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
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _Psi_ket, _ket_u, _psi__t_ket, _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 = Interval(one, _t)
sub_expr3 = Interval(Add(_t, one), Add(_t, _s))
expr = Equals(Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _psi__t_ket, part = sub_expr1), targets = sub_expr2), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr3), one, _s)], [ExprRange(sub_expr1, MultiQubitElem(element = Gate(operation = InverseFourierTransform(_t), part = sub_expr1), targets = sub_expr2), 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_expr2), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr3), 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 \psi_{t} \rangle} & \gate{{\mathrm {FT}}^{\dag}_{t}} & \qout{\lvert \Psi \rangle} \\
\qin{\lvert u \rangle} & { /^{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, 69
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: 69 end_index: 70
13	ExprRange	lambda_map: 19 start_index: 69 end_index: 71
14	ExprRange	lambda_map: 20 start_index: 69 end_index: 70
15	ExprRange	lambda_map: 21 start_index: 69 end_index: 71
16	ExprRange	lambda_map: 22 start_index: 69 end_index: 70
17	ExprRange	lambda_map: 23 start_index: 69 end_index: 71
18	Lambda	parameter: 60 body: 24
19	Lambda	parameter: 60 body: 25
20	Lambda	parameter: 60 body: 26
21	Lambda	parameter: 60 body: 27
22	Lambda	parameter: 60 body: 28
23	Lambda	parameter: 60 body: 30
24	Operation	operator: 36 operands: 31
25	Operation	operator: 36 operands: 32
26	Operation	operator: 36 operands: 33
27	Operation	operator: 48 operands: 34
28	Operation	operator: 36 operands: 35
29	ExprTuple	60
30	Operation	operator: 36 operands: 37
31	NamedExprs	element: 38 targets: 43
32	NamedExprs	element: 39 targets: 45
33	NamedExprs	element: 40 targets: 43
34	NamedExprs	operation: 41
35	NamedExprs	element: 42 targets: 43
36	Literal
37	NamedExprs	element: 44 targets: 45
38	Operation	operator: 47 operands: 46
39	Operation	operator: 47 operands: 53
40	Operation	operator: 48 operands: 49
41	Literal
42	Operation	operator: 52 operands: 50
43	Operation	operator: 54 operands: 51
44	Operation	operator: 52 operands: 53
45	Operation	operator: 54 operands: 55
46	NamedExprs	state: 56 part: 60
47	Literal
48	Literal
49	NamedExprs	operation: 57 part: 60
50	NamedExprs	state: 58 part: 60
51	ExprTuple	69, 70
52	Literal
53	NamedExprs	state: 59 part: 60
54	Literal
55	ExprTuple	61, 62
56	Operation	operator: 63 operand: 70
57	Operation	operator: 64 operand: 70
58	Literal
59	Literal
60	Variable
61	Operation	operator: 67 operands: 66
62	Operation	operator: 67 operands: 68
63	Literal
64	Literal
65	ExprTuple	70
66	ExprTuple	70, 69
67	Literal
68	ExprTuple	70, 71
69	Literal
70	Literal
71	Literal