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, TensorProd, VecAdd, VecSum
from proveit.numbers import Add, Exp, Interval, Mult, e, frac, 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
sub_expr1 = 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 = ScalarMult(frac(one, Exp(two, frac(Add(t, one), two))), VecAdd(TensorProd(ket0, sub_expr1), ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), TensorProd(ket1, sub_expr1))))

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 + 1}{2}}} \cdot \left(\left(\lvert 0 \rangle {\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) + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \left(\lvert 1 \rangle {\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)\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: 37 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 17 operands: 4
3	Operation	operator: 5 operands: 6
4	ExprTuple	69, 7
5	Literal
6	ExprTuple	8, 9
7	Operation	operator: 63 operands: 10
8	Operation	operator: 21 operands: 11
9	Operation	operator: 37 operands: 12
10	ExprTuple	67, 13
11	ExprTuple	14, 26
12	ExprTuple	15, 16
13	Operation	operator: 17 operands: 18
14	Operation	operator: 29 operand: 51
15	Operation	operator: 63 operands: 20
16	Operation	operator: 21 operands: 22
17	Literal
18	ExprTuple	23, 67
19	ExprTuple	51
20	ExprTuple	49, 24
21	Literal
22	ExprTuple	25, 26
23	Operation	operator: 55 operands: 27
24	Operation	operator: 53 operands: 28
25	Operation	operator: 29 operand: 69
26	Operation	operator: 30 operand: 32
27	ExprTuple	68, 69
28	ExprTuple	67, 57, 58, 59, 61
29	Literal
30	Literal
31	ExprTuple	32
32	Lambda	parameter: 60 body: 34
33	ExprTuple	60
34	Conditional	value: 35 condition: 36
35	Operation	operator: 37 operands: 38
36	Operation	operator: 39 operands: 40
37	Literal
38	ExprTuple	41, 42
39	Literal
40	ExprTuple	60, 43
41	Operation	operator: 63 operands: 44
42	Operation	operator: 45 operands: 46
43	Operation	operator: 47 operands: 48
44	ExprTuple	49, 50
45	Literal
46	ExprTuple	60, 68
47	Literal
48	ExprTuple	51, 52
49	Literal
50	Operation	operator: 53 operands: 54
51	Literal
52	Operation	operator: 55 operands: 56
53	Literal
54	ExprTuple	67, 57, 58, 59, 60
55	Literal
56	ExprTuple	61, 62
57	Literal
58	Literal
59	Literal
60	Variable
61	Operation	operator: 63 operands: 64
62	Operation	operator: 65 operand: 69
63	Literal
64	ExprTuple	67, 68
65	Literal
66	ExprTuple	69
67	Literal
68	Variable
69	Literal