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, VecAdd, 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, ket0, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Exp(e, Mult(two, pi, i, _phase, two_pow_t))
sub_expr2 = VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, t)), domain = Interval(zero, subtract(two_pow_t, one)))
expr = Equals(VecAdd(TensorProd(ket0, sub_expr2), ScalarMult(sub_expr1, TensorProd(ket1, sub_expr2))), TensorProd(VecAdd(ket0, ScalarMult(sub_expr1, ket1)), sub_expr2))

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(\left(\lvert 0 \rangle {\otimes} \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) + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \left(\lvert 1 \rangle {\otimes} \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)\right) = \left(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\right) {\otimes} \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)

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: 12 operands: 5
4	Operation	operator: 17 operands: 6
5	ExprTuple	7, 8
6	ExprTuple	9, 21
7	Operation	operator: 17 operands: 10
8	Operation	operator: 35 operands: 11
9	Operation	operator: 12 operands: 13
10	ExprTuple	15, 21
11	ExprTuple	22, 14
12	Literal
13	ExprTuple	15, 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 27 operand: 49
16	Operation	operator: 35 operands: 20
17	Literal
18	ExprTuple	23, 21
19	ExprTuple	49
20	ExprTuple	22, 23
21	Operation	operator: 24 operand: 28
22	Operation	operator: 61 operands: 26
23	Operation	operator: 27 operand: 67
24	Literal
25	ExprTuple	28
26	ExprTuple	47, 29
27	Literal
28	Lambda	parameter: 58 body: 31
29	Operation	operator: 51 operands: 32
30	ExprTuple	58
31	Conditional	value: 33 condition: 34
32	ExprTuple	65, 55, 56, 57, 59
33	Operation	operator: 35 operands: 36
34	Operation	operator: 37 operands: 38
35	Literal
36	ExprTuple	39, 40
37	Literal
38	ExprTuple	58, 41
39	Operation	operator: 61 operands: 42
40	Operation	operator: 43 operands: 44
41	Operation	operator: 45 operands: 46
42	ExprTuple	47, 48
43	Literal
44	ExprTuple	58, 66
45	Literal
46	ExprTuple	49, 50
47	Literal
48	Operation	operator: 51 operands: 52
49	Literal
50	Operation	operator: 53 operands: 54
51	Literal
52	ExprTuple	65, 55, 56, 57, 58
53	Literal
54	ExprTuple	59, 60
55	Literal
56	Literal
57	Literal
58	Variable
59	Operation	operator: 61 operands: 62
60	Operation	operator: 63 operand: 67
61	Literal
62	ExprTuple	65, 66
63	Literal
64	ExprTuple	67
65	Literal
66	Variable
67	Literal