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, _neg_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 = _neg_domain), Sum(index_or_indices = sub_expr2, summand = Exp(Abs(_rel_indexed_alpha), two), domain = _neg_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 \{-2^{t - 1} + 1~\ldotp \ldotp~-\left(e + 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 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 = -2^{t - 1} + 1}^{-\left(e + 1\right)} \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: 125 body: 11
10	Lambda	parameter: 125 body: 13
11	Conditional	value: 14 condition: 16
12	ExprTuple	125
13	Conditional	value: 15 condition: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 105 operands: 19
16	Operation	operator: 20 operands: 21
17	Literal
18	ExprTuple	22, 23, 24, 25
19	ExprTuple	26, 113
20	Literal
21	ExprTuple	125, 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: 94 operands: 38
28	ExprRange	lambda_map: 39 start_index: 128 end_index: 122
29	ExprRange	lambda_map: 40 start_index: 128 end_index: 117
30	ExprRange	lambda_map: 41 start_index: 128 end_index: 122
31	ExprRange	lambda_map: 41 start_index: 103 end_index: 104
32	ExprRange	lambda_map: 42 start_index: 128 end_index: 122
33	ExprRange	lambda_map: 43 start_index: 128 end_index: 117
34	ExprRange	lambda_map: 44 start_index: 128 end_index: 122
35	ExprRange	lambda_map: 45 start_index: 128 end_index: 117
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: 116
47	Operation	operator: 118 operands: 59
48	Operation	operator: 126 operand: 71
49	Operation	operator: 86 operands: 61
50	Operation	operator: 68 operands: 62
51	Operation	operator: 68 operands: 63
52	Operation	operator: 64 operands: 65
53	Operation	operator: 87 operands: 66
54	Operation	operator: 68 operands: 67
55	ExprTuple	102
56	Operation	operator: 68 operands: 69
57	Literal
58	ExprTuple	116
59	ExprTuple	70, 128
60	ExprTuple	71
61	NamedExprs	state: 72
62	NamedExprs	element: 73 targets: 81
63	NamedExprs	element: 74 targets: 75
64	Literal
65	NamedExprs	basis: 76
66	NamedExprs	operation: 77
67	NamedExprs	element: 78 targets: 79
68	Literal
69	NamedExprs	element: 80 targets: 81
70	Operation	operator: 126 operand: 96
71	Operation	operator: 118 operands: 83
72	Operation	operator: 84 operand: 98
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
83	ExprTuple	97, 128
84	Literal
85	ExprTuple	98
86	Literal
87	Literal
88	NamedExprs	operation: 99 part: 102
89	ExprTuple	128, 104
90	NamedExprs	state: 100 part: 102
91	ExprTuple	128, 122
92	Literal
93	NamedExprs	state: 101 part: 102
94	Literal
95	ExprTuple	103, 104
96	Operation	operator: 105 operands: 106
97	Variable
98	Literal
99	Operation	operator: 107 operands: 108
100	Operation	operator: 109 operands: 110
101	Literal
102	Variable
103	Operation	operator: 118 operands: 111
104	Operation	operator: 118 operands: 112
105	Literal
106	ExprTuple	113, 114
107	Literal
108	ExprTuple	115, 122
109	Literal
110	ExprTuple	116, 122
111	ExprTuple	122, 128
112	ExprTuple	122, 117
113	Literal
114	Operation	operator: 118 operands: 119
115	Literal
116	Operation	operator: 120 operands: 121
117	Literal
118	Literal
119	ExprTuple	122, 123
120	Literal
121	ExprTuple	124, 125
122	Literal
123	Operation	operator: 126 operand: 128
124	Literal
125	Variable
126	Literal
127	ExprTuple	128
128	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 ¶