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, TensorProd
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, e, i, one, pi, two
from proveit.physics.quantum import NumKet, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Add(k, two_pow_t)
expr = Equals(ScalarMult(Exp(e, Mult(two, pi, i, _phase, sub_expr1)), NumKet(sub_expr1, Add(t, one))), ScalarMult(Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Mult(two, pi, i, _phase, two_pow_t))), TensorProd(ket1, NumKet(k, t))))

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(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}}\right) \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\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: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 9
6	Literal
7	ExprTuple	10, 11
8	Operation	operator: 46 operands: 12
9	Operation	operator: 29 operands: 13
10	Operation	operator: 39 operands: 14
11	Operation	operator: 15 operands: 16
12	ExprTuple	33, 17
13	ExprTuple	31, 18
14	ExprTuple	19, 20
15	Literal
16	ExprTuple	21, 22
17	Operation	operator: 39 operands: 23
18	Operation	operator: 36 operands: 24
19	Operation	operator: 46 operands: 25
20	Operation	operator: 46 operands: 26
21	Operation	operator: 27 operand: 35
22	Operation	operator: 29 operands: 30
23	ExprTuple	48, 42, 43, 44, 31
24	ExprTuple	49, 35
25	ExprTuple	33, 32
26	ExprTuple	33, 34
27	Literal
28	ExprTuple	35
29	Literal
30	ExprTuple	41, 49
31	Operation	operator: 36 operands: 37
32	Operation	operator: 39 operands: 38
33	Literal
34	Operation	operator: 39 operands: 40
35	Literal
36	Literal
37	ExprTuple	41, 45
38	ExprTuple	48, 42, 43, 44, 41
39	Literal
40	ExprTuple	48, 42, 43, 44, 45
41	Variable
42	Literal
43	Literal
44	Literal
45	Operation	operator: 46 operands: 47
46	Literal
47	ExprTuple	48, 49
48	Literal
49	Variable