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, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr3 = TensorProd(ket1, NumKet(k, t))
sub_expr4 = Exp(e, Mult(two, pi, i, _phase, two_pow_t))
sub_expr5 = Interval(zero, subtract(two_pow_t, one))
expr = Equals(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Mult(sub_expr2, sub_expr4), sub_expr3), domain = sub_expr5), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Mult(sub_expr4, sub_expr2), sub_expr3), domain = sub_expr5))

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(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}}\right) \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\right)\right)\right) = \left(\sum_{k=0}^{2^{t} - 1} \left(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\right) \cdot \left(\lvert 1 \rangle {\otimes} \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: 57 body: 10
9	Lambda	parameter: 57 body: 12
10	Conditional	value: 13 condition: 15
11	ExprTuple	57
12	Conditional	value: 14 condition: 15
13	Operation	operator: 17 operands: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	ExprTuple	21, 23
17	Literal
18	ExprTuple	22, 23
19	Literal
20	ExprTuple	57, 24
21	Operation	operator: 49 operands: 25
22	Operation	operator: 49 operands: 26
23	Operation	operator: 27 operands: 28
24	Operation	operator: 29 operands: 30
25	ExprTuple	32, 31
26	ExprTuple	31, 32
27	Literal
28	ExprTuple	33, 34
29	Literal
30	ExprTuple	35, 36
31	Operation	operator: 59 operands: 37
32	Operation	operator: 59 operands: 38
33	Operation	operator: 39 operand: 58
34	Operation	operator: 40 operands: 41
35	Literal
36	Operation	operator: 42 operands: 43
37	ExprTuple	45, 44
38	ExprTuple	45, 46
39	Literal
40	Literal
41	ExprTuple	57, 62
42	Literal
43	ExprTuple	53, 47
44	Operation	operator: 49 operands: 48
45	Literal
46	Operation	operator: 49 operands: 50
47	Operation	operator: 51 operand: 58
48	ExprTuple	61, 54, 55, 56, 53
49	Literal
50	ExprTuple	61, 54, 55, 56, 57
51	Literal
52	ExprTuple	58
53	Operation	operator: 59 operands: 60
54	Literal
55	Literal
56	Literal
57	Variable
58	Literal
59	Literal
60	ExprTuple	61, 62
61	Literal
62	Variable