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.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, e, i, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _m_domain, _phase, _t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = NumBra(m, _t)
sub_expr3 = InverseFourierTransform(_t)
sub_expr4 = NumKet(k, _t)
sub_expr5 = Exp(e, Mult(two, pi, i, _phase, k))
expr = Equals(VecSum(index_or_indices = sub_expr1, summand = Qmult(sub_expr2, sub_expr3, ScalarMult(sub_expr5, sub_expr4)), domain = _m_domain), VecSum(index_or_indices = sub_expr1, summand = Mult(sub_expr5, Qmult(sub_expr2, sub_expr3, sub_expr4)), 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(\sum_{k=0}^{2^{t} - 1} \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right) = \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)

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, 4
3	Operation	operator: 6 operand: 8
4	Operation	operator: 6 operand: 9
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	Lambda	parameter: 58 body: 10
9	Lambda	parameter: 58 body: 12
10	Conditional	value: 13 condition: 15
11	ExprTuple	58
12	Conditional	value: 14 condition: 15
13	Operation	operator: 25 operands: 16
14	Operation	operator: 49 operands: 17
15	Operation	operator: 18 operands: 19
16	ExprTuple	30, 31, 20
17	ExprTuple	29, 21
18	Literal
19	ExprTuple	58, 22
20	Operation	operator: 23 operands: 24
21	Operation	operator: 25 operands: 26
22	Operation	operator: 27 operands: 28
23	Literal
24	ExprTuple	29, 32
25	Literal
26	ExprTuple	30, 31, 32
27	Literal
28	ExprTuple	33, 34
29	Operation	operator: 51 operands: 35
30	Operation	operator: 36 operands: 37
31	Operation	operator: 38 operand: 60
32	Operation	operator: 40 operands: 41
33	Literal
34	Operation	operator: 42 operands: 43
35	ExprTuple	44, 45
36	Literal
37	ExprTuple	46, 60
38	Literal
39	ExprTuple	60
40	Literal
41	ExprTuple	58, 60
42	Literal
43	ExprTuple	47, 48
44	Literal
45	Operation	operator: 49 operands: 50
46	Variable
47	Operation	operator: 51 operands: 52
48	Operation	operator: 53 operand: 61
49	Literal
50	ExprTuple	59, 55, 56, 57, 58
51	Literal
52	ExprTuple	59, 60
53	Literal
54	ExprTuple	61
55	Literal
56	Literal
57	Literal
58	Variable
59	Literal
60	Literal
61	Literal