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
from proveit.linear_algebra import ScalarMult, VecAdd, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Interval, Mult, e, i, one, pi, two, zero
from proveit.physics.quantum import Ket, ket0, ket1
from proveit.physics.quantum.QPE import _phase

# build up the expression from sub-expressions
expr = Equals(VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), Ket(k)), domain = Interval(zero, one)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase)), ket1)))

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(\sum_{k=0}^{1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle\right)\right) = \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert 1 \rangle\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: 5 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 46 body: 12
10	Operation	operator: 31 operand: 38
11	Operation	operator: 19 operands: 14
12	Conditional	value: 15 condition: 16
13	ExprTuple	38
14	ExprTuple	17, 18
15	Operation	operator: 19 operands: 20
16	Operation	operator: 21 operands: 22
17	Operation	operator: 29 operands: 23
18	Operation	operator: 31 operand: 39
19	Literal
20	ExprTuple	25, 26
21	Literal
22	ExprTuple	46, 27
23	ExprTuple	36, 28
24	ExprTuple	39
25	Operation	operator: 29 operands: 30
26	Operation	operator: 31 operand: 46
27	Operation	operator: 33 operands: 34
28	Operation	operator: 40 operands: 35
29	Literal
30	ExprTuple	36, 37
31	Literal
32	ExprTuple	46
33	Literal
34	ExprTuple	38, 39
35	ExprTuple	42, 43, 44, 45
36	Literal
37	Operation	operator: 40 operands: 41
38	Literal
39	Literal
40	Literal
41	ExprTuple	42, 43, 44, 45, 46
42	Literal
43	Literal
44	Literal
45	Literal
46	Variable