Expression of type Implies ¶

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, VecSum
from proveit.logic import CartExp, Equals, Implies, InSet
from proveit.numbers import Complex, Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, QubitSpace, ket1
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr3 = TensorProd(ket1, NumKet(k, t))
sub_expr4 = Interval(zero, subtract(two_pow_t, one))
sub_expr5 = Exp(e, Mult(two, pi, i, _phase, two_pow_t))
sub_expr6 = VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, sub_expr3), domain = sub_expr4)
expr = Implies(InSet(sub_expr6, TensorProd(QubitSpace, CartExp(Complex, two_pow_t))), Equals(ScalarMult(sub_expr5, sub_expr6), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Mult(sub_expr5, sub_expr2), sub_expr3), domain = sub_expr4)).with_wrapping_at(1)).with_wrapping_at(1)

# 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())

\begin{array}{c} \begin{array}{l} \left(\left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\right)\right)\right) \in \left(\mathbb{C}^{2} {\otimes} \mathbb{C}^{2^{t}}\right)\right) \\  \Rightarrow \left(\begin{array}{c} \begin{array}{l} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\right)\right)\right)\right) \\  = \left(\sum_{k=0}^{2^{t} - 1} \left(\left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\right) \cdot \left(\lvert 1 \rangle {\otimes} \lvert k \rangle_{t}\right)\right)\right) \end{array} \end{array}\right) \end{array} \end{array}

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

Expression of type Implies¶

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

Expression of type Implies ¶

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