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, frac, i, one, 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), frac(one, Exp(two, frac(_t, two))), 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 \frac{1}{2^{\frac{t}{2}}} \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: 11
4	Literal
5	ExprTuple	8, 9, 10
6	Literal
7	ExprTuple	11
8	Operation	operator: 12 operands: 13
9	Operation	operator: 32 operands: 14
10	Operation	operator: 15 operand: 58
11	Lambda	parameter: 50 body: 18
12	Literal
13	ExprTuple	19, 58
14	ExprTuple	59, 20
15	Literal
16	ExprTuple	58
17	ExprTuple	50
18	Conditional	value: 21 condition: 22
19	Variable
20	Operation	operator: 53 operands: 23
21	Operation	operator: 24 operands: 25
22	Operation	operator: 26 operands: 27
23	ExprTuple	57, 28
24	Literal
25	ExprTuple	29, 30
26	Literal
27	ExprTuple	50, 31
28	Operation	operator: 32 operands: 33
29	Operation	operator: 53 operands: 34
30	Operation	operator: 35 operands: 36
31	Operation	operator: 37 operands: 38
32	Literal
33	ExprTuple	58, 57
34	ExprTuple	39, 40
35	Literal
36	ExprTuple	50, 58
37	Literal
38	ExprTuple	41, 42
39	Literal
40	Operation	operator: 43 operands: 44
41	Literal
42	Operation	operator: 45 operands: 46
43	Literal
44	ExprTuple	57, 47, 48, 49, 50
45	Literal
46	ExprTuple	51, 52
47	Literal
48	Literal
49	Literal
50	Variable
51	Operation	operator: 53 operands: 54
52	Operation	operator: 55 operand: 59
53	Literal
54	ExprTuple	57, 58
55	Literal
56	ExprTuple	59
57	Literal
58	Literal
59	Literal