Theorem phase_kickback of type Forall

from the theory of proveit.physics.quantum.circuits

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.physics.quantum.circuits import phase_kickback

# check that the built expression is the same as the stored expression
assert phase_kickback.expr == stored_expr
assert phase_kickback.expr._style_id == stored_expr._style_id
print("Passed sanity check: phase_kickback matches stored_expr")

Passed sanity check: phase_kickback matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{m \in \mathbb{N}^+}~\left[\forall_{U \in \textrm{U}\left(2^{m}\right)}~\left[\forall_{\lvert u \rangle \in \mathbb{C}^{2^{m}}~|~\left \|\lvert u \rangle\right \| = 1}~\left[\forall_{\varphi~|~\left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right)}~\left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \control{} \qw \qwx[1] & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert u \rangle} & \gate{U} & \qout{\lvert u \rangle}
} \end{array}\right) = 1\right)\right]\right]\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 24 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 128 body: 4
3	ExprTuple	128
4	Conditional	value: 5 condition: 6
5	Operation	operator: 24 operand: 9
6	Operation	operator: 33 operands: 8
7	ExprTuple	9
8	ExprTuple	128, 10
9	Lambda	parameter: 111 body: 12
10	Literal
11	ExprTuple	111
12	Conditional	value: 13 condition: 14
13	Operation	operator: 24 operand: 17
14	Operation	operator: 33 operands: 16
15	ExprTuple	17
16	ExprTuple	111, 18
17	Lambda	parameter: 115 body: 19
18	Operation	operator: 20 operand: 51
19	Conditional	value: 22 condition: 23
20	Literal
21	ExprTuple	51
22	Operation	operator: 24 operand: 28
23	Operation	operator: 26 operands: 27
24	Literal
25	ExprTuple	28
26	Literal
27	ExprTuple	29, 30
28	Lambda	parameter: 143 body: 32
29	Operation	operator: 33 operands: 34
30	Operation	operator: 41 operands: 35
31	ExprTuple	143
32	Conditional	value: 36 condition: 37
33	Literal
34	ExprTuple	115, 38
35	ExprTuple	39, 137
36	Operation	operator: 41 operands: 40
37	Operation	operator: 41 operands: 42
38	Operation	operator: 43 operands: 44
39	Operation	operator: 45 operand: 115
40	ExprTuple	47, 137
41	Literal
42	ExprTuple	48, 49
43	Literal
44	ExprTuple	50, 51
45	Literal
46	ExprTuple	115
47	Operation	operator: 52 operand: 58
48	Operation	operator: 54 operands: 55
49	Operation	operator: 120 operands: 56
50	Literal
51	Operation	operator: 131 operands: 57
52	Literal
53	ExprTuple	58
54	Literal
55	ExprTuple	111, 115
56	ExprTuple	126, 115
57	ExprTuple	140, 128
58	Operation	operator: 59 operands: 60
59	Literal
60	ExprTuple	61, 62, 63
61	ExprTuple	64, 65
62	ExprTuple	66, 67
63	ExprTuple	68, 69
64	Operation	operator: 101 operands: 70
65	ExprRange	lambda_map: 71 start_index: 137 end_index: 76
66	Operation	operator: 92 operands: 72
67	ExprRange	lambda_map: 73 start_index: 137 end_index: 76
68	Operation	operator: 107 operands: 74
69	ExprRange	lambda_map: 75 start_index: 137 end_index: 76
70	NamedExprs	state: 77
71	Lambda	parameter: 116 body: 78
72	NamedExprs	element: 79 targets: 80
73	Lambda	parameter: 116 body: 81
74	NamedExprs	state: 82
75	Lambda	parameter: 116 body: 84
76	Operation	operator: 122 operands: 85
77	Operation	operator: 133 operand: 94
78	Operation	operator: 92 operands: 87
79	Literal
80	Operation	operator: 88 operand: 140
81	Operation	operator: 92 operands: 90
82	Operation	operator: 120 operands: 91
83	ExprTuple	116
84	Operation	operator: 92 operands: 93
85	ExprTuple	128, 137
86	ExprTuple	94
87	NamedExprs	element: 95 targets: 100
88	Literal
89	ExprTuple	140
90	NamedExprs	element: 96 targets: 100
91	ExprTuple	97, 98
92	Literal
93	NamedExprs	element: 99 targets: 100
94	Literal
95	Operation	operator: 101 operands: 108
96	Operation	operator: 102 operands: 103
97	Operation	operator: 129 operands: 104
98	Operation	operator: 105 operands: 106
99	Operation	operator: 107 operands: 108
100	Operation	operator: 109 operands: 110
101	Literal
102	Literal
103	NamedExprs	operation: 111 part: 116
104	ExprTuple	137, 112
105	Literal
106	ExprTuple	113, 114
107	Literal
108	NamedExprs	state: 115 part: 116
109	Literal
110	ExprTuple	140, 117
111	Variable
112	Operation	operator: 131 operands: 118
113	Operation	operator: 133 operand: 125
114	Operation	operator: 120 operands: 121
115	Variable
116	Variable
117	Operation	operator: 122 operands: 123
118	ExprTuple	140, 124
119	ExprTuple	125
120	Literal
121	ExprTuple	126, 127
122	Literal
123	ExprTuple	128, 140
124	Operation	operator: 129 operands: 130
125	Literal
126	Operation	operator: 131 operands: 132
127	Operation	operator: 133 operand: 137
128	Variable
129	Literal
130	ExprTuple	137, 140
131	Literal
132	ExprTuple	135, 136
133	Literal
134	ExprTuple	137
135	Literal
136	Operation	operator: 138 operands: 139
137	Literal
138	Literal
139	ExprTuple	140, 141, 142, 143
140	Literal
141	Literal
142	Literal
143	Variable