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 Variable, k, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd, VecSum
from proveit.numbers import Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, ket0, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
expr = ScalarMult(Variable("_a", latex_format = r"{_{-}a}"), TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1)), 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())

{_{-}a} \cdot \left(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\right) {\otimes} \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',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 25 operands: 1
1	ExprTuple	2, 3
2	Variable
3	Operation	operator: 4 operands: 5
4	Literal
5	ExprTuple	6, 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operand: 14
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14
12	Operation	operator: 24 operand: 41
13	Operation	operator: 25 operands: 16
14	Lambda	parameter: 50 body: 18
15	ExprTuple	41
16	ExprTuple	19, 20
17	ExprTuple	50
18	Conditional	value: 21 condition: 22
19	Operation	operator: 53 operands: 23
20	Operation	operator: 24 operand: 59
21	Operation	operator: 25 operands: 26
22	Operation	operator: 27 operands: 28
23	ExprTuple	39, 29
24	Literal
25	Literal
26	ExprTuple	30, 31
27	Literal
28	ExprTuple	50, 32
29	Operation	operator: 43 operands: 33
30	Operation	operator: 53 operands: 34
31	Operation	operator: 35 operands: 36
32	Operation	operator: 37 operands: 38
33	ExprTuple	57, 47, 48, 49, 51
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	Variable
59	Literal