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
from proveit.logic import Equals
from proveit.numbers import Add, Interval, one
from proveit.physics.quantum import I, ket_plus
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import QPE1, _Psi_ket, _U, _ket_u, _s, _t
from proveit.physics.quantum.circuits import Gate, Input, MultiQubitElem, Output, Qcircuit
from proveit.statistics import Prob

# 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(one, _t)
sub_expr5 = Interval(sub_expr2, sub_expr3)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE1(_U, _t), part = sub_expr1), targets = Interval(one, sub_expr3))
expr = Equals(Prob(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_expr2, sub_expr3)], [ExprRange(sub_expr1, MultiQubitElem(element = Gate(operation = InverseFourierTransform(_t), part = sub_expr1), targets = sub_expr4), one, _t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, _s)], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _Psi_ket, part = sub_expr1), targets = sub_expr4), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)]))), one)

expr:

# 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())

\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}_1\left(U, t\right)} & \multigate{3}{{\mathrm {FT}}^{\dag}_{t}} & \multiqout{3}{\lvert \Psi \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghost{{\mathrm {FT}}^{\dag}_{t}} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghost{{\mathrm {FT}}^{\dag}_{t}} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghost{{\mathrm {FT}}^{\dag}_{t}} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) = 1

stored_expr.style_options()

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')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 83
3	Operation	operator: 4 operand: 6
4	Literal
5	ExprTuple	6
6	Operation	operator: 7 operands: 8
7	Literal
8	ExprTuple	9, 10, 11, 12
9	ExprTuple	13, 14
10	ExprTuple	15, 16
11	ExprTuple	17, 18
12	ExprTuple	19, 20
13	ExprRange	lambda_map: 21 start_index: 83 end_index: 84
14	ExprRange	lambda_map: 22 start_index: 83 end_index: 85
15	ExprRange	lambda_map: 23 start_index: 83 end_index: 84
16	ExprRange	lambda_map: 23 start_index: 73 end_index: 74
17	ExprRange	lambda_map: 24 start_index: 83 end_index: 84
18	ExprRange	lambda_map: 25 start_index: 83 end_index: 85
19	ExprRange	lambda_map: 26 start_index: 83 end_index: 84
20	ExprRange	lambda_map: 27 start_index: 83 end_index: 85
21	Lambda	parameter: 72 body: 28
22	Lambda	parameter: 72 body: 29
23	Lambda	parameter: 72 body: 30
24	Lambda	parameter: 72 body: 31
25	Lambda	parameter: 72 body: 32
26	Lambda	parameter: 72 body: 33
27	Lambda	parameter: 72 body: 35
28	Operation	operator: 56 operands: 36
29	Operation	operator: 42 operands: 37
30	Operation	operator: 42 operands: 38
31	Operation	operator: 42 operands: 39
32	Operation	operator: 59 operands: 40
33	Operation	operator: 42 operands: 41
34	ExprTuple	72
35	Operation	operator: 42 operands: 43
36	NamedExprs	state: 44
37	NamedExprs	element: 45 targets: 53
38	NamedExprs	element: 46 targets: 47
39	NamedExprs	element: 48 targets: 51
40	NamedExprs	operation: 49
41	NamedExprs	element: 50 targets: 51
42	Literal
43	NamedExprs	element: 52 targets: 53
44	Operation	operator: 54 operand: 67
45	Operation	operator: 56 operands: 64
46	Operation	operator: 59 operands: 57
47	Operation	operator: 65 operands: 58
48	Operation	operator: 59 operands: 60
49	Literal
50	Operation	operator: 63 operands: 61
51	Operation	operator: 65 operands: 62
52	Operation	operator: 63 operands: 64
53	Operation	operator: 65 operands: 66
54	Literal
55	ExprTuple	67
56	Literal
57	NamedExprs	operation: 68 part: 72
58	ExprTuple	83, 74
59	Literal
60	NamedExprs	operation: 69 part: 72
61	NamedExprs	state: 70 part: 72
62	ExprTuple	83, 84
63	Literal
64	NamedExprs	state: 71 part: 72
65	Literal
66	ExprTuple	73, 74
67	Literal
68	Operation	operator: 75 operands: 76
69	Operation	operator: 77 operand: 84
70	Literal
71	Literal
72	Variable
73	Operation	operator: 80 operands: 79
74	Operation	operator: 80 operands: 81
75	Literal
76	ExprTuple	82, 84
77	Literal
78	ExprTuple	84
79	ExprTuple	84, 83
80	Literal
81	ExprTuple	84, 85
82	Literal
83	Literal
84	Literal
85	Literal