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(ScalarMult(sub_expr4, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, sub_expr3), domain = sub_expr5)), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Mult(sub_expr4, sub_expr2), sub_expr3), domain = sub_expr5)).with_wrapping_at(1)

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(\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) \\  = \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) \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.	()	(1)	('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: 20 operands: 5
4	Operation	operator: 9 operand: 8
5	ExprTuple	27, 7
6	ExprTuple	8
7	Operation	operator: 9 operand: 12
8	Lambda	parameter: 55 body: 11
9	Literal
10	ExprTuple	12
11	Conditional	value: 13 condition: 18
12	Lambda	parameter: 55 body: 15
13	Operation	operator: 20 operands: 16
14	ExprTuple	55
15	Conditional	value: 17 condition: 18
16	ExprTuple	19, 25
17	Operation	operator: 20 operands: 21
18	Operation	operator: 22 operands: 23
19	Operation	operator: 48 operands: 24
20	Literal
21	ExprTuple	28, 25
22	Literal
23	ExprTuple	55, 26
24	ExprTuple	27, 28
25	Operation	operator: 29 operands: 30
26	Operation	operator: 31 operands: 32
27	Operation	operator: 58 operands: 33
28	Operation	operator: 58 operands: 34
29	Literal
30	ExprTuple	35, 36
31	Literal
32	ExprTuple	37, 38
33	ExprTuple	40, 39
34	ExprTuple	40, 41
35	Operation	operator: 42 operand: 60
36	Operation	operator: 43 operands: 44
37	Literal
38	Operation	operator: 45 operands: 46
39	Operation	operator: 48 operands: 47
40	Literal
41	Operation	operator: 48 operands: 49
42	Literal
43	Literal
44	ExprTuple	55, 62
45	Literal
46	ExprTuple	51, 50
47	ExprTuple	61, 52, 53, 54, 51
48	Literal
49	ExprTuple	61, 52, 53, 54, 55
50	Operation	operator: 56 operand: 60
51	Operation	operator: 58 operands: 59
52	Literal
53	Literal
54	Literal
55	Variable
56	Literal
57	ExprTuple	60
58	Literal
59	ExprTuple	61, 62
60	Literal
61	Literal
62	Variable