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, m
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, e, i, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _m_domain, _phase, _t

# build up the expression from sub-expressions
sub_expr1 = NumBra(m, _t)
sub_expr2 = InverseFourierTransform(_t)
sub_expr3 = VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t)), domain = _m_domain)
expr = Equals(Qmult(sub_expr1, sub_expr2, sub_expr3), Qmult(Qmult(sub_expr1, sub_expr2), sub_expr3)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \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) =  \\ \left(\left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t}\right) \thinspace \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) \end{array} \end{array}

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.	()	(2)	('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: 9 operands: 5
4	Operation	operator: 9 operands: 6
5	ExprTuple	13, 14, 8
6	ExprTuple	7, 8
7	Operation	operator: 9 operands: 10
8	Operation	operator: 11 operand: 15
9	Literal
10	ExprTuple	13, 14
11	Literal
12	ExprTuple	15
13	Operation	operator: 16 operands: 17
14	Operation	operator: 18 operand: 56
15	Lambda	parameter: 48 body: 21
16	Literal
17	ExprTuple	22, 56
18	Literal
19	ExprTuple	56
20	ExprTuple	48
21	Conditional	value: 23 condition: 24
22	Variable
23	Operation	operator: 25 operands: 26
24	Operation	operator: 27 operands: 28
25	Literal
26	ExprTuple	29, 30
27	Literal
28	ExprTuple	48, 31
29	Operation	operator: 51 operands: 32
30	Operation	operator: 33 operands: 34
31	Operation	operator: 35 operands: 36
32	ExprTuple	37, 38
33	Literal
34	ExprTuple	48, 56
35	Literal
36	ExprTuple	39, 40
37	Literal
38	Operation	operator: 41 operands: 42
39	Literal
40	Operation	operator: 43 operands: 44
41	Literal
42	ExprTuple	55, 45, 46, 47, 48
43	Literal
44	ExprTuple	49, 50
45	Literal
46	Literal
47	Literal
48	Variable
49	Operation	operator: 51 operands: 52
50	Operation	operator: 53 operand: 57
51	Literal
52	ExprTuple	55, 56
53	Literal
54	ExprTuple	57
55	Literal
56	Literal
57	Literal