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 Variable, k, t
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, e, frac, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Add(t, one)
expr = Equals(ScalarMult(frac(one, Exp(two, frac(sub_expr1, two))), VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, sub_expr1)), domain = Interval(zero, subtract(Mult(two, two_pow_t), one)))), Variable("_b", latex_format = r"{_{-}b}"))

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^{\frac{t + 1}{2}}} \cdot \left(\sum_{k=0}^{\left(2 \cdot 2^{t}\right) - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t + 1}\right)\right)\right) = {_{-}b}

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