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 ExprRange, Variable, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, Neg, e, frac, i, one, pi, sqrt, 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}")
sub_expr2 = 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 = Equals(TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, two_pow_t, _phase)), ket1)), sub_expr2), TensorProd(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1)), sub_expr2))

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(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t} \cdot \varphi} \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) = \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',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

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