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
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, e, i, one, pi, two
from proveit.physics.quantum import NumKet
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Add(k, two_pow_t)
sub_expr2 = NumKet(sub_expr1, Add(t, one))
expr = Equals(ScalarMult(Exp(e, Mult(two, pi, i, _phase, sub_expr1)), sub_expr2), ScalarMult(Exp(e, Add(Mult(two, pi, i, _phase, k), Mult(two, pi, i, _phase, two_pow_t))), sub_expr2))

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(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot \left(k + 2^{t}\right)} \cdot \lvert k + 2^{t} \rangle_{t + 1}\right) = \left(\mathsf{e}^{\left(2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k\right) + \left(2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}\right)} \cdot \lvert k + 2^{t} \rangle_{t + 1}\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: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 10
6	Literal
7	ExprTuple	9, 10
8	Operation	operator: 36 operands: 11
9	Operation	operator: 36 operands: 12
10	Operation	operator: 13 operands: 14
11	ExprTuple	16, 15
12	ExprTuple	16, 17
13	Literal
14	ExprTuple	22, 18
15	Operation	operator: 29 operands: 19
16	Literal
17	Operation	operator: 26 operands: 20
18	Operation	operator: 26 operands: 21
19	ExprTuple	38, 32, 33, 34, 22
20	ExprTuple	23, 24
21	ExprTuple	39, 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 29 operands: 28
24	Operation	operator: 29 operands: 30
25	Literal
26	Literal
27	ExprTuple	31, 35
28	ExprTuple	38, 32, 33, 34, 31
29	Literal
30	ExprTuple	38, 32, 33, 34, 35
31	Variable
32	Literal
33	Literal
34	Literal
35	Operation	operator: 36 operands: 37
36	Literal
37	ExprTuple	38, 39
38	Literal
39	Variable