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 = NumKet(k, t)
sub_expr3 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr4 = Interval(zero, subtract(two_pow_t, one))
expr = Equals(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr3, TensorProd(ket1, sub_expr2)), domain = sub_expr4), TensorProd(ket1, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr3, sub_expr2), domain = sub_expr4)))

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(\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) = \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)

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