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 = frac(t, two)
expr = ScalarMult(Mult(frac(one, Exp(two, subtract(frac(Add(t, one), two), sub_expr1))), frac(one, Exp(two, sub_expr1))), 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())

\left(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}} \cdot \frac{1}{2^{\frac{t}{2}}}\right) \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: 36 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 63 operands: 4
3	Operation	operator: 5 operands: 6
4	ExprTuple	7, 8
5	Literal
6	ExprTuple	9, 10
7	Operation	operator: 61 operands: 11
8	Operation	operator: 61 operands: 12
9	Operation	operator: 13 operands: 14
10	Operation	operator: 15 operand: 21
11	ExprTuple	79, 17
12	ExprTuple	79, 18
13	Literal
14	ExprTuple	19, 20
15	Literal
16	ExprTuple	21
17	Operation	operator: 73 operands: 22
18	Operation	operator: 73 operands: 23
19	Operation	operator: 35 operand: 58
20	Operation	operator: 36 operands: 25
21	Lambda	parameter: 70 body: 27
22	ExprTuple	77, 28
23	ExprTuple	77, 55
24	ExprTuple	58
25	ExprTuple	29, 30
26	ExprTuple	70
27	Conditional	value: 31 condition: 32
28	Operation	operator: 65 operands: 33
29	Operation	operator: 73 operands: 34
30	Operation	operator: 35 operand: 79
31	Operation	operator: 36 operands: 37
32	Operation	operator: 38 operands: 39
33	ExprTuple	40, 41
34	ExprTuple	56, 42
35	Literal
36	Literal
37	ExprTuple	43, 44
38	Literal
39	ExprTuple	70, 45
40	Operation	operator: 61 operands: 46
41	Operation	operator: 75 operand: 55
42	Operation	operator: 63 operands: 48
43	Operation	operator: 73 operands: 49
44	Operation	operator: 50 operands: 51
45	Operation	operator: 52 operands: 53
46	ExprTuple	54, 77
47	ExprTuple	55
48	ExprTuple	77, 67, 68, 69, 71
49	ExprTuple	56, 57
50	Literal
51	ExprTuple	70, 78
52	Literal
53	ExprTuple	58, 59
54	Operation	operator: 65 operands: 60
55	Operation	operator: 61 operands: 62
56	Literal
57	Operation	operator: 63 operands: 64
58	Literal
59	Operation	operator: 65 operands: 66
60	ExprTuple	78, 79
61	Literal
62	ExprTuple	78, 77
63	Literal
64	ExprTuple	77, 67, 68, 69, 70
65	Literal
66	ExprTuple	71, 72
67	Literal
68	Literal
69	Literal
70	Variable
71	Operation	operator: 73 operands: 74
72	Operation	operator: 75 operand: 79
73	Literal
74	ExprTuple	77, 78
75	Literal
76	ExprTuple	79
77	Literal
78	Variable
79	Literal