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 Add, 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 = [k]
sub_expr2 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr3 = ScalarMult(sub_expr2, NumKet(k, Add(t, one)))
sub_expr4 = Interval(zero, subtract(two_pow_t, one))
sub_expr5 = VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, NumKet(k, t)), domain = sub_expr4)
sub_expr6 = VecSum(index_or_indices = sub_expr1, summand = sub_expr3, domain = Interval(zero, subtract(Mult(two, two_pow_t), one)))
sub_expr7 = ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), TensorProd(ket1, sub_expr5))
expr = Equals(Equals(sub_expr6, VecAdd(VecSum(index_or_indices = sub_expr1, summand = sub_expr3, domain = sub_expr4), sub_expr7)), Equals(sub_expr6, VecAdd(TensorProd(ket0, sub_expr5), sub_expr7)))

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(\sum_{k=0}^{\left(2 \cdot 2^{t}\right) - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t + 1}\right)\right) = \left(\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 + 1}\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)\right) = \left(\left(\sum_{k=0}^{\left(2 \cdot 2^{t}\right) - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t + 1}\right)\right) = \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)\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: 5 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 5 operands: 4
3	Operation	operator: 5 operands: 6
4	ExprTuple	8, 7
5	Literal
6	ExprTuple	8, 9
7	Operation	operator: 12 operands: 10
8	Operation	operator: 42 operand: 15
9	Operation	operator: 12 operands: 13
10	ExprTuple	14, 17
11	ExprTuple	15
12	Literal
13	ExprTuple	16, 17
14	Operation	operator: 42 operand: 22
15	Lambda	parameter: 80 body: 19
16	Operation	operator: 31 operands: 20
17	Operation	operator: 57 operands: 21
18	ExprTuple	22
19	Conditional	value: 33 condition: 23
20	ExprTuple	24, 37
21	ExprTuple	25, 26
22	Lambda	parameter: 80 body: 27
23	Operation	operator: 59 operands: 28
24	Operation	operator: 41 operand: 71
25	Operation	operator: 83 operands: 30
26	Operation	operator: 31 operands: 32
27	Conditional	value: 33 condition: 54
28	ExprTuple	80, 34
29	ExprTuple	71
30	ExprTuple	69, 35
31	Literal
32	ExprTuple	36, 37
33	Operation	operator: 57 operands: 38
34	Operation	operator: 67 operands: 39
35	Operation	operator: 73 operands: 40
36	Operation	operator: 41 operand: 89
37	Operation	operator: 42 operand: 46
38	ExprTuple	61, 44
39	ExprTuple	71, 45
40	ExprTuple	87, 77, 78, 79, 81
41	Literal
42	Literal
43	ExprTuple	46
44	Operation	operator: 65 operands: 47
45	Operation	operator: 75 operands: 48
46	Lambda	parameter: 80 body: 50
47	ExprTuple	80, 51
48	ExprTuple	52, 82
49	ExprTuple	80
50	Conditional	value: 53 condition: 54
51	Operation	operator: 75 operands: 55
52	Operation	operator: 73 operands: 56
53	Operation	operator: 57 operands: 58
54	Operation	operator: 59 operands: 60
55	ExprTuple	88, 89
56	ExprTuple	87, 81
57	Literal
58	ExprTuple	61, 62
59	Literal
60	ExprTuple	80, 63
61	Operation	operator: 83 operands: 64
62	Operation	operator: 65 operands: 66
63	Operation	operator: 67 operands: 68
64	ExprTuple	69, 70
65	Literal
66	ExprTuple	80, 88
67	Literal
68	ExprTuple	71, 72
69	Literal
70	Operation	operator: 73 operands: 74
71	Literal
72	Operation	operator: 75 operands: 76
73	Literal
74	ExprTuple	87, 77, 78, 79, 80
75	Literal
76	ExprTuple	81, 82
77	Literal
78	Literal
79	Literal
80	Variable
81	Operation	operator: 83 operands: 84
82	Operation	operator: 85 operand: 89
83	Literal
84	ExprTuple	87, 88
85	Literal
86	ExprTuple	89
87	Literal
88	Variable
89	Literal