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
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, Pfail, QPE, _U, _b_floor, _full_domain, _ket_u, _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 = Add(_t, one)
sub_expr3 = Add(_t, _s)
sub_expr4 = Interval(sub_expr2, sub_expr3)
sub_expr5 = MultiQubitElem(element = Gate(operation = QPE(_U, _t), part = sub_expr1), targets = Interval(one, sub_expr3))
expr = Equals(Pfail(e), ProbOfAll(instance_param_or_params = [l], 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_expr4), one, _s)], [ExprRange(sub_expr1, sub_expr5, one, _t), ExprRange(sub_expr1, sub_expr5, sub_expr2, sub_expr3)], [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_expr4), one, _s)])), domain = _full_domain, condition = greater(ModAbs(l, _two_pow_t), e)))

# 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())

\left[P_{\rm fail}\right]\left(e\right) = \left[\textrm{Prob}_{l \in \{-2^{t - 1} + 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]

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