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, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Interval, Mult, e, frac, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet
from proveit.physics.quantum.QPE import _phase, _psi_t_ket, two_pow_t

# build up the expression from sub-expressions
expr = Equals(_psi_t_ket, ScalarMult(frac(one, Exp(two, frac(t, two))), VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, t)), domain = Interval(zero, subtract(two_pow_t, one)))))

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())

\lvert \psi_{t} \rangle = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\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: 54
4	Operation	operator: 23 operands: 7
5	Literal
6	ExprTuple	54
7	ExprTuple	8, 9
8	Operation	operator: 21 operands: 10
9	Operation	operator: 11 operand: 14
10	ExprTuple	55, 13
11	Literal
12	ExprTuple	14
13	Operation	operator: 49 operands: 15
14	Lambda	parameter: 46 body: 17
15	ExprTuple	53, 18
16	ExprTuple	46
17	Conditional	value: 19 condition: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	Operation	operator: 25 operands: 26
21	Literal
22	ExprTuple	54, 53
23	Literal
24	ExprTuple	27, 28
25	Literal
26	ExprTuple	46, 29
27	Operation	operator: 49 operands: 30
28	Operation	operator: 31 operands: 32
29	Operation	operator: 33 operands: 34
30	ExprTuple	35, 36
31	Literal
32	ExprTuple	46, 54
33	Literal
34	ExprTuple	37, 38
35	Literal
36	Operation	operator: 39 operands: 40
37	Literal
38	Operation	operator: 41 operands: 42
39	Literal
40	ExprTuple	53, 43, 44, 45, 46
41	Literal
42	ExprTuple	47, 48
43	Literal
44	Literal
45	Literal
46	Variable
47	Operation	operator: 49 operands: 50
48	Operation	operator: 51 operand: 55
49	Literal
50	ExprTuple	53, 54
51	Literal
52	ExprTuple	55
53	Literal
54	Variable
55	Literal