Expression of type Less ¶

from the theory of proveit.physics.quantum.QPE ¶

In [1]:

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 m
from proveit.numbers import Less, one, subtract
from proveit.physics.quantum.QPE import _eps, _m_domain, _phase_est_circuit, _success_cond
from proveit.statistics import ProbOfAll

In [2]:

# build up the expression from sub-expressions
expr = Less(subtract(one, _eps), ProbOfAll(instance_param_or_params = [m], instance_element = _phase_est_circuit, domain = _m_domain, condition = _success_cond).with_wrapping())

expr:

In [3]:

# 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

In [4]:

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

\left(1 - \epsilon\right) < \left[\begin{array}{l}\textrm{Prob}_{m \in \{0~\ldotp \ldotp~2^{t} - 1\}~|~\left|\frac{m}{2^{t}} - \varphi\right|_{\textup{mod}\thinspace 1} \leq 2^{-n}}~\\
\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]

In [5]:

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')

In [6]:

# display the expression information
stored_expr.expr_info()

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

Expression of type Less¶

from the theory of proveit.physics.quantum.QPE¶

Expression of type Less ¶

from the theory of proveit.physics.quantum.QPE ¶