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
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, Neg, Sum, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
expr = Equals(Mult(frac(one, _two_pow_t), Sum(index_or_indices = [k], summand = Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, Mult(_two_pow_t, _phase)), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, k))), domain = _m_domain)), 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())

\left(\frac{1}{2^{t}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot \left(2^{t} \cdot \varphi\right)}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\right)\right)\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, 41
3	Operation	operator: 47 operands: 4
4	ExprTuple	5, 6
5	Operation	operator: 36 operands: 7
6	Operation	operator: 8 operand: 10
7	ExprTuple	41, 49
8	Literal
9	ExprTuple	10
10	Lambda	parameter: 45 body: 12
11	ExprTuple	45
12	Conditional	value: 13 condition: 14
13	Operation	operator: 47 operands: 15
14	Operation	operator: 16 operands: 17
15	ExprTuple	18, 19
16	Literal
17	ExprTuple	45, 20
18	Operation	operator: 51 operands: 21
19	Operation	operator: 51 operands: 22
20	Operation	operator: 23 operands: 24
21	ExprTuple	26, 25
22	ExprTuple	26, 27
23	Literal
24	ExprTuple	28, 29
25	Operation	operator: 38 operand: 34
26	Literal
27	Operation	operator: 47 operands: 31
28	Literal
29	Operation	operator: 32 operands: 33
30	ExprTuple	34
31	ExprTuple	53, 43, 44, 50, 45
32	Literal
33	ExprTuple	49, 35
34	Operation	operator: 36 operands: 37
35	Operation	operator: 38 operand: 41
36	Literal
37	ExprTuple	40, 49
38	Literal
39	ExprTuple	41
40	Operation	operator: 47 operands: 42
41	Literal
42	ExprTuple	53, 43, 44, 45, 46
43	Literal
44	Literal
45	Variable
46	Operation	operator: 47 operands: 48
47	Literal
48	ExprTuple	49, 50
49	Operation	operator: 51 operands: 52
50	Literal
51	Literal
52	ExprTuple	53, 54
53	Literal
54	Literal