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 Abs, Add, Exp, Interval, ModAbs, Sum, greater, 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, _pos_domain, _rel_indexed_alpha, _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(Sum(index_or_indices = sub_expr2, summand = Exp(Abs(_rel_indexed_alpha), two), domain = _neg_domain), 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(\left(\sum_{l = -2^{t - 1} + 1}^{-\left(e + 1\right)} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right) + \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: 13 operand: 7
4	Operation	operator: 146 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameter: 149 body: 10
8	Operation	operator: 11 operand: 16
9	Operation	operator: 13 operand: 17
10	Conditional	value: 27 condition: 15
11	Literal
12	ExprTuple	16
13	Literal
14	ExprTuple	17
15	Operation	operator: 18 operands: 19
16	Lambda	parameter: 149 body: 20
17	Lambda	parameter: 149 body: 22
18	Literal
19	ExprTuple	23, 24
20	Conditional	value: 25 condition: 26
21	ExprTuple	149
22	Conditional	value: 27 condition: 28
23	Operation	operator: 36 operands: 29
24	Operation	operator: 30 operands: 31
25	Operation	operator: 137 operands: 32
26	Operation	operator: 36 operands: 33
27	Operation	operator: 34 operands: 35
28	Operation	operator: 36 operands: 37
29	ExprTuple	149, 38
30	Literal
31	ExprTuple	130, 39
32	ExprTuple	40, 142
33	ExprTuple	149, 59
34	Literal
35	ExprTuple	41, 42, 43, 44
36	Literal
37	ExprTuple	149, 60
38	Operation	operator: 45 operands: 46
39	Operation	operator: 47 operands: 48
40	Operation	operator: 49 operand: 62
41	ExprTuple	51, 52
42	ExprTuple	53, 54
43	ExprTuple	55, 56
44	ExprTuple	57, 58
45	Literal
46	ExprTuple	59, 60
47	Literal
48	ExprTuple	149, 61
49	Literal
50	ExprTuple	62
51	ExprRange	lambda_map: 63 start_index: 154 end_index: 150
52	ExprRange	lambda_map: 64 start_index: 154 end_index: 141
53	ExprRange	lambda_map: 65 start_index: 154 end_index: 150
54	ExprRange	lambda_map: 65 start_index: 127 end_index: 128
55	ExprRange	lambda_map: 66 start_index: 154 end_index: 150
56	ExprRange	lambda_map: 67 start_index: 154 end_index: 141
57	ExprRange	lambda_map: 68 start_index: 154 end_index: 150
58	ExprRange	lambda_map: 69 start_index: 154 end_index: 141
59	Operation	operator: 118 operands: 70
60	Operation	operator: 118 operands: 71
61	Operation	operator: 137 operands: 72
62	Operation	operator: 73 operand: 140
63	Lambda	parameter: 126 body: 75
64	Lambda	parameter: 126 body: 76
65	Lambda	parameter: 126 body: 77
66	Lambda	parameter: 126 body: 78
67	Lambda	parameter: 126 body: 79
68	Lambda	parameter: 126 body: 80
69	Lambda	parameter: 126 body: 82
70	ExprTuple	83, 84
71	ExprTuple	107, 129
72	ExprTuple	142, 150
73	Literal
74	ExprTuple	140
75	Operation	operator: 110 operands: 85
76	Operation	operator: 92 operands: 86
77	Operation	operator: 92 operands: 87
78	Operation	operator: 88 operands: 89
79	Operation	operator: 111 operands: 90
80	Operation	operator: 92 operands: 91
81	ExprTuple	126
82	Operation	operator: 92 operands: 93
83	Operation	operator: 146 operands: 94
84	Operation	operator: 152 operand: 107
85	NamedExprs	state: 96
86	NamedExprs	element: 97 targets: 105
87	NamedExprs	element: 98 targets: 99
88	Literal
89	NamedExprs	basis: 100
90	NamedExprs	operation: 101
91	NamedExprs	element: 102 targets: 103
92	Literal
93	NamedExprs	element: 104 targets: 105
94	ExprTuple	106, 154
95	ExprTuple	107
96	Operation	operator: 108 operand: 122
97	Operation	operator: 110 operands: 117
98	Operation	operator: 111 operands: 112
99	Operation	operator: 118 operands: 113
100	Literal
101	Literal
102	Operation	operator: 116 operands: 114
103	Operation	operator: 118 operands: 115
104	Operation	operator: 116 operands: 117
105	Operation	operator: 118 operands: 119
106	Operation	operator: 152 operand: 129
107	Operation	operator: 146 operands: 121
108	Literal
109	ExprTuple	122
110	Literal
111	Literal
112	NamedExprs	operation: 123 part: 126
113	ExprTuple	154, 128
114	NamedExprs	state: 124 part: 126
115	ExprTuple	154, 150
116	Literal
117	NamedExprs	state: 125 part: 126
118	Literal
119	ExprTuple	127, 128
120	ExprTuple	129
121	ExprTuple	130, 154
122	Literal
123	Operation	operator: 131 operands: 132
124	Operation	operator: 133 operands: 134
125	Literal
126	Variable
127	Operation	operator: 146 operands: 135
128	Operation	operator: 146 operands: 136
129	Operation	operator: 137 operands: 138
130	Variable
131	Literal
132	ExprTuple	139, 150
133	Literal
134	ExprTuple	140, 150
135	ExprTuple	150, 154
136	ExprTuple	150, 141
137	Literal
138	ExprTuple	142, 143
139	Literal
140	Operation	operator: 144 operands: 145
141	Literal
142	Literal
143	Operation	operator: 146 operands: 147
144	Literal
145	ExprTuple	148, 149
146	Literal
147	ExprTuple	150, 151
148	Literal
149	Variable
150	Literal
151	Operation	operator: 152 operand: 154
152	Literal
153	ExprTuple	154
154	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 ¶