Expression of type Forall

from the theory of proveit.physics.quantum.circuits

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import U
from proveit.linear_algebra import ScalarMult
from proveit.logic import Equals, Forall
from proveit.numbers import Exp, Mult, e, i, one, pi, two
from proveit.physics.quantum import Qmult, m_ket_domain, normalized_var_ket_u, var_ket_u, varphi
from proveit.physics.quantum.circuits import phase_kickback_circuit
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [var_ket_u], instance_expr = Forall(instance_param_or_params = [varphi], instance_expr = Equals(Prob(phase_kickback_circuit), one), condition = Equals(Qmult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, varphi)), var_ket_u))), domain = m_ket_domain, condition = normalized_var_ket_u)

expr:

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

Passed sanity check: expr matches stored_expr

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

\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]

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