Expression of type Equals

from the theory of proveit.physics.quantum.QPE

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 e
from proveit.logic import Equals
from proveit.physics.quantum.QPE import Pfail, _fail_prob_e

# build up the expression from sub-expressions
expr = Equals(Pfail(e), _fail_prob_e)

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())

\left[P_{\rm fail}\right]\left(e\right) = \left[\begin{array}{l}\textrm{Prob}_{m \in \{0~\ldotp \ldotp~2^{t} - 1\}~|~\left|m - b_{\textit{f}}\right|_{\textup{mod}\thinspace 2^{t}} > e}~\\
\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert m \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert m \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right)\end{array}\right]

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

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