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 k, m
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, Neg, Sum, e, frac, i, one, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _alpha_m, _m_domain, _phase, _t, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr3 = frac(one, Exp(two, frac(_t, two)))
expr = Equals(Equals(_alpha_m, Mult(sub_expr3, Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr2, Qmult(NumBra(m, _t), InverseFourierTransform(_t), NumKet(k, _t))), domain = _m_domain))), Equals(_alpha_m, Mult(Exp(sub_expr3, two), Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr2, Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))), domain = _m_domain))))

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())

\left(\alpha_{m} = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \lvert k \rangle_{t}\right)\right)\right)\right)\right) = \left(\alpha_{m} = \left(\left(\frac{1}{2^{\frac{t}{2}}}\right)^{2} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right)\right)\right)

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: 5 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 5 operands: 4
3	Operation	operator: 5 operands: 6
4	ExprTuple	8, 7
5	Literal
6	ExprTuple	8, 9
7	Operation	operator: 77 operands: 10
8	Operation	operator: 11 operand: 84
9	Operation	operator: 77 operands: 13
10	ExprTuple	22, 14
11	Literal
12	ExprTuple	84
13	ExprTuple	15, 16
14	Operation	operator: 19 operand: 21
15	Operation	operator: 79 operands: 18
16	Operation	operator: 19 operand: 23
17	ExprTuple	21
18	ExprTuple	22, 85
19	Literal
20	ExprTuple	23
21	Lambda	parameter: 83 body: 24
22	Operation	operator: 70 operands: 25
23	Lambda	parameter: 83 body: 27
24	Conditional	value: 28 condition: 31
25	ExprTuple	76, 29
26	ExprTuple	83
27	Conditional	value: 30 condition: 31
28	Operation	operator: 77 operands: 32
29	Operation	operator: 79 operands: 33
30	Operation	operator: 77 operands: 34
31	Operation	operator: 35 operands: 36
32	ExprTuple	39, 37
33	ExprTuple	85, 38
34	ExprTuple	39, 40
35	Literal
36	ExprTuple	83, 41
37	Operation	operator: 42 operands: 43
38	Operation	operator: 70 operands: 44
39	Operation	operator: 79 operands: 45
40	Operation	operator: 79 operands: 46
41	Operation	operator: 47 operands: 48
42	Literal
43	ExprTuple	49, 50, 51
44	ExprTuple	86, 85
45	ExprTuple	53, 52
46	ExprTuple	53, 54
47	Literal
48	ExprTuple	55, 56
49	Operation	operator: 57 operands: 58
50	Operation	operator: 59 operand: 86
51	Operation	operator: 61 operands: 62
52	Operation	operator: 77 operands: 63
53	Literal
54	Operation	operator: 72 operand: 68
55	Literal
56	Operation	operator: 65 operands: 66
57	Literal
58	ExprTuple	84, 86
59	Literal
60	ExprTuple	86
61	Literal
62	ExprTuple	83, 86
63	ExprTuple	85, 81, 82, 67, 83
64	ExprTuple	68
65	Literal
66	ExprTuple	75, 69
67	Literal
68	Operation	operator: 70 operands: 71
69	Operation	operator: 72 operand: 76
70	Literal
71	ExprTuple	74, 75
72	Literal
73	ExprTuple	76
74	Operation	operator: 77 operands: 78
75	Operation	operator: 79 operands: 80
76	Literal
77	Literal
78	ExprTuple	85, 81, 82, 83, 84
79	Literal
80	ExprTuple	85, 86
81	Literal
82	Literal
83	Variable
84	Variable
85	Literal
86	Literal