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_expr4, sub_expr2), sub_expr3), domain = sub_expr5), ScalarMult(sub_expr4, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(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 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) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left(\lvert 1 \rangle {\otimes} \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: 10 operand: 7
4	Operation	operator: 21 operands: 6
5	ExprTuple	7
6	ExprTuple	25, 8
7	Lambda	parameter: 53 body: 9
8	Operation	operator: 10 operand: 13
9	Conditional	value: 12 condition: 19
10	Literal
11	ExprTuple	13
12	Operation	operator: 21 operands: 14
13	Lambda	parameter: 53 body: 16
14	ExprTuple	17, 27
15	ExprTuple	53
16	Conditional	value: 18 condition: 19
17	Operation	operator: 43 operands: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	ExprTuple	25, 26
21	Literal
22	ExprTuple	26, 27
23	Literal
24	ExprTuple	53, 28
25	Operation	operator: 56 operands: 29
26	Operation	operator: 56 operands: 30
27	Operation	operator: 31 operands: 32
28	Operation	operator: 33 operands: 34
29	ExprTuple	36, 35
30	ExprTuple	36, 37
31	Literal
32	ExprTuple	38, 39
33	Literal
34	ExprTuple	40, 41
35	Operation	operator: 43 operands: 42
36	Literal
37	Operation	operator: 43 operands: 44
38	Operation	operator: 45 operand: 62
39	Operation	operator: 46 operands: 47
40	Literal
41	Operation	operator: 48 operands: 49
42	ExprTuple	60, 50, 51, 52, 54
43	Literal
44	ExprTuple	60, 50, 51, 52, 53
45	Literal
46	Literal
47	ExprTuple	53, 61
48	Literal
49	ExprTuple	54, 55
50	Literal
51	Literal
52	Literal
53	Variable
54	Operation	operator: 56 operands: 57
55	Operation	operator: 58 operand: 62
56	Literal
57	ExprTuple	60, 61
58	Literal
59	ExprTuple	62
60	Literal
61	Variable
62	Literal