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, i, pi, sqrt, two
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase

# build up the expression from sub-expressions
expr = Equals(Norm(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Variable("_a", latex_format = r"{_{-}a}"))), _phase)), ket1))), 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 \|\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-{_{-}a}} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right \| = \sqrt{\left(\left \|\lvert 0 \rangle\right \|^{2} + \left \|\lvert 1 \rangle\right \|^{2}\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: 28 operand: 7
4	Operation	operator: 45 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Operation	operator: 10 operands: 11
8	Operation	operator: 12 operands: 13
9	Operation	operator: 14 operands: 15
10	Literal
11	ExprTuple	32, 16
12	Literal
13	ExprTuple	17, 18
14	Literal
15	ExprTuple	44, 47
16	Operation	operator: 19 operands: 20
17	Operation	operator: 45 operands: 21
18	Operation	operator: 45 operands: 22
19	Literal
20	ExprTuple	23, 33
21	ExprTuple	24, 47
22	ExprTuple	25, 47
23	Operation	operator: 45 operands: 26
24	Operation	operator: 28 operand: 32
25	Operation	operator: 28 operand: 33
26	ExprTuple	30, 31
27	ExprTuple	32
28	Literal
29	ExprTuple	33
30	Literal
31	Operation	operator: 34 operands: 35
32	Operation	operator: 37 operand: 43
33	Operation	operator: 37 operand: 44
34	Literal
35	ExprTuple	47, 39, 40, 41, 42
36	ExprTuple	43
37	Literal
38	ExprTuple	44
39	Literal
40	Literal
41	Operation	operator: 45 operands: 46
42	Literal
43	Literal
44	Literal
45	Literal
46	ExprTuple	47, 48
47	Literal
48	Operation	operator: 49 operand: 51
49	Literal
50	ExprTuple	51
51	Variable