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, t
from proveit.logic import Equals
from proveit.numbers import Add, Interval, one
from proveit.physics.quantum import ket_plus
from proveit.physics.quantum.QPE import QPE1, _U, _ket_u, _psi_t_ket, _s
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(sub_expr2, sub_expr3)
sub_expr5 = 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_expr4), one, _s)], [ExprRange(sub_expr1, sub_expr5, one, t), ExprRange(sub_expr1, sub_expr5, sub_expr2, sub_expr3)], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _psi_t_ket, 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)]))), 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)} & \multiqout{3}{\lvert \psi_{t} \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \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, 70
3	Operation	operator: 4 operand: 6
4	Literal
5	ExprTuple	6
6	Operation	operator: 7 operands: 8
7	Literal
8	ExprTuple	9, 10, 11
9	ExprTuple	12, 13
10	ExprTuple	14, 15
11	ExprTuple	16, 17
12	ExprRange	lambda_map: 18 start_index: 70 end_index: 71
13	ExprRange	lambda_map: 19 start_index: 70 end_index: 72
14	ExprRange	lambda_map: 20 start_index: 70 end_index: 71
15	ExprRange	lambda_map: 20 start_index: 60 end_index: 61
16	ExprRange	lambda_map: 21 start_index: 70 end_index: 71
17	ExprRange	lambda_map: 22 start_index: 70 end_index: 72
18	Lambda	parameter: 59 body: 23
19	Lambda	parameter: 59 body: 24
20	Lambda	parameter: 59 body: 25
21	Lambda	parameter: 59 body: 26
22	Lambda	parameter: 59 body: 28
23	Operation	operator: 45 operands: 29
24	Operation	operator: 33 operands: 30
25	Operation	operator: 33 operands: 31
26	Operation	operator: 33 operands: 32
27	ExprTuple	59
28	Operation	operator: 33 operands: 34
29	NamedExprs	state: 35
30	NamedExprs	element: 36 targets: 42
31	NamedExprs	element: 37 targets: 38
32	NamedExprs	element: 39 targets: 40
33	Literal
34	NamedExprs	element: 41 targets: 42
35	Operation	operator: 43 operand: 55
36	Operation	operator: 45 operands: 52
37	Operation	operator: 46 operands: 47
38	Operation	operator: 53 operands: 48
39	Operation	operator: 51 operands: 49
40	Operation	operator: 53 operands: 50
41	Operation	operator: 51 operands: 52
42	Operation	operator: 53 operands: 54
43	Literal
44	ExprTuple	55
45	Literal
46	Literal
47	NamedExprs	operation: 56 part: 59
48	ExprTuple	70, 61
49	NamedExprs	state: 57 part: 59
50	ExprTuple	70, 71
51	Literal
52	NamedExprs	state: 58 part: 59
53	Literal
54	ExprTuple	60, 61
55	Literal
56	Operation	operator: 62 operands: 63
57	Operation	operator: 64 operand: 71
58	Literal
59	Variable
60	Operation	operator: 67 operands: 66
61	Operation	operator: 67 operands: 68
62	Literal
63	ExprTuple	69, 71
64	Literal
65	ExprTuple	71
66	ExprTuple	71, 70
67	Literal
68	ExprTuple	71, 72
69	Literal
70	Literal
71	Variable
72	Literal