Expression of type Equals ¶

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.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = Add(t, one)
sub_expr3 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr4 = Add(k, two_pow_t)
sub_expr5 = ScalarMult(sub_expr3, NumKet(k, sub_expr2))
sub_expr6 = Interval(zero, subtract(two_pow_t, one))
sub_expr7 = VecSum(index_or_indices = sub_expr1, summand = sub_expr5, domain = sub_expr6)
sub_expr8 = VecSum(index_or_indices = sub_expr1, summand = sub_expr5, domain = Interval(zero, subtract(Mult(two, two_pow_t), one)))
expr = Equals(Equals(sub_expr8, VecAdd(sub_expr7, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, sub_expr4)), NumKet(sub_expr4, sub_expr2)), domain = sub_expr6))), Equals(sub_expr8, VecAdd(sub_expr7, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), TensorProd(ket1, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr3, NumKet(k, t)), domain = sub_expr6))))))

# 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(\left(\sum_{k=0}^{\left(2 \cdot 2^{t}\right) - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t + 1}\right)\right) = \left(\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 + 1}\right)\right) + \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot \left(k + 2^{t}\right)} \cdot \lvert k + 2^{t} \rangle_{t + 1}\right)\right)\right)\right) = \left(\left(\sum_{k=0}^{\left(2 \cdot 2^{t}\right) - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t + 1}\right)\right) = \left(\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 + 1}\right)\right) + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \left(\lvert 1 \rangle {\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)\right)\right)\right)

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

	core type	sub-expressions
0	Operation	operator: 5 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 5 operands: 4
3	Operation	operator: 5 operands: 6
4	ExprTuple	8, 7
5	Literal
6	ExprTuple	8, 9
7	Operation	operator: 12 operands: 10
8	Operation	operator: 44 operand: 15
9	Operation	operator: 12 operands: 13
10	ExprTuple	16, 14
11	ExprTuple	15
12	Literal
13	ExprTuple	16, 17
14	Operation	operator: 44 operand: 22
15	Lambda	parameter: 90 body: 19
16	Operation	operator: 44 operand: 24
17	Operation	operator: 65 operands: 21
18	ExprTuple	22
19	Conditional	value: 35 condition: 23
20	ExprTuple	24
21	ExprTuple	25, 26
22	Lambda	parameter: 90 body: 27
23	Operation	operator: 67 operands: 28
24	Lambda	parameter: 90 body: 29
25	Operation	operator: 93 operands: 30
26	Operation	operator: 31 operands: 32
27	Conditional	value: 33 condition: 61
28	ExprTuple	90, 34
29	Conditional	value: 35 condition: 61
30	ExprTuple	79, 36
31	Literal
32	ExprTuple	37, 38
33	Operation	operator: 65 operands: 39
34	Operation	operator: 77 operands: 40
35	Operation	operator: 65 operands: 41
36	Operation	operator: 83 operands: 42
37	Operation	operator: 43 operand: 99
38	Operation	operator: 44 operand: 50
39	ExprTuple	46, 47
40	ExprTuple	81, 48
41	ExprTuple	70, 49
42	ExprTuple	97, 87, 88, 89, 91
43	Literal
44	Literal
45	ExprTuple	50
46	Operation	operator: 93 operands: 51
47	Operation	operator: 75 operands: 52
48	Operation	operator: 85 operands: 53
49	Operation	operator: 75 operands: 54
50	Lambda	parameter: 90 body: 56
51	ExprTuple	79, 57
52	ExprTuple	69, 59
53	ExprTuple	58, 92
54	ExprTuple	90, 59
55	ExprTuple	90
56	Conditional	value: 60 condition: 61
57	Operation	operator: 83 operands: 62
58	Operation	operator: 83 operands: 63
59	Operation	operator: 85 operands: 64
60	Operation	operator: 65 operands: 66
61	Operation	operator: 67 operands: 68
62	ExprTuple	97, 87, 88, 89, 69
63	ExprTuple	97, 91
64	ExprTuple	98, 99
65	Literal
66	ExprTuple	70, 71
67	Literal
68	ExprTuple	90, 72
69	Operation	operator: 85 operands: 73
70	Operation	operator: 93 operands: 74
71	Operation	operator: 75 operands: 76
72	Operation	operator: 77 operands: 78
73	ExprTuple	90, 91
74	ExprTuple	79, 80
75	Literal
76	ExprTuple	90, 98
77	Literal
78	ExprTuple	81, 82
79	Literal
80	Operation	operator: 83 operands: 84
81	Literal
82	Operation	operator: 85 operands: 86
83	Literal
84	ExprTuple	97, 87, 88, 89, 90
85	Literal
86	ExprTuple	91, 92
87	Literal
88	Literal
89	Literal
90	Variable
91	Operation	operator: 93 operands: 94
92	Operation	operator: 95 operand: 99
93	Literal
94	ExprTuple	97, 98
95	Literal
96	ExprTuple	99
97	Literal
98	Variable
99	Literal

Expression of type Equals¶

from the theory of proveit.physics.quantum.QPE¶

Expression of type Equals ¶

from the theory of proveit.physics.quantum.QPE ¶