Expression of type Qmult

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.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
expr = Qmult(Qmult(NumBra(m, _t), InverseFourierTransform(_t)), VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t)), domain = _m_domain))

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({_{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)

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',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 4 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 4 operands: 5
3	Operation	operator: 6 operand: 10
4	Literal
5	ExprTuple	8, 9
6	Literal
7	ExprTuple	10
8	Operation	operator: 11 operands: 12
9	Operation	operator: 13 operand: 51
10	Lambda	parameter: 43 body: 16
11	Literal
12	ExprTuple	17, 51
13	Literal
14	ExprTuple	51
15	ExprTuple	43
16	Conditional	value: 18 condition: 19
17	Variable
18	Operation	operator: 20 operands: 21
19	Operation	operator: 22 operands: 23
20	Literal
21	ExprTuple	24, 25
22	Literal
23	ExprTuple	43, 26
24	Operation	operator: 46 operands: 27
25	Operation	operator: 28 operands: 29
26	Operation	operator: 30 operands: 31
27	ExprTuple	32, 33
28	Literal
29	ExprTuple	43, 51
30	Literal
31	ExprTuple	34, 35
32	Literal
33	Operation	operator: 36 operands: 37
34	Literal
35	Operation	operator: 38 operands: 39
36	Literal
37	ExprTuple	50, 40, 41, 42, 43
38	Literal
39	ExprTuple	44, 45
40	Literal
41	Literal
42	Literal
43	Variable
44	Operation	operator: 46 operands: 47
45	Operation	operator: 48 operand: 52
46	Literal
47	ExprTuple	50, 51
48	Literal
49	ExprTuple	52
50	Literal
51	Literal
52	Literal