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 Variable
from proveit.linear_algebra import Norm, ScalarMult, VecAdd
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, Neg, e, frac, i, one, pi, sqrt, two
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase

# build up the expression from sub-expressions
sub_expr1 = frac(one, sqrt(two))
expr = Equals(Norm(ScalarMult(sub_expr1, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Variable("_a", latex_format = r"{_{-}a}"))), _phase)), ket1)))), Mult(sub_expr1, sqrt(Add(Exp(Norm(ket0), two), Exp(Norm(ket1), two)))))

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}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-{_{-}a}} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right \| = \left(\frac{1}{\sqrt{2}} \cdot \sqrt{\left(\left \|\lvert 0 \rangle\right \|^{2} + \left \|\lvert 1 \rangle\right \|^{2}\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: 36 operand: 7
4	Operation	operator: 42 operands: 6
5	ExprTuple	7
6	ExprTuple	11, 8
7	Operation	operator: 24 operands: 9
8	Operation	operator: 53 operands: 10
9	ExprTuple	11, 12
10	ExprTuple	13, 28
11	Operation	operator: 32 operands: 14
12	Operation	operator: 15 operands: 16
13	Operation	operator: 17 operands: 18
14	ExprTuple	52, 19
15	Literal
16	ExprTuple	40, 20
17	Literal
18	ExprTuple	21, 22
19	Operation	operator: 53 operands: 23
20	Operation	operator: 24 operands: 25
21	Operation	operator: 53 operands: 26
22	Operation	operator: 53 operands: 27
23	ExprTuple	55, 28
24	Literal
25	ExprTuple	29, 41
26	ExprTuple	30, 55
27	ExprTuple	31, 55
28	Operation	operator: 32 operands: 33
29	Operation	operator: 53 operands: 34
30	Operation	operator: 36 operand: 40
31	Operation	operator: 36 operand: 41
32	Literal
33	ExprTuple	52, 55
34	ExprTuple	38, 39
35	ExprTuple	40
36	Literal
37	ExprTuple	41
38	Literal
39	Operation	operator: 42 operands: 43
40	Operation	operator: 45 operand: 51
41	Operation	operator: 45 operand: 52
42	Literal
43	ExprTuple	55, 47, 48, 49, 50
44	ExprTuple	51
45	Literal
46	ExprTuple	52
47	Literal
48	Literal
49	Operation	operator: 53 operands: 54
50	Literal
51	Literal
52	Literal
53	Literal
54	ExprTuple	55, 56
55	Literal
56	Operation	operator: 57 operand: 59
57	Literal
58	ExprTuple	59
59	Variable