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, frac, 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 = Add(t, one)
sub_expr3 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr4 = frac(one, Exp(two, frac(sub_expr2, two)))
sub_expr5 = VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr3, NumKet(k, t)), domain = Interval(zero, subtract(two_pow_t, one)))
expr = Equals(ScalarMult(sub_expr4, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr3, NumKet(k, sub_expr2)), domain = Interval(zero, subtract(Mult(two, two_pow_t), one)))), ScalarMult(sub_expr4, VecAdd(TensorProd(ket0, sub_expr5), ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), TensorProd(ket1, 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(\frac{1}{2^{\frac{t + 1}{2}}} \cdot \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)\right) = \left(\frac{1}{2^{\frac{t + 1}{2}}} \cdot \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: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 58 operands: 5
4	Operation	operator: 58 operands: 6
5	ExprTuple	8, 7
6	ExprTuple	8, 9
7	Operation	operator: 45 operand: 14
8	Operation	operator: 30 operands: 11
9	Operation	operator: 12 operands: 13
10	ExprTuple	14
11	ExprTuple	90, 15
12	Literal
13	ExprTuple	16, 17
14	Lambda	parameter: 81 body: 18
15	Operation	operator: 84 operands: 19
16	Operation	operator: 34 operands: 20
17	Operation	operator: 58 operands: 21
18	Conditional	value: 22 condition: 23
19	ExprTuple	88, 24
20	ExprTuple	25, 40
21	ExprTuple	26, 27
22	Operation	operator: 58 operands: 28
23	Operation	operator: 60 operands: 29
24	Operation	operator: 30 operands: 31
25	Operation	operator: 44 operand: 72
26	Operation	operator: 84 operands: 33
27	Operation	operator: 34 operands: 35
28	ExprTuple	62, 36
29	ExprTuple	81, 37
30	Literal
31	ExprTuple	47, 88
32	ExprTuple	72
33	ExprTuple	70, 38
34	Literal
35	ExprTuple	39, 40
36	Operation	operator: 66 operands: 41
37	Operation	operator: 68 operands: 42
38	Operation	operator: 74 operands: 43
39	Operation	operator: 44 operand: 90
40	Operation	operator: 45 operand: 49
41	ExprTuple	81, 47
42	ExprTuple	72, 48
43	ExprTuple	88, 78, 79, 80, 82
44	Literal
45	Literal
46	ExprTuple	49
47	Operation	operator: 76 operands: 50
48	Operation	operator: 76 operands: 51
49	Lambda	parameter: 81 body: 53
50	ExprTuple	89, 90
51	ExprTuple	54, 83
52	ExprTuple	81
53	Conditional	value: 55 condition: 56
54	Operation	operator: 74 operands: 57
55	Operation	operator: 58 operands: 59
56	Operation	operator: 60 operands: 61
57	ExprTuple	88, 82
58	Literal
59	ExprTuple	62, 63
60	Literal
61	ExprTuple	81, 64
62	Operation	operator: 84 operands: 65
63	Operation	operator: 66 operands: 67
64	Operation	operator: 68 operands: 69
65	ExprTuple	70, 71
66	Literal
67	ExprTuple	81, 89
68	Literal
69	ExprTuple	72, 73
70	Literal
71	Operation	operator: 74 operands: 75
72	Literal
73	Operation	operator: 76 operands: 77
74	Literal
75	ExprTuple	88, 78, 79, 80, 81
76	Literal
77	ExprTuple	82, 83
78	Literal
79	Literal
80	Literal
81	Variable
82	Operation	operator: 84 operands: 85
83	Operation	operator: 86 operand: 90
84	Literal
85	ExprTuple	88, 89
86	Literal
87	ExprTuple	90
88	Literal
89	Variable
90	Literal