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 LessEq, ModAbs, greater, one, subtract
from proveit.physics.quantum.QPE import _b_floor, _e_value, _eps, _m_domain, _phase_est_circuit, _two_pow_t
from proveit.statistics import ProbOfAll

In [2]:

# build up the expression from sub-expressions
expr = greater(ProbOfAll(instance_param_or_params = [m], instance_element = _phase_est_circuit, domain = _m_domain, condition = LessEq(ModAbs(subtract(m, _b_floor), _two_pow_t), _e_value)).with_wrapping(), subtract(one, _eps))

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[\begin{array}{l}\textrm{Prob}_{m \in \{0~\ldotp \ldotp~2^{t} - 1\}~|~\left|m - b_{\textit{f}}\right|_{\textup{mod}\thinspace 2^{t}} \leq \left(2^{t - n} - 1\right)}~\\
\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] > \left(1 - \epsilon\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	reversed	('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	126, 8
6	Literal
7	ExprTuple	9
8	Operation	operator: 129 operand: 13
9	Lambda	parameter: 125 body: 12
10	ExprTuple	13
11	ExprTuple	125
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: 126 end_index: 127
27	ExprRange	lambda_map: 39 start_index: 126 end_index: 128
28	ExprRange	lambda_map: 40 start_index: 126 end_index: 127
29	ExprRange	lambda_map: 40 start_index: 107 end_index: 108
30	ExprRange	lambda_map: 41 start_index: 126 end_index: 127
31	ExprRange	lambda_map: 42 start_index: 126 end_index: 128
32	ExprRange	lambda_map: 43 start_index: 126 end_index: 127
33	ExprRange	lambda_map: 44 start_index: 126 end_index: 128
34	Literal
35	ExprTuple	125, 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: 119 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, 98
59	ExprTuple	72, 99
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: 109 operands: 85
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	98, 99
84	ExprTuple	125, 100
85	ExprTuple	121, 101
86	Literal
87	ExprTuple	102
88	Literal
89	Literal
90	NamedExprs	operation: 103 part: 106
91	ExprTuple	126, 108
92	NamedExprs	state: 104 part: 106
93	ExprTuple	126, 127
94	Literal
95	NamedExprs	state: 105 part: 106
96	Literal
97	ExprTuple	107, 108
98	Operation	operator: 109 operands: 110
99	Operation	operator: 129 operand: 126
100	Operation	operator: 129 operand: 122
101	Operation	operator: 119 operands: 113
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	Literal
110	ExprTuple	121, 127
111	ExprTuple	126
112	ExprTuple	122
113	ExprTuple	127, 123
114	Literal
115	ExprTuple	124, 127
116	Literal
117	ExprTuple	125, 127
118	ExprTuple	127, 126
119	Literal
120	ExprTuple	127, 128
121	Literal
122	Literal
123	Operation	operator: 129 operand: 131
124	Literal
125	Variable
126	Literal
127	Literal
128	Literal
129	Literal
130	ExprTuple	131
131	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 ¶