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

In [2]:

# build up the expression from sub-expressions
expr = Equals(Psuccess(_e_value), 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())

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 success}\right]\left(2^{t - n} - 1\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}} \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]

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