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, subtract, 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}")
expr = ScalarMult(frac(one, Exp(two, subtract(frac(Add(t, one), two), frac(t, two)))), TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1)), ExprRange(sub_expr1, ScalarMult(frac(one, sqrt(two)), 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}{2^{\frac{t + 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(\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: 46 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 54 operands: 4
3	Operation	operator: 5 operands: 6
4	ExprTuple	62, 7
5	Literal
6	ExprTuple	8, 9
7	Operation	operator: 69 operands: 10
8	Operation	operator: 32 operands: 11
9	ExprRange	lambda_map: 12 start_index: 13 end_index: 50
10	ExprTuple	71, 14
11	ExprTuple	38, 15
12	Lambda	parameter: 75 body: 16
13	Operation	operator: 40 operands: 17
14	Operation	operator: 40 operands: 18
15	Operation	operator: 46 operands: 19
16	Operation	operator: 46 operands: 20
17	ExprTuple	21, 62
18	ExprTuple	22, 23
19	ExprTuple	24, 52
20	ExprTuple	25, 26
21	Operation	operator: 73 operand: 59
22	Operation	operator: 54 operands: 28
23	Operation	operator: 73 operand: 35
24	Operation	operator: 69 operands: 30
25	Operation	operator: 54 operands: 31
26	Operation	operator: 32 operands: 33
27	ExprTuple	59
28	ExprTuple	34, 71
29	ExprTuple	35
30	ExprTuple	60, 36
31	ExprTuple	62, 37
32	Literal
33	ExprTuple	38, 39
34	Operation	operator: 40 operands: 41
35	Operation	operator: 54 operands: 42
36	Operation	operator: 63 operands: 43
37	Operation	operator: 69 operands: 44
38	Operation	operator: 57 operand: 50
39	Operation	operator: 46 operands: 47
40	Literal
41	ExprTuple	59, 62
42	ExprTuple	59, 71
43	ExprTuple	71, 65, 66, 68, 48
44	ExprTuple	71, 49
45	ExprTuple	50
46	Literal
47	ExprTuple	51, 52
48	Operation	operator: 69 operands: 53
49	Operation	operator: 54 operands: 55
50	Literal
51	Operation	operator: 69 operands: 56
52	Operation	operator: 57 operand: 62
53	ExprTuple	71, 59
54	Literal
55	ExprTuple	62, 71
56	ExprTuple	60, 61
57	Literal
58	ExprTuple	62
59	Variable
60	Literal
61	Operation	operator: 63 operands: 64
62	Literal
63	Literal
64	ExprTuple	71, 65, 66, 67, 68
65	Literal
66	Literal
67	Operation	operator: 69 operands: 70
68	Literal
69	Literal
70	ExprTuple	71, 72
71	Literal
72	Operation	operator: 73 operand: 75
73	Literal
74	ExprTuple	75
75	Variable