit"]], "What\u2019s a method?": [[0, "what-s-a-method"], [42, "what-s-a-method"]], "Basics of encoding an input": [[0, "basics-of-encoding-an-input"], [42, "basics-of-encoding-an-input"]], "Hands-on session 1": [[1, "hands-on-session-1"], [37, "hands-on-session-1"]], "Session Recording": [[1, "session-recording"], [3, "session-recording"], [5, "session-recording"], [7, "session-recording"], [9, "session-recording"], [11, "session-recording"], [13, "session-recording"], [18, "session-recording"], [21, "session-recording"], [21, "id1"], [37, "session-recording"], [60, "session-recording"], [62, "session-recording"], [64, "session-recording"], [66, "session-recording"], [68, "session-recording"], [71, "session-recording"], [73, "session-recording"], [41, "session-recording"], [43, "session-recording"], [45, "session-recording"], [47, "session-recording"], [49, "session-recording"], [51, "session-recording"], [53, "session-recording"], [20, "session-recording"], [39, "session-recording"], [55, "session-recording"]], "Lab book": [[1, "lab-book"], [3, "lab-book"], [5, "lab-book"], [7, "lab-book"], [9, "lab-book"], [11, "lab-book"], [13, "lab-book"], [18, "lab-book"], [37, "lab-book"], [60, "lab-book"], [62, "lab-book"], [64, "lab-book"], [66, "lab-book"], [68, "lab-book"], [71, "lab-book"], [73, "lab-book"], [41, "lab-book"], [43, "lab-book"], [45, "lab-book"], [47, "lab-book"], [49, "lab-book"], [51, "lab-book"], [53, "lab-book"], [20, "lab-book"], [39, "lab-book"], [55, "lab-book"]], "Hands-on 2": [[2, "hands-on-2"]], "Representing Multi-Qubit States": [[2, "representing-multi-qubit-states"], [42, "representing-multi-qubit-states"]], "Single Qubit Gates on Multi-Qubit Statevectors": [[2, "single-qubit-gates-on-multi-qubit-statevectors"], [42, "single-qubit-gates-on-multi-qubit-statevectors"]], "Send it after class 1": [[2, "send-it-after-class-1"], [4, "send-it-after-class-1"], [10, "send-it-after-class-1"], [22, "send-it-after-class-1"], [23, "send-it-after-class-1"], [24, "send-it-after-class-1"], [25, "send-it-after-class-1"], [27, "send-it-after-class-1"], [28, "send-it-after-class-1"], [29, "send-it-after-class-1"], [30, "send-it-after-class-1"], [30, "id2"], [42, "send-it-after-class-1"]], "Multi-Qubit Gates": [[2, "multi-qubit-gates"], [42, "multi-qubit-gates"]], "The CNOT-Gate": [[2, "the-cnot-gate"], [42, "the-cnot-gate"]], "Entangled States": [[2, "entangled-states"], [42, "entangled-states"]], "Visualizing Entangled States": [[2, "visualizing-entangled-states"], [42, "visualizing-entangled-states"]], "Send it after class 2": [[2, "send-it-after-class-2"], [4, "send-it-after-class-2"], [8, "send-it-after-class-2"], [10, "send-it-after-class-2"], [22, "send-it-after-class-2"], [23, "send-it-after-class-2"], [24, "send-it-after-class-2"], [25, "send-it-after-class-2"], [27, "send-it-after-class-2"], [28, "send-it-after-class-2"], [29, "send-it-after-class-2"], [30, "send-it-after-class-2"], [30, "id3"], [42, "send-it-after-class-2"]], "Exploring the CNOT-Gate": [[2, "exploring-the-cnot-gate"], [42, "exploring-the-cnot-gate"]], "Phase Kickback": [[2, "phase-kickback"], [42, "phase-kickback"]], "Explaining the CNOT Circuit Identity": [[2, "explaining-the-cnot-circuit-identity"], [42, "explaining-the-cnot-circuit-identity"]], "Kickback with the T-gate <a id=\"kickback-t-gate\"></a>": [[2, "kickback-with-the-t-gate-a-id-kickback-t-gate-a"], [42, "kickback-with-the-t-gate-a-id-kickback-t-gate-a"]], "Send it after class 3:": [[2, "send-it-after-class-3"]], "Hands-on session 2": [[3, "hands-on-session-2"], [41, "hands-on-session-2"]], "Hands on 3": [[4, "hands-on-3"], [75, null]], "Single Qubit Gates": [[4, "single-qubit-gates"], [42, "single-qubit-gates"]], "Contents": [[4, "contents"], [65, "contents"], [65, "id8"], [72, "contents"], [72, "id5"], [74, "contents"], [74, "id7"], [74, "id9"]], "1. The Pauli Gates <a id=\"pauli\"></a>": [[4, "the-pauli-gates-a-id-pauli-a"]], "1.1 The X-Gate <a id=\"xgate\"></a>": [[4, "the-x-gate-a-id-xgate-a"]], "1.2 The Y & Z-gates <a id=\"ynzgatez\"></a>": [[4, "the-y-z-gates-a-id-ynzgatez-a"]], "2. Digression: The X, Y & Z-Bases <a id=\"xyzbases\"></a>": [[4, "digression-the-x-y-z-bases-a-id-xyzbases-a"]], "Reminder: Eigenvectors of Matrices": [[4, "reminder-eigenvectors-of-matrices"], [42, "reminder-eigenvectors-of-matrices"]], "3. The Hadamard Gate <a id=\"hgate\"></a>": [[4, "the-hadamard-gate-a-id-hgate-a"]], "4. Digression: Measuring in Different Bases <a id=\"measuring\"></a>": [[4, "digression-measuring-in-different-bases-a-id-measuring-a"]], "Send it after class 3": [[4, "send-it-after-class-3"], [10, "send-it-after-class-3"], [22, "send-it-after-class-3"], [23, "send-it-after-class-3"], [24, "send-it-after-class-3"], [25, "send-it-after-class-3"], [27, "send-it-after-class-3"], [28, "send-it-after-class-3"], [29, "send-it-after-class-3"], [30, "send-it-after-class-3"], [30, "id5"]], "5. The P-gate <a id=\"rzgate\"></a>": [[4, "the-p-gate-a-id-rzgate-a"]], "6. The I, S and T-gates <a id=\"istgates\"></a>": [[4, "the-i-s-and-t-gates-a-id-istgates-a"]], "6.1 The I-gate <a id=\"igate\"></a>": [[4, "the-i-gate-a-id-igate-a"]], "Send it after class 4": [[4, "send-it-after-class-4"], [10, "send-it-after-class-4"], [22, "send-it-after-class-4"], [23, "send-it-after-class-4"], [24, "send-it-after-class-4"], [25, "send-it-after-class-4"], [27, "send-it-after-class-4"], [28, "send-it-after-class-4"], [29, "send-it-after-class-4"], [30, "send-it-after-class-4"], [30, "id8"]], "6.2 The S-gates <a id=\"sgate\"></a>": [[4, "the-s-gates-a-id-sgate-a"]], "6.3 The T-gate <a id=\"tgate\"></a>": [[4, "the-t-gate-a-id-tgate-a"]], "7. The U-gate <a id=\"generalU\"></a>": [[4, "the-u-gate-a-id-generalu-a"]], "Quantum Measurements": [[4, "quantum-measurements"], [42, "quantum-measurements"]], "Part 1: Measuring the state of a qubit": [[4, "part-1-measuring-the-state-of-a-qubit"], [42, "part-1-measuring-the-state-of-a-qubit"]], "1. Express the expectation values of the Pauli operators for an arbitrary qubit state |q\\rangle in the computational basis.": [[4, "express-the-expectation-values-of-the-pauli-operators-for-an-arbitrary-qubit-state-q-rangle-in-the-computational-basis"], [42, "express-the-expectation-values-of-the-pauli-operators-for-an-arbitrary-qubit-state-q-rangle-in-the-computational-basis"]], "2. Measure the Bloch sphere coordinates of a qubit using the Aer simulator and plot the vector on the Bloch sphere.": [[4, "measure-the-bloch-sphere-coordinates-of-a-qubit-using-the-aer-simulator-and-plot-the-vector-on-the-bloch-sphere"], [42, "measure-the-bloch-sphere-coordinates-of-a-qubit-using-the-aer-simulator-and-plot-the-vector-on-the-bloch-sphere"]], "Step A. Create a qubit state using the circuit method, <code>initialize</code> with two random complex numbers as the parameter.": [[4, "step-a-create-a-qubit-state-using-the-circuit-method-code-initialize-code-with-two-random-complex-numbers-as-the-parameter"], [42, "step-a-create-a-qubit-state-using-the-circuit-method-code-initialize-code-with-two-random-complex-numbers-as-the-parameter"]], "Step B. Build the circuits to measure the expectation values of X, Y, Z gate based on your answers to the question 1. Run the cell below to estimate the Bloch sphere coordinates of the qubit from step A using the Aer simulator.": [[4, "step-b-build-the-circuits-to-measure-the-expectation-values-of-x-y-z-gate-based-on-your-answers-to-the-question-1-run-the-cell-below-to-estimate-the-bloch-sphere-coordinates-of-the-qubit-from-step-a-using-the-aer-simulator"], [42, "step-b-build-the-circuits-to-measure-the-expectation-values-of-x-y-z-gate-based-on-your-answers-to-the-question-1-run-the-cell-below-to-estimate-the-bloch-sphere-coordinates-of-the-qubit-from-step-a-using-the-aer-simulator"]], "Step C. Plot the vector on the Bloch sphere.": [[4, "step-c-plot-the-vector-on-the-bloch-sphere"], [42, "step-c-plot-the-vector-on-the-bloch-sphere"]], "Part 2: Measuring Energy": [[4, "part-2-measuring-energy"], [42, "part-2-measuring-energy"]], "1. Express the expectation value of each term in the Hamiltonian for an arbitrary two qubit state |\\psi \\rangle in the computational basis.": [[4, "express-the-expectation-value-of-each-term-in-the-hamiltonian-for-an-arbitrary-two-qubit-state-psi-rangle-in-the-computational-basis"], [42, "express-the-expectation-value-of-each-term-in-the-hamiltonian-for-an-arbitrary-two-qubit-state-psi-rangle-in-the-computational-basis"]], "2. Measure the expected energy of the system using the Aer simulator when two qubits are entangled. Regard the bell basis, four different entangled states.": [[4, "measure-the-expected-energy-of-the-system-using-the-aer-simulator-when-two-qubits-are-entangled-regard-the-bell-basis-four-different-entangled-states"], [42, "measure-the-expected-energy-of-the-system-using-the-aer-simulator-when-two-qubits-are-entangled-regard-the-bell-basis-four-different-entangled-states"]], "Step A. Construct the circuits to prepare four different bell states.": [[4, "step-a-construct-the-circuits-to-prepare-four-different-bell-states"], [42, "step-a-construct-the-circuits-to-prepare-four-different-bell-states"]], "Step B. Create the circuits to measure the expectation value of each term in the Hamiltonian based on your answer to the question 1.": [[4, "step-b-create-the-circuits-to-measure-the-expectation-value-of-each-term-in-the-hamiltonian-based-on-your-answer-to-the-question-1"], [42, "step-b-create-the-circuits-to-measure-the-expectation-value-of-each-term-in-the-hamiltonian-based-on-your-answer-to-the-question-1"]], "Step C. Execute the circuits on Aer simulator by running the cell below and evaluate the energy expectation value for each state.": [[4, "step-c-execute-the-circuits-on-aer-simulator-by-running-the-cell-below-and-evaluate-the-energy-expectation-value-for-each-state"], [42, "step-c-execute-the-circuits-on-aer-simulator-by-running-the-cell-below-and-evaluate-the-energy-expectation-value-for-each-state"]], "Step D. Understanding the result.": [[4, "step-d-understanding-the-result"], [42, "step-d-understanding-the-result"]], "Part 3: Execute the circuits on Quantum Computer": [[4, "part-3-execute-the-circuits-on-quantum-computer"], [42, "part-3-execute-the-circuits-on-quantum-computer"]], "Step A. Run the following cells to load your account and select the backend": [[4, "step-a-run-the-following-cells-to-load-your-account-and-select-the-backend"], [42, "step-a-run-the-following-cells-to-load-your-account-and-select-the-backend"]], "Step B. Execute the circuits on the quantum system.": [[4, "step-b-execute-the-circuits-on-the-quantum-system"], [42, "step-b-execute-the-circuits-on-the-quantum-system"]], "Step C. Estimate the ground state energy levels from the results of the previous step by executing the cells below.": [[4, "step-c-estimate-the-ground-state-energy-levels-from-the-results-of-the-previous-step-by-executing-the-cells-below"], [42, "step-c-estimate-the-ground-state-energy-levels-from-the-results-of-the-previous-step-by-executing-the-cells-below"]], "Step D. Measurement error mitigation.": [[4, "step-d-measurement-error-mitigation"], [42, "step-d-measurement-error-mitigation"]], "Step E. Interpret the result.": [[4, "step-e-interpret-the-result"], [42, "step-e-interpret-the-result"]], "Hands-on session 3": [[5, "hands-on-session-3"], [43, "hands-on-session-3"]], "Hands on 4": [[6, "hands-on-4"], [75, null]], "Using your IBM credentials locally": [[6, "using-your-ibm-credentials-locally"], [42, "using-your-ibm-credentials-locally"]], "Qiskit Visualization tools": [[6, "qiskit-visualization-tools"], [42, "qiskit-visualization-tools"]], "Plot a state": [[6, "plot-a-state"], [42, "plot-a-state"]], "Plot Bloch Vector": [[6, "plot-bloch-vector"], [42, "plot-bloch-vector"]], "Circuit Visualization": [[6, "circuit-visualization"], [42, "circuit-visualization"]], "Disable Plot Barriers and Reversing Bit Order": [[6, "disable-plot-barriers-and-reversing-bit-order"], [42, "disable-plot-barriers-and-reversing-bit-order"]], "Backend specific costumizations": [[6, "backend-specific-costumizations"], [42, "backend-specific-costumizations"]], "Simulators": [[6, "simulators"], [42, "simulators"]], "Simulation methods": [[6, "simulation-methods"], [42, "simulation-methods"]], "GPU simulation": [[6, "gpu-simulation"], [42, "gpu-simulation"]], "Simulation precision": [[6, "simulation-precision"], [42, "simulation-precision"]], "Can we simulate noise?": [[6, "can-we-simulate-noise"], [42, "can-we-simulate-noise"]], "Device backend noise model simulations": [[6, "device-backend-noise-model-simulations"], [42, "device-backend-noise-model-simulations"]], "2. Introduction to Quantum Gates and Circuits. Building Circuits with multiple components.": [[6, "introduction-to-quantum-gates-and-circuits-building-circuits-with-multiple-components"], [42, "introduction-to-quantum-gates-and-circuits-building-circuits-with-multiple-components"]], "Quantum Circuit Properties": [[6, "quantum-circuit-properties"], [42, "quantum-circuit-properties"]], "Final Statevector and Unitary": [[6, "final-statevector-and-unitary"], [42, "final-statevector-and-unitary"]], "Setting a custom statevector": [[6, "setting-a-custom-statevector"], [42, "setting-a-custom-statevector"]], "Other components": [[6, "other-components"], [42, "other-components"]], "How to create an operator?": [[6, "how-to-create-an-operator"]], "3. Run on real hardware. Compiling Circuits": [[6, "run-on-real-hardware-compiling-circuits"], [42, "run-on-real-hardware-compiling-circuits"]], "Optimization level": [[6, "optimization-level"], [42, "optimization-level"]], "Send it after class": [[6, "send-it-after-class"], [12, "send-it-after-class"], [42, "id2"], [42, "id4"], [44, "send-it-after-class"], [44, "id1"], [46, "send-it-after-class"], [46, "id1"], [52, "send-it-after-class"], [48, "send-it-after-class"], [48, "id1"], [48, "id2"], [48, "id3"], [48, "id4"], [48, "id6"]], "Hands-on session 4": [[7, "hands-on-session-4"], [45, "hands-on-session-4"]], "Hands-on 5": [[8, "hands-on-5"]], "Quantum Coin Game": [[8, "quantum-coin-game"], [42, "quantum-coin-game"]], "What are the rules of this game ?": [[8, "what-are-the-rules-of-this-game"], [42, "what-are-the-rules-of-this-game"]], "Analogy <a id=\"analogy\"></a>": [[8, "analogy-a-id-analogy-a"]], "Approach <a id=\"approach\"></a>": [[8, "approach-a-id-approach-a"]], "Optimal Strategy": [[8, "optimal-strategy"], [42, "optimal-strategy"]], "Lets play this version using Qiskit": [[8, "lets-play-this-version-using-qiskit"], [42, "lets-play-this-version-using-qiskit"]], "Building the initial circuit": [[8, "building-the-initial-circuit"], [42, "building-the-initial-circuit"]], "Turn 1. Quantum Computer": [[8, "turn-1-quantum-computer"], [42, "turn-1-quantum-computer"]], "Turn 2. Classical Human": [[8, "turn-2-classical-human"], [42, "turn-2-classical-human"]], "Turn 3. Quantum Computer": [[8, "turn-3-quantum-computer"], [42, "turn-3-quantum-computer"]], "Measurement <a id=\"quantum_measurement\"></a>": [[8, "measurement-a-id-quantum-measurement-a"], [42, "measurement-a-id-quantum-measurement-a"]], "QASM_Simulator <a id=\"quantum_qasm\"></a>": [[8, "qasm-simulator-a-id-quantum-qasm-a"], [42, "qasm-simulator-a-id-quantum-qasm-a"]], "Lets see who wins <a id=\"quantum_wins\"></a>": [[8, "lets-see-who-wins-a-id-quantum-wins-a"], [42, "lets-see-who-wins-a-id-quantum-wins-a"]], "Running on Quantum Computer <a id=\"real_qc\"></a>": [[8, "running-on-quantum-computer-a-id-real-qc-a"], [42, "running-on-quantum-computer-a-id-real-qc-a"]], "Conclusion": [[8, "conclusion"], [19, "conclusion"], [42, "conclusion"], [54, "conclusion"]], "Send it after class 1:": [[8, "send-it-after-class-1"]], "Local Reality and the CHSH inequality": [[8, "local-reality-and-the-chsh-inequality"], [42, "local-reality-and-the-chsh-inequality"]], "The CHSH inequality": [[8, "the-chsh-inequality"], [42, "the-chsh-inequality"]], "Hands-on session 5": [[9, "hands-on-session-5"], [47, "hands-on-session-5"]], "Hands-on 6 :": [[10, "hands-on-6"]], "Entanglement Purification: The BBPSSW Protocol": [[10, "entanglement-purification-the-bbpssw-protocol"]], "Part I: Introduction to Entanglement Purification and the BBPSSW Protocol": [[10, "part-i-introduction-to-entanglement-purification-and-the-bbpssw-protocol"]], "1.1 Concept": [[10, "concept"]], "1.2 Quantum Protocol": [[10, "quantum-protocol"]], "Step 1: Creation of a Werner State through Random Bilateral Rotation": [[10, "step-1-creation-of-a-werner-state-through-random-bilateral-rotation"]], "Step 2: Exchange of |\\Psi^-\\rangle and |\\Phi^+\\rangle Components": [[10, "step-2-exchange-of-psi-rangle-and-phi-rangle-components"]], "Step 3: Application of Bilateral CNOT Gate": [[10, "step-3-application-of-bilateral-cnot-gate"]], "Step 4: Measurement of Target Qubits": [[10, "step-4-measurement-of-target-qubits"]], "Step 5: Restoration of Werner State": [[10, "step-5-restoration-of-werner-state"]], "Part II: Implementation of Quantum Circuit for BBPSSW Protocol in Qiskit": [[10, "part-ii-implementation-of-quantum-circuit-for-bbpssw-protocol-in-qiskit"]], "2.1 Basic Code Imports": [[10, "basic-code-imports"]], "2.2 Definitions": [[10, "definitions"]], "2.3 Quantum Register and Quantum Circuit": [[10, "quantum-register-and-quantum-circuit"]], "2.3.1 Quantum Register": [[10, "quantum-register"]], "2.3.2 Circuit for Initialization of Source State": [[10, "circuit-for-initialization-of-source-state"]], "2.3.3 Implementation of Twirl Operation by Random Bilateral Rotation": [[10, "implementation-of-twirl-operation-by-random-bilateral-rotation"]], "2.3.4 Exchange of |\\Psi^-\\rangle and |\\Phi^+\\rangle Components": [[10, "exchange-of-psi-rangle-and-phi-rangle-components"]], "2.3.5 Application of Bilateral CNOT Gate": [[10, "application-of-bilateral-cnot-gate"]], "2.3.6 Concatenation of Quantum Circuits": [[10, "concatenation-of-quantum-circuits"]], "2.4 (Conditional) Quantum State Tomography": [[10, "conditional-quantum-state-tomography"]], "Part III: Simulation of the BBPSSW Protocol using Qiskit": [[10, "part-iii-simulation-of-the-bbpssw-protocol-using-qiskit"]], "3.1 Single Quantum System": [[10, "single-quantum-system"]], "3.2 Ensemble Average": [[10, "ensemble-average"]], "3.2.1 Definition of Auxiliary Functions": [[10, "definition-of-auxiliary-functions"]], "3.2.2 Simulation": [[10, "simulation"]], "Part IV: Including Noise in the Simulation Model": [[10, "part-iv-including-noise-in-the-simulation-model"]], "4.1 IBM Q Account": [[10, "ibm-q-account"]], "4.2 Creation of Noise Model in Qiskit": [[10, "creation-of-noise-model-in-qiskit"]], "4.3 Simulation Including Noise": [[10, "simulation-including-noise"]], "4.3.1 Initial State and Initial Fidelity": [[10, "initial-state-and-initial-fidelity"]], "4.3.2 Entanglement Purification in the Presence of Noise": [[10, "entanglement-purification-in-the-presence-of-noise"]], "Part V: Entanglement Purification on a Real IBM Q Device": [[10, "part-v-entanglement-purification-on-a-real-ibm-q-device"]], "5.1 IBM Q Experience": [[10, "ibm-q-experience"]], "5.1.1 Device Queue, Job Status, and Job Monitor": [[10, "device-queue-job-status-and-job-monitor"]], "5.1.2 Personal Credits": [[10, "personal-credits"]], "5.2 Initial State and Initial Fidelity": [[10, "id1"]], "5.3 Execution of the BBPSSW Protocol on a Real Device": [[10, "execution-of-the-bbpssw-protocol-on-a-real-device"]], "Part VI: Conclusion and Further Ideas": [[10, "part-vi-conclusion-and-further-ideas"]], "References": [[10, "references"], [12, "references"], [14, "references"], [15, "references"], [19, "references"], [19, "id1"], [19, "id2"], [19, "id5"], [19, "id8"], [19, "id9"], [48, "references"]], "Hands-on session 6": [[11, "hands-on-session-6"], [49, "hands-on-session-6"]], "Hands-on 7": [[12, "hands-on-7"]], "Grover\u2019s Algorithm and Amplitude Amplification": [[12, "grover-s-algorithm-and-amplitude-amplification"]], "Grover\u2019s algorithm": [[12, "grover-s-algorithm"], [30, "grover-s-algorithm"], [30, "id1"]], "Running Grover\u2019s algorithm": [[12, "running-grover-s-algorithm"]], "Using the different types of classes as the oracle of Grover": [[12, "using-the-different-types-of-classes-as-the-oracle-of-grover"]], "Amplitude amplification": [[12, "amplitude-amplification"]], "State preparation": [[12, "state-preparation"]], "Full flexibility": [[12, "full-flexibility"]], "Dive into other arguments of Grover": [[12, "dive-into-other-arguments-of-grover"]], "Specifying good_state": [[12, "specifying-good-state"]], "The number of iterations": [[12, "the-number-of-iterations"]], "Applying post_processing": [[12, "applying-post-processing"]], "Grover\u2019s algorithm examples": [[12, "grover-s-algorithm-examples"]], "Finding solutions to 3-SAT problems": [[12, "finding-solutions-to-3-sat-problems"]], "Boolean Logical Expressions": [[12, "boolean-logical-expressions"]], "Quantum Amplitude Estimation": [[12, "quantum-amplitude-estimation"]], "Qiskit\u2019s Amplitude Estimation workflow": [[12, "qiskit-s-amplitude-estimation-workflow"]], "Canonical AE": [[12, "canonical-ae"]], "Iterative Amplitude Estimation": [[12, "iterative-amplitude-estimation"]], "Maximum Likelihood Amplitude Estimation": [[12, "maximum-likelihood-amplitude-estimation"]], "Faster Amplitude Estimation": [[12, "faster-amplitude-estimation"]], "Hands-on session 7": [[13, "hands-on-session-7"], [51, "hands-on-session-7"]], "Portfolio Diversification": [[14, "portfolio-diversification"], [15, "portfolio-diversification"]], "Introduction": [[14, "introduction"], [15, "introduction"], [22, "introduction"]], "The Model": [[14, "the-model"], [15, "the-model"]], "A Hybrid Approach": [[14, "a-hybrid-approach"], [15, "a-hybrid-approach"]], "Construct a binary polynomial optimization": [[14, "construct-a-binary-polynomial-optimization"], [15, "construct-a-binary-polynomial-optimization"]], "Construct the Ising Hamiltonian": [[14, "construct-the-ising-hamiltonian"], [15, "construct-the-ising-hamiltonian"]], "From Hamiltonian to Quadratic Programming (QP) formulation": [[14, "from-hamiltonian-to-quadratic-programming-qp-formulation"], [15, "from-hamiltonian-to-quadratic-programming-qp-formulation"]], "The Implementation": [[14, "the-implementation"], [15, "the-implementation"]], "Classical solution using IBM ILOG CPLEX": [[14, "classical-solution-using-ibm-ilog-cplex"], [15, "classical-solution-using-ibm-ilog-cplex"]], "Quantum Computing with IBM Q": [[14, "quantum-computing-with-ibm-q"], [15, "quantum-computing-with-ibm-q"]], "Step 1": [[14, "step-1"], [15, "step-1"], [74, "step-1"], [74, "id1"]], "Step 2": [[14, "step-2"], [15, "step-2"], [74, "step-2"], [74, "id3"]], "Step 3": [[14, "step-3"], [15, "step-3"], [74, "step-3"]], "Step 4": [[14, "step-4"], [15, "step-4"], [74, "step-4"], [74, "id5"]], "Step 5": [[14, "step-5"], [15, "step-5"]], "Hands-on 8": [[15, "hands-on-8"]], "One qubit QARBM": [[15, "one-qubit-qarbm"]], "Do it yourself": [[15, "do-it-yourself"]], "Hands-on session 8": [[18, "hands-on-session-8"], [53, "hands-on-session-8"]], "Quantum information with polarized light": [[19, "quantum-information-with-polarized-light"]], "Manipulating polarization degrees of freedom": [[19, "manipulating-polarization-degrees-of-freedom"]], "Half-wave plates": [[19, "half-wave-plates"]], "Polarizing beamsplitters": [[19, "polarizing-beamsplitters"]], "Mapping to Strawberry Fields gates": [[19, "mapping-to-strawberry-fields-gates"]], "Simulating the generation of event-ready photon pairs": [[19, "simulating-the-generation-of-event-ready-photon-pairs"]], "Postselection": [[19, "postselection"]], "Quantum teleportation": [[19, "quantum-teleportation"]], "Background theory": [[19, "background-theory"]], "Importing Strawberry Fields": [[19, "importing-strawberry-fields"]], "Program initialization": [[19, "program-initialization"]], "Circuit construction": [[19, "circuit-construction"]], "Executing the program": [[19, "executing-the-program"]], "Results and visualization": [[19, "results-and-visualization"]], "Full program": [[19, "full-program"]], "Footnotes": [[19, "footnotes"]], "CV quantum gate visualizations": [[19, "cv-quantum-gate-visualizations"]], "Vacuum State": [[19, "vacuum-state"]], "Squeezing Gate": [[19, "squeezing-gate"]], "Rotation Gate": [[19, "rotation-gate"]], "Displacement Gate": [[19, "displacement-gate"]], "Cubic Phase Gate": [[19, "cubic-phase-gate"]], "Kerr Gate": [[19, "kerr-gate"]], "Gate teleportation": [[19, "gate-teleportation"]], "Implementation": [[19, "implementation"], [19, "id3"], [19, "id6"]], "Code": [[19, "code"], [19, "id4"], [19, "id7"]], "Gaussian cloning": [[19, "gaussian-cloning"]], "Circuit analysis": [[19, "circuit-analysis"]], "Coherent average fidelity": [[19, "coherent-average-fidelity"]], "Displaced squeezed states": [[19, "displaced-squeezed-states"]], "Hamiltonian simulation": [[19, "hamiltonian-simulation"]], "Time-domain photonic circuits": [[19, "time-domain-photonic-circuits"]], "An instruction set for the time-domain architecture": [[19, "an-instruction-set-for-the-time-domain-architecture"]], "Reducing simulation resources: shifting modes": [[19, "reducing-simulation-resources-shifting-modes"]], "The TDMProgram": [[19, "the-tdmprogram"]], "A single-loop architecture": [[19, "a-single-loop-architecture"]], "Generation of EPR states": [[19, "generation-of-epr-states"]], "Generating GHZ states": [[19, "generating-ghz-states"]], "Conclusion and final remarks": [[19, "conclusion-and-final-remarks"]], "Lecture 1 online material": [[21, "lecture-1-online-material"]], "Lecture 2 online material": [[21, "lecture-2-online-material"]], "Lecture 3 online material": [[21, "lecture-3-online-material"]], "Lecture 4 online material": [[21, "lecture-4-online-material"]], "Ch1-Lecture 1": [[22, "ch1-lecture-1"]], "Jupyter": [[22, "jupyter"]], "Markdown and LATEX": [[22, "markdown-and-latex"]], "Anaconda": [[22, "anaconda"]], "Exercise 1:": [[22, "exercise-1"]], "IBM Quantum Cloud": [[22, "ibm-quantum-cloud"]], "Some useful links": [[22, "some-useful-links"]], "Orthonormal bases": [[22, "orthonormal-bases"]], "Linear maps.": [[22, "linear-maps"]], "Send it after class 5": [[22, "send-it-after-class-5"], [23, "send-it-after-class-5"], [24, "send-it-after-class-5"], [28, "send-it-after-class-5"], [29, "send-it-after-class-5"], [30, "send-it-after-class-5"], [30, "id10"]], "Ch1-Lecture 2": [[23, "ch1-lecture-2"]], "The Image, Rank and Null space of a matrix": [[23, "the-image-rank-and-null-space-of-a-matrix"]], "Answer": [[23, "answer"], [23, "id1"], [23, "id2"], [23, "id3"], [23, "id4"], [23, "id5"], [61, "answer"], [61, "id1"], [61, "id2"], [63, "answer"], [63, "id1"], [63, "id2"], [63, "id3"], [63, "id4"], [63, "id5"], [63, "id6"], [63, "id7"], [63, "id8"], [63, "id9"], [65, "answer"], [65, "id1"], [65, "id2"], [65, "id3"], [65, "id4"], [65, "id5"], [65, "id6"], [65, "id7"], [65, "id9"], [65, "id10"], [67, "answer"], [67, "id1"], [67, "id2"], [67, "id5"], [69, "answer"], [69, "id1"], [69, "id2"], [69, "id3"], [69, "id4"], [69, "id5"], [69, "id6"], [69, "id7"], [69, "id8"], [70, "answer"], [70, "id1"], [70, "id2"], [70, "id3"], [70, "id4"], [70, "id5"], [70, "id6"], [70, "id7"], [70, "id8"], [72, "answer"], [72, "id1"], [72, "id2"], [72, "id3"], [72, "id4"], [74, "answer"], [74, "id2"], [74, "id4"], [74, "id6"], [74, "id8"], [74, "id19"]], "Matrix Operations": [[23, "matrix-operations"]], "Outer products.": [[23, "outer-products"]], "Eigenvalues and eigenvectors": [[23, "eigenvalues-and-eigenvectors"]], "The four postulates of Quantum Mechanics": [[23, "the-four-postulates-of-quantum-mechanics"], [42, "the-four-postulates-of-quantum-mechanics"]], "Postulate 1: Individual Quantum systems": [[23, "postulate-1-individual-quantum-systems"], [42, "postulate-1-individual-quantum-systems"]], "Postulate 2: Quantum operations": [[23, "postulate-2-quantum-operations"], [42, "postulate-2-quantum-operations"]], "X Gate": [[23, "x-gate"], [42, "x-gate"]], "Send it after class 6": [[23, "send-it-after-class-6"], [24, "send-it-after-class-6"], [28, "send-it-after-class-6"], [29, "send-it-after-class-6"]], "Ch1-Lecture 3": [[24, "ch1-lecture-3"]], "Z Gate": [[24, "z-gate"], [42, "z-gate"]], "Hadamard Gate": [[24, "hadamard-gate"], [42, "hadamard-gate"]], "Quantum Circuits": [[24, "quantum-circuits"], [42, "quantum-circuits"]], "Postulate 3: Composite Quantum Systems": [[24, "postulate-3-composite-quantum-systems"], [42, "postulate-3-composite-quantum-systems"]], "Quantum Entanglement": [[24, "quantum-entanglement"], [42, "quantum-entanglement"]], "Two-qubit quantum gates.": [[24, "two-qubit-quantum-gates"], [42, "two-qubit-quantum-gates"]], "Send it after class 7": [[24, "send-it-after-class-7"], [29, "send-it-after-class-7"]], "Ch1-Lecture 4": [[25, "ch1-lecture-4"]], "Delayed choice quantum eraser": [[25, "delayed-choice-quantum-eraser"], [42, "delayed-choice-quantum-eraser"]], "Postulate 4: Measurement": [[25, "postulate-4-measurement"], [42, "postulate-4-measurement"]], "Projective Measurements.": [[25, "projective-measurements"], [42, "projective-measurements"]], "Quantum Teleportation": [[25, "quantum-teleportation"], [42, "quantum-teleportation"]], "Lecture 5 online material": [[26, "lecture-5-online-material"]], "Lecture 6 online material": [[26, "lecture-6-online-material"]], "Lecture 7 online material": [[26, "lecture-7-online-material"]], "Ch2-Lecture 5": [[27, "ch2-lecture-5"]], "No cloning theorem": [[27, "no-cloning-theorem"]], "Quantum entanglement": [[27, "quantum-entanglement"]], "Bipartite entanglement": [[27, "bipartite-entanglement"]], "The Schmidt decomposition.": [[27, "the-schmidt-decomposition"]], "Density Matrices": [[27, "density-matrices"]], "The partial trace operation": [[27, "the-partial-trace-operation"]], "Using partial trace to detect entanglement": [[27, "using-partial-trace-to-detect-entanglement"]], "How the postulates of quantum mechanics apply to density operators": [[27, "how-the-postulates-of-quantum-mechanics-apply-to-density-operators"]], "Ch2-Lecture 6": [[28, "ch2-lecture-6"]], "Non-Local Games": [[28, "non-local-games"]], "Does entanglement allow superluminal signalling?": [[28, "does-entanglement-allow-superluminal-signalling"]], "Non-local games": [[28, "id1"]], "The CHSH game": [[28, "the-chsh-game"]], "Limits on classical strategies": [[28, "limits-on-classical-strategies"]], "A quantum strategy": [[28, "a-quantum-strategy"]], "The magic square game": [[28, "the-magic-square-game"]], "Ch2-Lecture 7": [[29, "ch2-lecture-7"]], "Entropy and Entanglement Distillation": [[29, "entropy-and-entanglement-distillation"]], "Entropy": [[29, "entropy"]], "Shannon entropy": [[29, "shannon-entropy"]], "Von Neumann Entropy": [[29, "von-neumann-entropy"]], "Properties of the von Neumann entropy": [[29, "properties-of-the-von-neumann-entropy"]], "Quantifying entanglement in composite quantum systems": [[29, "quantifying-entanglement-in-composite-quantum-systems"]], "Send it after class 8": [[29, "send-it-after-class-8"]], "Entanglement distillation": [[29, "entanglement-distillation"]], "Send it after class 9": [[29, "send-it-after-class-9"]], "Ch3-Lecture 8": [[30, "ch3-lecture-8"]], "Functions as oracles": [[30, "functions-as-oracles"]], "The phase kickback trick": [[30, "the-phase-kickback-trick"]], "Grover search": [[30, "grover-search"]], "The unstructured search problem": [[30, "the-unstructured-search-problem"]], "The start state \\left|\\psi\\right\\rangle ": [[30, "the-start-state-left-psi-right-rangle"], [30, "id4"]], "The goal.": [[30, "the-goal"], [30, "id6"]], "Two reflections make a rotation.": [[30, "two-reflections-make-a-rotation"], [30, "id7"]], "The number of iterations required.": [[30, "the-number-of-iterations-required"], [30, "id9"]], "Approximate counting": [[30, "approximate-counting"], [30, "id11"]], "Intuition": [[30, "intuition"], [30, "id12"]], "Quantizing the basic idea.": [[30, "quantizing-the-basic-idea"], [30, "id13"]], "Ch3-Lecture 9": [[31, "ch3-lecture-9"]], "Information Encoding": [[31, "information-encoding"]], "Basis Encoding": [[31, "basis-encoding"]], "Computing in Basis Encoding": [[31, "computing-in-basis-encoding"]], "Sampling from a Qubit": [[31, "sampling-from-a-qubit"]], "Preparing Superpositions of Inputs": [[31, "preparing-superpositions-of-inputs"], [31, "id1"]], "Phys 437": [[32, "phys-437"]], "General information": [[32, "general-information"], [57, "general-information"]], "Theory sessions": [[32, "theory-sessions"]], "Text": [[32, null]], "Online material": [[32, null], [32, null]], "Hands-on sessions": [[32, "hands-on-sessions"]], "Pre-compiled Jupyter": [[32, null]], "Forms": [[33, "forms"], [84, "forms"]], "Syllabus": [[34, "syllabus"], [58, "syllabus"], [86, "syllabus"]], "Course syllabus": [[34, "id1"], [58, "id1"], [86, "id1"]], "Tentative Course Contents": [[34, "tentative-course-contents"], [58, "tentative-course-contents"], [86, "tentative-course-contents"]], "Chapter 1: Review & Mathematical Foundation": [[34, "chapter-1-review-mathematical-foundation"], [58, "chapter-1-review-mathematical-foundation"]], "Chapter 2: Quantum Circuit Diagrams": [[34, "chapter-2-quantum-circuit-diagrams"], [58, "chapter-2-quantum-circuit-diagrams"]], "Chapter 3: Complexity Theory; Entropy and Entanglement Distillation": [[34, "chapter-3-complexity-theory-entropy-and-entanglement-distillation"], [58, "chapter-3-complexity-theory-entropy-and-entanglement-distillation"]], "Chapter 4: The Deutsch-Josza and Berstein-Vazirani algorithms": [[34, "chapter-4-the-deutsch-josza-and-berstein-vazirani-algorithms"], [58, "chapter-4-the-deutsch-josza-and-berstein-vazirani-algorithms"]], "Chapter 5: Strategies of Input Encoding": [[34, "chapter-5-strategies-of-input-encoding"], [58, "chapter-5-strategies-of-input-encoding"]], "Chapter 6: Simon\u2019s algorithm and applications to cryptography": [[34, "chapter-6-simons-algorithm-and-applications-to-cryptography"], [58, "chapter-6-simons-algorithm-and-applications-to-cryptography"]], "Chapter 7: The Quantum Fourier Transform": [[34, "chapter-7-the-quantum-fourier-transform"], [58, "chapter-7-the-quantum-fourier-transform"]], "Chapter 8: Shor\u2019s quantum factoring algorithm": [[34, "chapter-8-shors-quantum-factoring-algorithm"], [58, "chapter-8-shors-quantum-factoring-algorithm"]], "Chapter 9: Variational Circuits as Machine Learning Models (time permitting)": [[34, "chapter-9-variational-circuits-as-machine-learning-models-time-permitting"], [58, "chapter-9-variational-circuits-as-machine-learning-models-time-permitting"]], "Hands-On sessions": [[34, "hands-on-sessions"], [58, "hands-on-sessions"], [86, "hands-on-sessions"]], "Midterm": [[34, "midterm"], [58, "midterm"]], "Term projects": [[34, "term-projects"], [58, "term-projects"], [86, "term-projects"]], "Textbooks": [[34, "textbooks"], [58, "textbooks"], [86, "textbooks"]], "Theory Content:": [[34, "theory-content"], [58, "theory-content"]], "Lab Content:": [[34, "lab-content"], [58, "lab-content"]], "Optional content (time permitting):": [[34, "optional-content-time-permitting"], [58, "optional-content-time-permitting"]], "Term Project Ideas": [[35, "term-project-ideas"], [87, "term-project-ideas"]], "Preliminary setup": [[38, "preliminary-setup"]], "Anaconda/Miniconda": [[38, "anaconda-miniconda"]], "Aptainer (formerly Singularity)": [[38, "aptainer-formerly-singularity"]], "Basic regression: Predict fuel efficiency": [[38, "basic-regression-predict-fuel-efficiency"]], "The Auto MPG dataset": [[38, "the-auto-mpg-dataset"]], "Get the data": [[38, "get-the-data"]], "Clean the data": [[38, "clean-the-data"]], "Split the data into training and test sets": [[38, "split-the-data-into-training-and-test-sets"]], "Inspect the data": [[38, "inspect-the-data"]], "Split features from labels": [[38, "split-features-from-labels"]], "Normalization": [[38, "normalization"]], "The Normalization layer": [[38, "the-normalization-layer"]], "Linear regression": [[38, "linear-regression"]], "Linear regression with one variable": [[38, "linear-regression-with-one-variable"]], "Linear regression with multiple inputs": [[38, "linear-regression-with-multiple-inputs"]], "Regression with a deep neural network (DNN)": [[38, "regression-with-a-deep-neural-network-dnn"]], "Regression using a DNN and a single input": [[38, "regression-using-a-dnn-and-a-single-input"]], "Regression using a DNN and multiple inputs": [[38, "regression-using-a-dnn-and-multiple-inputs"]], "Performance": [[38, "performance"]], "Make predictions": [[38, "make-predictions"]], "Takeaways:": [[38, "takeaways"]], "DO IT YOURSELF 1:": [[38, "do-it-yourself-1"]], "Overfit and underfit": [[38, "overfit-and-underfit"]], "Setup": [[38, "setup"], [38, "id2"]], "The Higgs dataset": [[38, "the-higgs-dataset"]], "Demonstrate overfitting": [[38, "demonstrate-overfitting"]], "Training procedure": [[38, "training-procedure"]], "Tiny model": [[38, "tiny-model"]], "Small model": [[38, "small-model"]], "Medium model": [[38, "medium-model"]], "Large model": [[38, "large-model"]], "Plot the training and validation losses": [[38, "plot-the-training-and-validation-losses"]], "View in TensorBoard": [[38, "view-in-tensorboard"], [38, "id1"]], "Strategies to prevent overfitting": [[38, "strategies-to-prevent-overfitting"]], "Add weight regularization": [[38, "add-weight-regularization"]], "More info": [[38, "more-info"]], "Add dropout": [[38, "add-dropout"]], "Combined L2 + dropout": [[38, "combined-l2-dropout"]], "Takeaway": [[38, "takeaway"]], "DO IT YOURSELF 2:": [[38, "do-it-yourself-2"]], "Save and load models": [[38, "save-and-load-models"]], "Options": [[38, "options"]], "Installs and imports": [[38, "installs-and-imports"]], "Get an example dataset": [[38, "get-an-example-dataset"]], "Define a model": [[38, "define-a-model"]], "Save checkpoints during training": [[38, "save-checkpoints-during-training"]], "Checkpoint callback usage": [[38, "checkpoint-callback-usage"]], "Checkpoint callback options": [[38, "checkpoint-callback-options"]], "What are these files?": [[38, "what-are-these-files"]], "Manually save weights": [[38, "manually-save-weights"]], "Save the entire model": [[38, "save-the-entire-model"]], "SavedModel format": [[38, "savedmodel-format"]], "HDF5 format": [[38, "hdf5-format"]], "Saving custom objects": [[38, "saving-custom-objects"]], "TFP Probabilistic Layers: Regression": [[38, "tfp-probabilistic-layers-regression"]], "Dependencies & Prerequisites": [[38, "dependencies-prerequisites"]], "Make things Fast!": [[38, "make-things-fast"]], "Motivation": [[38, "motivation"]], "Case 1: No Uncertainty": [[38, "case-1-no-uncertainty"]], "Case 2: Aleatoric Uncertainty": [[38, "case-2-aleatoric-uncertainty"]], "Case 3: Epistemic Uncertainty": [[38, "case-3-epistemic-uncertainty"]], "Case 4: Aleatoric & Epistemic Uncertainty": [[38, "case-4-aleatoric-epistemic-uncertainty"]], "Case 5: Functional Uncertainty": [[38, "case-5-functional-uncertainty"]], "Linux Terminal Primer": [[59, "linux-terminal-primer"]], "Getting your computer ready": [[59, "getting-your-computer-ready"]], "Conda": [[59, "conda"]], "Qiskit": [[59, "qiskit"]], "Dilhan": [[59, "dilhan"]], "Connecting to Dilhan": [[59, "connecting-to-dilhan"]], "Using the ssh-key": [[59, "using-the-ssh-key"]], "Which software is installed on Dilhan": [[59, "which-software-is-installed-on-dilhan"]], "Using Qiskit with Conda in Dilhan": [[59, "using-qiskit-with-conda-in-dilhan"]], "Batch system": [[59, "batch-system"]], "Creating and submitting a job script": [[59, "creating-and-submitting-a-job-script"]], "Finding queuing information with squeue": [[59, "finding-queuing-information-with-squeue"]], "Stopping jobs with scancel": [[59, "stopping-jobs-with-scancel"]], "Helper scripts": [[59, "helper-scripts"]], "Example Slurm job script for running a qiskit python script": [[59, "example-slurm-job-script-for-running-a-qiskit-python-script"]], "Learning Outcomes": [[61, "learning-outcomes"], [63, "learning-outcomes"], [65, "learning-outcomes"], [67, "learning-outcomes"], [69, "learning-outcomes"], [70, "learning-outcomes"], [72, "learning-outcomes"], [74, "learning-outcomes"]], "Single Qubit Quantum Gates": [[61, "single-qubit-quantum-gates"]], "Exercise 1": [[61, "exercise-1"], [63, "exercise-1"], [65, "exercise-1"], [67, "exercise-1"], [69, "exercise-1"], [70, "exercise-1"], [72, "exercise-1"], [74, "exercise-1"]], "Exercise 2": [[61, "exercise-2"], [63, "exercise-2"], [65, "exercise-2"], [67, "exercise-2"], [69, "exercise-2"], [70, "exercise-2"], [72, "exercise-2"], [74, "exercise-2"]], "Exercise 3": [[61, "exercise-3"], [63, "exercise-3"], [65, "exercise-3"], [67, "exercise-3"], [69, "exercise-3"], [70, "exercise-3"], [72, "exercise-3"], [74, "exercise-3"]], "Using Qiskit": [[61, "using-qiskit"]], "Important Remarks about Backends": [[61, "important-remarks-about-backends"]], "Important Remarks about Results": [[61, "important-remarks-about-results"]], "Congratulations": [[61, "congratulations"]], "Exercise 4": [[61, "exercise-4"], [63, "exercise-4"], [65, "exercise-4"], [67, "exercise-4"], [69, "exercise-4"], [70, "exercise-4"], [72, "exercise-4"], [74, "exercise-4"]], "Exercise 5": [[61, "exercise-5"], [63, "exercise-5"], [65, "exercise-5"], [67, "exercise-5"], [69, "exercise-5"], [70, "exercise-5"], [72, "exercise-5"], [74, "exercise-5"]], "Exercise 6": [[61, "exercise-6"], [63, "exercise-6"], [65, "exercise-6"], [67, "exercise-6"], [69, "exercise-6"], [70, "exercise-6"], [74, "exercise-6"]], "Exercise 7": [[61, "exercise-7"], [63, "exercise-7"], [65, "exercise-7"], [67, "exercise-7"], [69, "exercise-7"], [70, "exercise-7"], [74, "exercise-7"]], "Exercise 8": [[61, "exercise-8"], [63, "exercise-8"], [65, "exercise-8"], [67, "exercise-8"], [69, "exercise-8"], [70, "exercise-8"], [74, "exercise-8"]], "Exercise 9": [[61, "exercise-9"], [63, "exercise-9"], [65, "exercise-9"], [69, "exercise-9"], [70, "exercise-9"], [74, "exercise-9"]], "Non-standard Basis": [[63, "non-standard-basis"]], "Exercise 3.2": [[63, "exercise-3-2"]], "Multiple Qubit States": [[63, "multiple-qubit-states"]], "Answers": [[63, "answers"], [67, "answers"], [67, "id3"], [67, "id4"]], "Product and Entangled States": [[63, "product-and-entangled-states"]], "CNOT Gate": [[63, "cnot-gate"]], "Exercise 10": [[63, "exercise-10"], [74, "exercise-10"]], "Exercise 11": [[63, "exercise-11"]], "Exercise 12": [[63, "exercise-12"]], "Exercise 13": [[63, "exercise-13"]], "Chapter 1": [[76, "chapter-1"]], "Chapter 2": [[77, "chapter-2"]], "Chapter 3": [[78, "chapter-3"]], "Chapter 4": [[79, "chapter-4"]], "Chapter 5": [[80, "chapter-5"]], "Chapter 6": [[81, "chapter-6"]], "Chapter 7": [[82, "chapter-7"]], "Lectures and Lecture notes": [[83, "lectures-and-lecture-notes"]], "Course schedule": [[85, "course-schedule"]], "Schedule (last updated 07/04/2021)": [[85, "id1"]], "Theoretical sections": [[86, "theoretical-sections"]], "Archives": [[88, "archives"]], "PHYS 343": [[88, "phys-343"]], "Courses": [[89, "courses"]], "Using IBM quantum Cloud": [[90, "using-ibm-quantum-cloud"]], "Invitation e-mail": [[90, "invitation-e-mail"]], "IBM Quantum Computing cloud": [[90, "ibm-quantum-computing-cloud"]], "IBM Quantum Lab": [[90, "ibm-quantum-lab"]], "HOWTOs": [[91, "howtos"]], "Send it after class :": [[42, "send-it-after-class"]], "1. The Pauli Gates": [[42, "the-pauli-gates"]], "1.1 The X-Gate": [[42, "the-x-gate"]], "1.2 The Y & Z-gates": [[42, "the-y-z-gates"]], "2. Digression: The X, Y & Z-Bases": [[42, "digression-the-x-y-z-bases"]], "3. The Hadamard Gate": [[42, "the-hadamard-gate"]], "5. The P-gate": [[42, "the-p-gate"]], "6. The I, S and T-gates": [[42, "the-i-s-and-t-gates"]], "6.1 The I-gate": [[42, "the-i-gate"]], "6.2 The S-gates": [[42, "the-s-gates"]], "6.3 The T-gate": [[42, "the-t-gate"]], "7. The U-gate": [[42, "the-u-gate"]], "Analogy": [[42, "analogy"]], "Approach": [[42, "approach"]], "Send it after class:": [[42, "id3"], [54, "send-it-after-class"], [40, "send-it-after-class"], [40, "id1"]], "Preliminaries": [[44, "preliminaries"]], "How to embed data into a quantum state": [[44, "how-to-embed-data-into-a-quantum-state"]], "Different embedding types": [[44, "different-embedding-types"]], "1. Discrete data, represented as binary or integer values": [[44, "discrete-data-represented-as-binary-or-integer-values"]], "2. Continuous data, represented as floating-point values": [[44, "continuous-data-represented-as-floating-point-values"]], "3. Continuous data, represented as complex values": [[44, "continuous-data-represented-as-complex-values"]], "Templates": [[44, "templates"]], "Embedding templates": [[44, "embedding-templates"]], "Using the Ising model for embedding": [[44, "using-the-ising-model-for-embedding"]], "Training the features": [[44, "training-the-features"]], "Simulating Nature": [[46, "simulating-nature"]], "Unitaries": [[46, "unitaries"]], "Hamiltonians": [[46, "hamiltonians"]], "Approximating exponentials": [[46, "approximating-exponentials"]], "send it after class": [[46, "id2"], [46, "id3"], [46, "id4"]], "Parameter-shift rules": [[50, "parameter-shift-rules"]], "A more technical explanation": [[50, "a-more-technical-explanation"]], "A single parameterized gate": [[50, "a-single-parameterized-gate"]], "Multiple parameterized gates": [[50, "multiple-parameterized-gates"]], "Pauli gate example": [[50, "pauli-gate-example"]], "Gaussian gate example": [[50, "gaussian-gate-example"]], "Quantum gradients with backpropagation": [[50, "quantum-gradients-with-backpropagation"]], "The parameter-shift rule example": [[50, "the-parameter-shift-rule-example"]], "Benchmarking": [[50, "benchmarking"], [50, "id1"]], "Backpropagation example": [[50, "backpropagation-example"]], "Send it after class: Time comparison": [[50, "send-it-after-class-time-comparison"]], "Question 1: Quantum natural gradient": [[36, "question-1-quantum-natural-gradient"]], "Background": [[36, "background"], [36, "id1"], [52, "background"], [48, "background"], [48, "id5"]], "The natural gradient": [[36, "the-natural-gradient"]], "The quantum analog": [[36, "the-quantum-analog"]], "Block-diagonal metric tensor": [[36, "block-diagonal-metric-tensor"]], "Tasks": [[36, "tasks"], [36, "id2"]], "Task 1": [[36, "task-1"], [36, "id3"]], "Task 2": [[36, "task-2"], [36, "id4"]], "Task 3": [[36, "task-3"]], "Note": [[36, "note"]], "Question 2: Quantum transfer learning": [[36, "question-2-quantum-transfer-learning"]], "Classical-to-quantum transfer learning": [[36, "classical-to-quantum-transfer-learning"]], "General setup": [[36, "general-setup"]], "Setting of the main hyper-parameters of the model": [[36, "setting-of-the-main-hyper-parameters-of-the-model"]], "Dataset loading": [[36, "dataset-loading"]], "Variational quantum circuit": [[36, "variational-quantum-circuit"]], "Dressed quantum circuit": [[36, "dressed-quantum-circuit"]], "Hybrid classical-quantum model": [[36, "hybrid-classical-quantum-model"]], "Training and results": [[36, "training-and-results"]], "Visualizing the model predictions": [[36, "visualizing-the-model-predictions"]], "Training and evaluating quantum kernels": [[52, "training-and-evaluating-quantum-kernels"]], "What are kernel methods?": [[52, "what-are-kernel-methods"]], "A toy problem": [[52, "a-toy-problem"]], "Defining a Quantum Embedding Kernel": [[52, "defining-a-quantum-embedding-kernel"]], "Using the Quantum Embedding Kernel for predictions": [[52, "using-the-quantum-embedding-kernel-for-predictions"]], "Training the Quantum Embedding Kernel": [[52, "training-the-quantum-embedding-kernel"]], "How to approximate a classical kernel with a quantum computer": [[52, "how-to-approximate-a-classical-kernel-with-a-quantum-computer"]], "Kernel-based Machine Learning": [[52, "kernel-based-machine-learning"]], "Shift-invariant kernels": [[52, "shift-invariant-kernels"]], "Warm up: Implementing the Gaussian kernel": [[52, "warm-up-implementing-the-gaussian-kernel"]], "Fourier analysis of the Gaussian kernel": [[52, "fourier-analysis-of-the-gaussian-kernel"]], "Designing a suitable QK": [[52, "designing-a-suitable-qk"]], "Computing the QK function on a quantum device": [[52, "computing-the-qk-function-on-a-quantum-device"]], "How to find the amplitudes emulating a Gaussian kernel": [[52, "how-to-find-the-amplitudes-emulating-a-gaussian-kernel"]], "Finding the solution": [[52, "finding-the-solution"]], "Visualizing the solution": [[52, "visualizing-the-solution"]], "Kernel-based training of quantum models with scikit-learn": [[52, "kernel-based-training-of-quantum-models-with-scikit-learn"]], "Kernel-based training": [[52, "kernel-based-training"]], "A similar example using variational training": [[52, "a-similar-example-using-variational-training"]], "Which method scales best?": [[52, "which-method-scales-best"]], "Quantum Feature Maps with PennyLane": [[54, "quantum-feature-maps-with-pennylane"]], "Main idea of feature map": [[54, "main-idea-of-feature-map"]], "Circles": [[54, "circles"]], "XOR": [[54, "xor"]], "Transformation of random points": [[54, "transformation-of-random-points"]], "2d order Quantum Feature Map": [[54, "d-order-quantum-feature-map"]], "Implementation in PennyLane": [[54, "implementation-in-pennylane"]], "Circuit": [[54, "circuit"]], "Solution of circles problem": [[54, "solution-of-circles-problem"]], "Send it after class: YY-projection": [[54, "send-it-after-class-yy-projection"]], "Solution of XOR problem": [[54, "solution-of-xor-problem"]], "Machine learning for quantum many-body problems": [[54, "machine-learning-for-quantum-many-body-problems"]], "Building the 2D Heisenberg Model": [[54, "building-the-2d-heisenberg-model"]], "Constructing Classical Shadows": [[54, "constructing-classical-shadows"]], "Training Classical Machine Learning Models": [[54, "training-classical-machine-learning-models"]], "3-qubit Ising model in PyTorch": [[56, "qubit-ising-model-in-pytorch"]], "PennyLane implementation": [[56, "pennylane-implementation"]], "Sanity check": [[56, "sanity-check"]], "Random initialization": [[56, "random-initialization"]], "Optimization": [[56, "optimization"]], "Local minimum": [[56, "local-minimum"]], "Send it after class |": [[56, "send-it-after-class"]], "Two Types of Quantum Computers": [[56, "two-types-of-quantum-computers"]], "Gate Model": [[56, "gate-model"]], "Quantum Annealing": [[56, "quantum-annealing"]], "Formulating Problem for QA": [[56, "formulating-problem-for-qa"]], "Implementation Using Quantum Annealer": [[56, "implementation-using-quantum-annealer"]], "Preparation": [[56, "preparation"]], "Problem Setting": [[56, "problem-setting"]], "Dependencies": [[56, "dependencies"]], "Cost and Constraint Function": [[56, "cost-and-constraint-function"]], "1. Import historical stock price data from the internet": [[56, "import-historical-stock-price-data-from-the-internet"]], "2. Prepare data by cleansing and feature engineering": [[56, "prepare-data-by-cleansing-and-feature-engineering"]], "Feature Engineering": [[56, "feature-engineering"]], "Constraint Terms": [[56, "constraint-terms"]], "Cost Functions": [[56, "cost-functions"]], "Prepare QUBO": [[56, "prepare-qubo"]], "4. Sample optimisation results from a D-Wave solver": [[56, "sample-optimisation-results-from-a-d-wave-solver"]], "5. Sample optimisation results from a simulated quantum annealing solver": [[56, "sample-optimisation-results-from-a-simulated-quantum-annealing-solver"]], "Hands-on session 9": [[20, "hands-on-session-9"], [55, "hands-on-session-9"]], "Hands-on session 10": [[39, "hands-on-session-10"]], "Introduction to Geometric Quantum Machine Learning": [[40, "introduction-to-geometric-quantum-machine-learning"]], "Representation theory in circuits": [[40, "representation-theory-in-circuits"]], "Categories of Geometric Deep Learning": [[40, "categories-of-geometric-deep-learning"]], "Noughts and Crosses": [[40, "noughts-and-crosses"]], "The Quantum Graph Recurrent Neural Network": [[40, "the-quantum-graph-recurrent-neural-network"]], "The Idea": [[40, "the-idea"]], "Using the QGRNN": [[40, "using-the-qgrnn"]], "Learning an Ising Model with the QGRNN": [[40, "learning-an-ising-model-with-the-qgrnn"]], "Getting Started": [[40, "getting-started"]], "Preparing Quantum Data": [[40, "preparing-quantum-data"]], "Learning the Hamiltonian": [[40, "learning-the-hamiltonian"]], "Quantum models as Fourier series": [[48, "quantum-models-as-fourier-series"]], "Goal of this demonstration": [[48, "goal-of-this-demonstration"]], "Part I: Fitting Fourier series with serial Pauli-rotation encoding": [[48, "part-i-fitting-fourier-series-with-serial-pauli-rotation-encoding"]], "Define a target function": [[48, "define-a-target-function"]], "Define the serial quantum model": [[48, "define-the-serial-quantum-model"]], "Fit the model to the target": [[48, "fit-the-model-to-the-target"]], "Part II: Fitting Fourier series with parallel Pauli-rotation encoding": [[48, "part-ii-fitting-fourier-series-with-parallel-pauli-rotation-encoding"]], "Define the parallel quantum model": [[48, "define-the-parallel-quantum-model"]], "Send it after class: Training the model": [[48, "send-it-after-class-training-the-model"]], "Part III: Sampling Fourier coefficients": [[48, "part-iii-sampling-fourier-coefficients"]], "Define your quantum model": [[48, "define-your-quantum-model"]], "Continuous-variable model": [[48, "continuous-variable-model"]], "Data-reuploading classifier": [[48, "data-reuploading-classifier"]], "Transforming quantum states using unitary operations": [[48, "transforming-quantum-states-using-unitary-operations"]], "Data loading using unitaries": [[48, "data-loading-using-unitaries"]], "Model parameters with data re-uploading": [[48, "model-parameters-with-data-re-uploading"]], "The cost function and \u201cnonlinear collapse\u201d": [[48, "the-cost-function-and-nonlinear-collapse"]], "Multiple qubits, entanglement and Deep Neural Networks": [[48, "multiple-qubits-entanglement-and-deep-neural-networks"]], "\u201cTalk is cheap. Show me the code.\u201d - Linus Torvalds": [[48, "talk-is-cheap-show-me-the-code-linus-torvalds"]], "Simple classifier with data reloading and fidelity loss": [[48, "simple-classifier-with-data-reloading-and-fidelity-loss"]], "Utility functions for testing and creating batches": [[48, "utility-functions-for-testing-and-creating-batches"]], "Train a quantum classifier on the circle dataset": [[48, "train-a-quantum-classifier-on-the-circle-dataset"]], "Results": [[48, "results"]], "Phys 710": [[57, "phys-710"]], "Lecture Notes / Recordings": [[57, "lecture-notes-recordings"]], "Homeworks": [[57, "homeworks"]], "Hands on sessions": [[57, null]], "1. Exploring the CNOT-Gate <a id=\"exploring-cnot\"></a>": [[65, "exploring-the-cnot-gate-a-id-exploring-cnot-a"]], "2. Phase Kickback <a id=\"kickback\"></a>": [[65, "phase-kickback-a-id-kickback-a"]], "2.1 Explaining the CNOT Circuit Identity <a id=\"explaining-identity\"></a>": [[65, "explaining-the-cnot-circuit-identity-a-id-explaining-identity-a"]], "Exercise 2.5": [[65, "exercise-2-5"]], "2.2 Kickback with the T-gate <a id=\"kickback-t-gate\"></a>": [[65, "kickback-with-the-t-gate-a-id-kickback-t-gate-a"]], "Quick Exercises:": [[65, "quick-exercises"]], "1. Making a Controlled-Z from a CNOT <a id=\"c-from-cnot\"></a>": [[65, "making-a-controlled-z-from-a-cnot-a-id-c-from-cnot-a"]], "2. Swapping Qubits <a id=\"swapping\"></a>": [[65, "swapping-qubits-a-id-swapping-a"]], "Quick Exercise:": [[65, "quick-exercise"]], "3. Controlled Rotations <a id=\"controlled-rotations\"></a>": [[65, "controlled-rotations-a-id-controlled-rotations-a"]], "6. References <a id=\"references\"></a>": [[65, "references-a-id-references-a"]], "Outer Product": [[67, "outer-product"]], "Measuring Time Complexity": [[67, "measuring-time-complexity"]], "Deutsch\u2019s Algorithm": [[67, "deutsch-s-algorithm"]], "SWAP Gate": [[69, "swap-gate"], [70, "swap-gate"]], "No Clonning Theorem": [[69, "no-clonning-theorem"], [70, "no-clonning-theorem"]], "Density Matrix Formalism": [[69, "density-matrix-formalism"], [70, "density-matrix-formalism"]], "Properties of Density Operator": [[69, "properties-of-density-operator"], [70, "properties-of-density-operator"]], "Reduced Density Operator": [[69, "reduced-density-operator"], [70, "reduced-density-operator"]], "Schmidt Decomposition": [[69, "schmidt-decomposition"], [70, "schmidt-decomposition"]], "1. Introduction <a id='introduction'></a>": [[72, "introduction-a-id-introduction-a"]], "1.1 Deutsch-Jozsa Problem <a id='djproblem'> </a>": [[72, "deutsch-jozsa-problem-a-id-djproblem-a"]], "1.2 The Classical Solution <a id='classical-solution'> </a>": [[72, "the-classical-solution-a-id-classical-solution-a"], [72, "id6"]], "1.3 Quantum Solution <a id='quantum-solution'> </a>": [[72, "quantum-solution-a-id-quantum-solution-a"]], "1.4 Why Does This Work? <a id='why-does-this-work'> </a>": [[72, "why-does-this-work-a-id-why-does-this-work-a"]], "2. Worked Example <a id='example'></a>": [[72, "worked-example-a-id-example-a"]], "3. Creating Quantum Oracles <a id='creating-quantum-oracles'> </a>": [[72, "creating-quantum-oracles-a-id-creating-quantum-oracles-a"]], "4. Qiskit Implementation <a id='implementation'></a>": [[72, "qiskit-implementation-a-id-implementation-a"]], "4.1 Constant Oracle <a id='const_oracle'></a>": [[72, "constant-oracle-a-id-const-oracle-a"]], "4.2 Balanced Oracle <a id='balanced_oracle'></a>": [[72, "balanced-oracle-a-id-balanced-oracle-a"]], "4.3 The Full Algorithm <a id='full_alg'></a>": [[72, "the-full-algorithm-a-id-full-alg-a"]], "4.4 Generalised Circuits <a id='general_circs'></a>": [[72, "generalised-circuits-a-id-general-circs-a"]], "5. Experiment with Real Devices <a id='device'></a>": [[72, "experiment-with-real-devices-a-id-device-a"]], "7. References <a id='references'></a>": [[72, "references-a-id-references-a"]], "1. The Bernstein-Vazirani Algorithm<a id='algorithm'></a>": [[72, "the-bernstein-vazirani-algorithm-a-id-algorithm-a"]], "1.1 The Bernstein-Vazirani Problem <a id='bvproblem'> </a>": [[72, "the-bernstein-vazirani-problem-a-id-bvproblem-a"]], "1.3 The Quantum Solution <a id='quantum-solution'> </a>": [[72, "the-quantum-solution-a-id-quantum-solution-a"]], "2. Example <a id='example'></a>": [[72, "example-a-id-example-a"]], "3. Qiskit Implementation <a id='implementation'></a>": [[72, "id7"]], "3a. Experiment with Simulators <a id='simulation'></a>": [[72, "a-experiment-with-simulators-a-id-simulation-a"]], "3b. Experiment with Real Devices <a id='device'></a>": [[72, "b-experiment-with-real-devices-a-id-device-a"]], "5. References <a id='references'></a>": [[72, "id8"]], "1. Overview <a id='overview'></a>": [[74, "overview-a-id-overview-a"]], "2. The Quantum Teleportation Protocol <a id='how'></a>": [[74, "the-quantum-teleportation-protocol-a-id-how-a"]], "3. Simulating the Teleportation Protocol <a id='simulating'></a>": [[74, "simulating-the-teleportation-protocol-a-id-simulating-a"]], "3.1 How Will We Test the Protocol on a Quantum Computer? <a id='testing'></a>": [[74, "how-will-we-test-the-protocol-on-a-quantum-computer-a-id-testing-a"]], "3.2 Using the Statevector Simulator <a id='simulating-sv'></a>": [[74, "using-the-statevector-simulator-a-id-simulating-sv-a"]], "3.3 Using the QASM Simulator <a id='simulating-qs'></a>": [[74, "using-the-qasm-simulator-a-id-simulating-qs-a"]], "4. Understanding Quantum Teleportation <a id=\"understanding-qt\">": [[74, "understanding-quantum-teleportation-a-id-understanding-qt"]], "5. Teleportation on a Real Quantum Computer <a id='real_qc'></a>": [[74, "teleportation-on-a-real-quantum-computer-a-id-real-qc-a"]], "5.1 IBM hardware and Deferred Measurement <a id='deferred-measurement'></a>": [[74, "ibm-hardware-and-deferred-measurement-a-id-deferred-measurement-a"]], "5.2 Executing <a id='executing'></a>": [[74, "executing-a-id-executing-a"]], "1. The Difference between Superdense Coding and Quantum Teleportation <a id='difference'></a>": [[74, "the-difference-between-superdense-coding-and-quantum-teleportation-a-id-difference-a"]], "2. The Process <a id='process'></a>": [[74, "the-process-a-id-process-a"]], "2.1 Step 1 <a id='step1'></a>": [[74, "step-1-a-id-step1-a"]], "Outcome States": [[74, "outcome-states"]], "2.2 Step 2 <a id='step2'></a>": [[74, "step-2-a-id-step2-a"]], "Encoding Rules for Superdense Coding (Alice protocol):": [[74, "encoding-rules-for-superdense-coding-alice-protocol"]], "2.3 Step 3 <a id='step3'></a>": [[74, "step-3-a-id-step3-a"]], "3. Simulating the Superdense Coding Protocol <a id='simulating'></a>": [[74, "simulating-the-superdense-coding-protocol-a-id-simulating-a"]], "3.1 Visualizing Our Measurements <a id='visualising'></a>": [[74, "visualizing-our-measurements-a-id-visualising-a"]], "4. Superdense Coding on a Real Quantum Computer <a id='real_qc'></a>": [[74, "superdense-coding-on-a-real-quantum-computer-a-id-real-qc-a"]], "1. Introduction": [[74, "introduction"]], "2. Protocol Overview": [[74, "protocol-overview"]], "3. Qiskit Example: Without Interception": [[74, "qiskit-example-without-interception"]], "3.1 Step 1:": [[74, "id10"]], "3.2 Step 2:": [[74, "id11"]], "3.3 Step 3:": [[74, "id12"]], "3.4 Step 4:": [[74, "id13"]], "3.5 Step 5:": [[74, "step-5"]], "4. Qiskit Example: With Interception": [[74, "qiskit-example-with-interception"]], "4.1 Step 1:": [[74, "id14"]], "4.2 Step 2:": [[74, "id15"]], "Interception!": [[74, "interception"]], "4.3 Step 3:": [[74, "id16"]], "4.4 Step 4:": [[74, "id17"]], "4.5 Step 5:": [[74, "id18"]], "5. Risk Analysis": [[74, "risk-analysis"]], "Hands On Sessions": [[75, "hands-on-sessions"]], "Hands on 0 (Preliminaries)": [[75, null]], "Hands on 1": [[75, null]], "Hands on 2": [[75, null]], "Hands on 5": [[75, null]], "Hands on 6": [[75, null]], "Hands on 7": [[75, null]], "Introduction to Quantum Computing for Physicists": [[92, "introduction-to-quantum-computing-for-physicists"]], "Useful Links": [[92, "useful-links"]]}, "indexentries": {}})