Expression of type Equals ¶

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 e, m
from proveit.logic import Equals
from proveit.numbers import LessEq, ModAbs, one, subtract
from proveit.physics.quantum.QPE import Pfail, _b_floor, _m_domain, _phase_est_circuit, _two_pow_t
from proveit.statistics import ProbOfAll

In [2]:

# build up the expression from sub-expressions
expr = Equals(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)), subtract(one, Pfail(e))).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left[\textrm{Prob}_{m \in \{0~\ldotp \ldotp~2^{t} - 1\}~|~\left|m - b_{\textit{f}}\right|_{\textup{mod}\thinspace 2^{t}} \leq 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)\right] =  \\ \left(1 - \left[P_{\rm fail}\right]\left(e\right)\right) \end{array} \end{array}

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

Expression of type Equals¶

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

Expression of type Equals ¶

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