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, VecSum
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, e, i, one, pi, two, zero
from proveit.physics.quantum import Ket, 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(Add(zero, one), one)), 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}^{1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle\right)\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert 1 \rangle\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: 8
4	Operation	operator: 17 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 42 body: 11
9	Operation	operator: 25 operands: 12
10	Operation	operator: 27 operand: 44
11	Conditional	value: 14 condition: 15
12	ExprTuple	31, 16
13	ExprTuple	44
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	Operation	operator: 34 operands: 21
17	Literal
18	ExprTuple	22, 23
19	Literal
20	ExprTuple	42, 24
21	ExprTuple	38, 39, 40, 41
22	Operation	operator: 25 operands: 26
23	Operation	operator: 27 operand: 42
24	Operation	operator: 29 operands: 30
25	Literal
26	ExprTuple	31, 32
27	Literal
28	ExprTuple	42
29	Literal
30	ExprTuple	33, 44
31	Literal
32	Operation	operator: 34 operands: 35
33	Operation	operator: 36 operands: 37
34	Literal
35	ExprTuple	38, 39, 40, 41, 42
36	Literal
37	ExprTuple	43, 44
38	Literal
39	Literal
40	Literal
41	Literal
42	Variable
43	Literal
44	Literal