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, SetOfAll
from proveit.numbers import ModAbs, greater, subtract
from proveit.physics.quantum.QPE import Pfail, _b_floor, _m_domain, _phase_est_circuit, _two_pow_t
from proveit.statistics import Prob

In [2]:

# build up the expression from sub-expressions
expr = Equals(Pfail(e), Prob(SetOfAll(instance_param_or_params = [m], instance_element = _phase_est_circuit, domain = _m_domain, condition = greater(ModAbs(subtract(m, _b_floor), _two_pow_t), e))))

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[P_{\rm fail}\right]\left(e\right) = \textrm{Pr}\left(\left\{\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)~|~\left|m - b_{\textit{f}}\right|_{\textup{mod}\thinspace 2^{t}} > e\right\}_{m \in \{0~\ldotp \ldotp~2^{t} - 1\}}\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: 5 operand: 47
4	Operation	operator: 7 operand: 9
5	Literal
6	ExprTuple	47
7	Literal
8	ExprTuple	9
9	Operation	operator: 10 operand: 12
10	Literal
11	ExprTuple	12
12	Lambda	parameter: 121 body: 14
13	ExprTuple	121
14	Conditional	value: 15 condition: 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22, 23, 24
19	Literal
20	ExprTuple	25, 26
21	ExprTuple	27, 28
22	ExprTuple	29, 30
23	ExprTuple	31, 32
24	ExprTuple	33, 34
25	Operation	operator: 35 operands: 36
26	Operation	operator: 37 operands: 38
27	ExprRange	lambda_map: 39 start_index: 122 end_index: 123
28	ExprRange	lambda_map: 40 start_index: 122 end_index: 124
29	ExprRange	lambda_map: 41 start_index: 122 end_index: 123
30	ExprRange	lambda_map: 41 start_index: 104 end_index: 105
31	ExprRange	lambda_map: 42 start_index: 122 end_index: 123
32	ExprRange	lambda_map: 43 start_index: 122 end_index: 124
33	ExprRange	lambda_map: 44 start_index: 122 end_index: 123
34	ExprRange	lambda_map: 45 start_index: 122 end_index: 124
35	Literal
36	ExprTuple	121, 46
37	Literal
38	ExprTuple	47, 48
39	Lambda	parameter: 103 body: 49
40	Lambda	parameter: 103 body: 50
41	Lambda	parameter: 103 body: 51
42	Lambda	parameter: 103 body: 52
43	Lambda	parameter: 103 body: 53
44	Lambda	parameter: 103 body: 54
45	Lambda	parameter: 103 body: 56
46	Operation	operator: 94 operands: 57
47	Variable
48	Operation	operator: 58 operands: 59
49	Operation	operator: 86 operands: 60
50	Operation	operator: 67 operands: 61
51	Operation	operator: 67 operands: 62
52	Operation	operator: 63 operands: 64
53	Operation	operator: 87 operands: 65
54	Operation	operator: 67 operands: 66
55	ExprTuple	103
56	Operation	operator: 67 operands: 68
57	ExprTuple	69, 70
58	Literal
59	ExprTuple	71, 96
60	NamedExprs	state: 72
61	NamedExprs	element: 73 targets: 81
62	NamedExprs	element: 74 targets: 75
63	Literal
64	NamedExprs	basis: 76
65	NamedExprs	operation: 77
66	NamedExprs	element: 78 targets: 79
67	Literal
68	NamedExprs	element: 80 targets: 81
69	Literal
70	Operation	operator: 116 operands: 82
71	Operation	operator: 116 operands: 83
72	Operation	operator: 84 operand: 99
73	Operation	operator: 86 operands: 93
74	Operation	operator: 87 operands: 88
75	Operation	operator: 94 operands: 89
76	Literal
77	Literal
78	Operation	operator: 92 operands: 90
79	Operation	operator: 94 operands: 91
80	Operation	operator: 92 operands: 93
81	Operation	operator: 94 operands: 95
82	ExprTuple	96, 97
83	ExprTuple	121, 98
84	Literal
85	ExprTuple	99
86	Literal
87	Literal
88	NamedExprs	operation: 100 part: 103
89	ExprTuple	122, 105
90	NamedExprs	state: 101 part: 103
91	ExprTuple	122, 123
92	Literal
93	NamedExprs	state: 102 part: 103
94	Literal
95	ExprTuple	104, 105
96	Operation	operator: 106 operands: 107
97	Operation	operator: 109 operand: 122
98	Operation	operator: 109 operand: 119
99	Literal
100	Operation	operator: 111 operands: 112
101	Operation	operator: 113 operands: 114
102	Literal
103	Variable
104	Operation	operator: 116 operands: 115
105	Operation	operator: 116 operands: 117
106	Literal
107	ExprTuple	118, 123
108	ExprTuple	122
109	Literal
110	ExprTuple	119
111	Literal
112	ExprTuple	120, 123
113	Literal
114	ExprTuple	121, 123
115	ExprTuple	123, 122
116	Literal
117	ExprTuple	123, 124
118	Literal
119	Literal
120	Literal
121	Variable
122	Literal
123	Literal
124	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 ¶