Expression of type Equals ¶

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

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 ExprRange, Variable, VertExprArray, l
from proveit.logic import Equals
from proveit.numbers import Abs, Add, Exp, Interval, Sum, one, two
from proveit.physics.quantum import NumKet, Z, ket_plus
from proveit.physics.quantum.QPE import ModAdd, QPE, _U, _b_floor, _ket_u, _pos_domain, _rel_indexed_alpha, _s, _s_wire, _t
from proveit.physics.quantum.circuits import Gate, Input, Measure, MultiQubitElem, Output, Qcircuit
from proveit.statistics import ProbOfAll

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = [l]
sub_expr3 = Add(_t, one)
sub_expr4 = Add(_t, _s)
sub_expr5 = Interval(sub_expr3, sub_expr4)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE(_U, _t), part = sub_expr1), targets = Interval(one, sub_expr4))
expr = Equals(ProbOfAll(instance_param_or_params = sub_expr2, instance_element = Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Input(state = ket_plus), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)], [ExprRange(sub_expr1, sub_expr6, one, _t), ExprRange(sub_expr1, sub_expr6, sub_expr3, sub_expr4)], [ExprRange(sub_expr1, Measure(basis = Z), one, _t), _s_wire], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(ModAdd(_b_floor, l), _t), part = sub_expr1), targets = Interval(one, _t)), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)])), domain = _pos_domain), Sum(index_or_indices = sub_expr2, summand = Exp(Abs(_rel_indexed_alpha), two), domain = _pos_domain)).with_wrapping_at(2)

# 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

# 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}_{l \in \{e + 1~\ldotp \ldotp~2^{t - 1}\}}~\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert b_{\textit{f}} \oplus l \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 b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert b_{\textit{f}} \oplus l \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right)\right] =  \\ \left(\sum_{l = e + 1}^{2^{t - 1}} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right) \end{array} \end{array}

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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')

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

Expression of type Equals¶

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

Expression of type Equals ¶

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