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, 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

# build up the expression from sub-expressions
expr = Equals(_alpha_m, Mult(frac(one, Exp(two, frac(_t, two))), Qmult(NumBra(m, _t), InverseFourierTransform(_t), VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _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())

\alpha_{m} = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{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: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operand: 25
4	Operation	operator: 49 operands: 7
5	Literal
6	ExprTuple	25
7	ExprTuple	8, 9
8	Operation	operator: 27 operands: 10
9	Operation	operator: 11 operands: 12
10	ExprTuple	65, 13
11	Literal
12	ExprTuple	14, 15, 16
13	Operation	operator: 59 operands: 17
14	Operation	operator: 18 operands: 19
15	Operation	operator: 20 operand: 64
16	Operation	operator: 22 operand: 26
17	ExprTuple	63, 24
18	Literal
19	ExprTuple	25, 64
20	Literal
21	ExprTuple	64
22	Literal
23	ExprTuple	26
24	Operation	operator: 27 operands: 28
25	Variable
26	Lambda	parameter: 56 body: 30
27	Literal
28	ExprTuple	64, 63
29	ExprTuple	56
30	Conditional	value: 31 condition: 32
31	Operation	operator: 33 operands: 34
32	Operation	operator: 35 operands: 36
33	Literal
34	ExprTuple	37, 38
35	Literal
36	ExprTuple	56, 39
37	Operation	operator: 59 operands: 40
38	Operation	operator: 41 operands: 42
39	Operation	operator: 43 operands: 44
40	ExprTuple	45, 46
41	Literal
42	ExprTuple	56, 64
43	Literal
44	ExprTuple	47, 48
45	Literal
46	Operation	operator: 49 operands: 50
47	Literal
48	Operation	operator: 51 operands: 52
49	Literal
50	ExprTuple	63, 53, 54, 55, 56
51	Literal
52	ExprTuple	57, 58
53	Literal
54	Literal
55	Literal
56	Variable
57	Operation	operator: 59 operands: 60
58	Operation	operator: 61 operand: 65
59	Literal
60	ExprTuple	63, 64
61	Literal
62	ExprTuple	65
63	Literal
64	Literal
65	Literal