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
expr = ScalarMult(frac(one, Exp(two, frac(Add(t, one), two))), 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())

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