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 ExprRange, Variable, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd
from proveit.numbers import Add, Exp, Mult, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = frac(one, sqrt(two))
expr = ScalarMult(sub_expr2, TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, two_pow_t, _phase)), ket1)), ExprRange(sub_expr1, ScalarMult(sub_expr2, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1))), Add(Neg(t), one), zero).with_decreasing_order()))

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