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 = frac(t, two)
sub_expr2 = frac(Add(t, one), two)
sub_expr3 = TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1)), 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(ScalarMult(Mult(frac(one, Exp(two, subtract(sub_expr2, sub_expr1))), frac(one, Exp(two, sub_expr1))), sub_expr3), ScalarMult(frac(one, Exp(two, sub_expr2)), sub_expr3))

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