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, Forall, Implies, InSet
from proveit.numbers import Complex, Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet, QubitSpace, ket0
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = NumKet(k, t)
sub_expr3 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr4 = ScalarMult(sub_expr3, sub_expr2)
sub_expr5 = Interval(zero, subtract(two_pow_t, one))
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(TensorProd(ket0, sub_expr4), TensorProd(QubitSpace, CartExp(Complex, two_pow_t))), domain = sub_expr5), Equals(TensorProd(ket0, VecSum(index_or_indices = sub_expr1, summand = sub_expr4, domain = sub_expr5)), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr3, TensorProd(ket0, sub_expr2)), domain = sub_expr5)).with_wrapping_at(1)).with_wrapping_at(2)

# 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[\forall_{k \in \{0~\ldotp \ldotp~2^{t} - 1\}}~\left(\left(\lvert 0 \rangle {\otimes} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right) \in \left(\mathbb{C}^{2} {\otimes} \mathbb{C}^{2^{t}}\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(\lvert 0 \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) \\  = \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left(\lvert 0 \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.	()	(2)	('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: 5 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 68 body: 12
10	Operation	operator: 43 operands: 13
11	Operation	operator: 19 operand: 17
12	Conditional	value: 15 condition: 34
13	ExprTuple	48, 16
14	ExprTuple	17
15	Operation	operator: 41 operands: 18
16	Operation	operator: 19 operand: 24
17	Lambda	parameter: 68 body: 21
18	ExprTuple	22, 23
19	Literal
20	ExprTuple	24
21	Conditional	value: 25 condition: 34
22	Operation	operator: 43 operands: 26
23	Operation	operator: 43 operands: 27
24	Lambda	parameter: 68 body: 29
25	Operation	operator: 39 operands: 30
26	ExprTuple	48, 33
27	ExprTuple	31, 32
28	ExprTuple	68
29	Conditional	value: 33 condition: 34
30	ExprTuple	46, 35
31	Operation	operator: 37 operands: 36
32	Operation	operator: 37 operands: 38
33	Operation	operator: 39 operands: 40
34	Operation	operator: 41 operands: 42
35	Operation	operator: 43 operands: 44
36	ExprTuple	45, 75
37	Literal
38	ExprTuple	45, 69
39	Literal
40	ExprTuple	46, 49
41	Literal
42	ExprTuple	68, 47
43	Literal
44	ExprTuple	48, 49
45	Literal
46	Operation	operator: 71 operands: 50
47	Operation	operator: 51 operands: 52
48	Operation	operator: 53 operand: 60
49	Operation	operator: 55 operands: 56
50	ExprTuple	57, 58
51	Literal
52	ExprTuple	60, 59
53	Literal
54	ExprTuple	60
55	Literal
56	ExprTuple	68, 76
57	Literal
58	Operation	operator: 61 operands: 62
59	Operation	operator: 63 operands: 64
60	Literal
61	Literal
62	ExprTuple	75, 65, 66, 67, 68
63	Literal
64	ExprTuple	69, 70
65	Literal
66	Literal
67	Literal
68	Variable
69	Operation	operator: 71 operands: 72
70	Operation	operator: 73 operand: 77
71	Literal
72	ExprTuple	75, 76
73	Literal
74	ExprTuple	77
75	Literal
76	Variable
77	Literal

Expression of type Implies¶

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

Expression of type Implies ¶

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