Expression of type ScalarMult

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.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, two_pow_t

# build up the expression from sub-expressions
expr = 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())

\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)

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