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, e, l
from proveit.logic import Equals, Union
from proveit.numbers import Add, Interval, ModAbs, greater, one
from proveit.physics.quantum import NumKet, Z, ket_plus
from proveit.physics.quantum.QPE import ModAdd, QPE, _U, _b_floor, _ket_u, _neg_domain, _pos_domain, _s, _s_wire, _t, _two_pow_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))
sub_expr7 = 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)]))
expr = Equals(ProbOfAll(instance_param_or_params = sub_expr2, instance_element = sub_expr7, domain = Union(_neg_domain, _pos_domain), condition = greater(ModAbs(l, _two_pow_t), e)), Add(Variable("_b", latex_format = r"{_{-}b}"), ProbOfAll(instance_param_or_params = sub_expr2, instance_element = sub_expr7, 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 \{-2^{t - 1} + 1~\ldotp \ldotp~-\left(e + 1\right)\} \cup \{e + 1~\ldotp \ldotp~2^{t - 1}\}~|~\left|l\right|_{\textup{mod}\thinspace 2^{t}} > 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 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({_{-}b} + \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]\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: 11 operand: 7
4	Operation	operator: 132 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameter: 135 body: 10
8	Variable
9	Operation	operator: 11 operand: 14
10	Conditional	value: 21 condition: 13
11	Literal
12	ExprTuple	14
13	Operation	operator: 15 operands: 16
14	Lambda	parameter: 135 body: 18
15	Literal
16	ExprTuple	19, 20
17	ExprTuple	135
18	Conditional	value: 21 condition: 22
19	Operation	operator: 28 operands: 23
20	Operation	operator: 24 operands: 25
21	Operation	operator: 26 operands: 27
22	Operation	operator: 28 operands: 29
23	ExprTuple	135, 30
24	Literal
25	ExprTuple	116, 31
26	Literal
27	ExprTuple	32, 33, 34, 35
28	Literal
29	ExprTuple	135, 49
30	Operation	operator: 36 operands: 37
31	Operation	operator: 38 operands: 39
32	ExprTuple	40, 41
33	ExprTuple	42, 43
34	ExprTuple	44, 45
35	ExprTuple	46, 47
36	Literal
37	ExprTuple	48, 49
38	Literal
39	ExprTuple	135, 50
40	ExprRange	lambda_map: 51 start_index: 140 end_index: 136
41	ExprRange	lambda_map: 52 start_index: 140 end_index: 127
42	ExprRange	lambda_map: 53 start_index: 140 end_index: 136
43	ExprRange	lambda_map: 53 start_index: 113 end_index: 114
44	ExprRange	lambda_map: 54 start_index: 140 end_index: 136
45	ExprRange	lambda_map: 55 start_index: 140 end_index: 127
46	ExprRange	lambda_map: 56 start_index: 140 end_index: 136
47	ExprRange	lambda_map: 57 start_index: 140 end_index: 127
48	Operation	operator: 104 operands: 58
49	Operation	operator: 104 operands: 59
50	Operation	operator: 123 operands: 60
51	Lambda	parameter: 112 body: 61
52	Lambda	parameter: 112 body: 62
53	Lambda	parameter: 112 body: 63
54	Lambda	parameter: 112 body: 64
55	Lambda	parameter: 112 body: 65
56	Lambda	parameter: 112 body: 66
57	Lambda	parameter: 112 body: 68
58	ExprTuple	69, 70
59	ExprTuple	93, 115
60	ExprTuple	128, 136
61	Operation	operator: 96 operands: 71
62	Operation	operator: 78 operands: 72
63	Operation	operator: 78 operands: 73
64	Operation	operator: 74 operands: 75
65	Operation	operator: 97 operands: 76
66	Operation	operator: 78 operands: 77
67	ExprTuple	112
68	Operation	operator: 78 operands: 79
69	Operation	operator: 132 operands: 80
70	Operation	operator: 138 operand: 93
71	NamedExprs	state: 82
72	NamedExprs	element: 83 targets: 91
73	NamedExprs	element: 84 targets: 85
74	Literal
75	NamedExprs	basis: 86
76	NamedExprs	operation: 87
77	NamedExprs	element: 88 targets: 89
78	Literal
79	NamedExprs	element: 90 targets: 91
80	ExprTuple	92, 140
81	ExprTuple	93
82	Operation	operator: 94 operand: 108
83	Operation	operator: 96 operands: 103
84	Operation	operator: 97 operands: 98
85	Operation	operator: 104 operands: 99
86	Literal
87	Literal
88	Operation	operator: 102 operands: 100
89	Operation	operator: 104 operands: 101
90	Operation	operator: 102 operands: 103
91	Operation	operator: 104 operands: 105
92	Operation	operator: 138 operand: 115
93	Operation	operator: 132 operands: 107
94	Literal
95	ExprTuple	108
96	Literal
97	Literal
98	NamedExprs	operation: 109 part: 112
99	ExprTuple	140, 114
100	NamedExprs	state: 110 part: 112
101	ExprTuple	140, 136
102	Literal
103	NamedExprs	state: 111 part: 112
104	Literal
105	ExprTuple	113, 114
106	ExprTuple	115
107	ExprTuple	116, 140
108	Literal
109	Operation	operator: 117 operands: 118
110	Operation	operator: 119 operands: 120
111	Literal
112	Variable
113	Operation	operator: 132 operands: 121
114	Operation	operator: 132 operands: 122
115	Operation	operator: 123 operands: 124
116	Variable
117	Literal
118	ExprTuple	125, 136
119	Literal
120	ExprTuple	126, 136
121	ExprTuple	136, 140
122	ExprTuple	136, 127
123	Literal
124	ExprTuple	128, 129
125	Literal
126	Operation	operator: 130 operands: 131
127	Literal
128	Literal
129	Operation	operator: 132 operands: 133
130	Literal
131	ExprTuple	134, 135
132	Literal
133	ExprTuple	136, 137
134	Literal
135	Variable
136	Literal
137	Operation	operator: 138 operand: 140
138	Literal
139	ExprTuple	140
140	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 ¶