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