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.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 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 = InverseFourierTransform(_t)
sub_expr2 = frac(one, Exp(two, frac(_t, two)))
sub_expr3 = VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t)), domain = _m_domain)
expr = Equals(Qmult(sub_expr1, ScalarMult(sub_expr2, sub_expr3)), Qmult(sub_expr1, sub_expr2, sub_expr3)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left({\mathrm {FT}}^{\dag}_{t} \thinspace \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 \lvert k \rangle_{t}\right)\right)\right)\right) =  \\ \left({\mathrm {FT}}^{\dag}_{t} \thinspace \frac{1}{2^{\frac{t}{2}}} \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) \end{array} \end{array}

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