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, 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 = Exp(e, Mult(two, pi, i, _phase, two_pow_t))
sub_expr2 = frac(one, Exp(two, frac(Add(t, one), two)))
sub_expr3 = 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 = Equals(ScalarMult(sub_expr2, VecAdd(TensorProd(ket0, sub_expr3), ScalarMult(sub_expr1, TensorProd(ket1, sub_expr3)))), ScalarMult(sub_expr2, TensorProd(VecAdd(ket0, ScalarMult(sub_expr1, ket1)), sub_expr3)))

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}}} \cdot \left(\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(\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) = \left(\frac{1}{2^{\frac{t + 1}{2}}} \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)\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: 48 operands: 5
4	Operation	operator: 48 operands: 6
5	ExprTuple	8, 7
6	ExprTuple	8, 9
7	Operation	operator: 20 operands: 10
8	Operation	operator: 28 operands: 11
9	Operation	operator: 26 operands: 12
10	ExprTuple	13, 14
11	ExprTuple	80, 15
12	ExprTuple	16, 32
13	Operation	operator: 26 operands: 17
14	Operation	operator: 48 operands: 18
15	Operation	operator: 74 operands: 19
16	Operation	operator: 20 operands: 21
17	ExprTuple	24, 32
18	ExprTuple	34, 22
19	ExprTuple	78, 23
20	Literal
21	ExprTuple	24, 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 28 operands: 29
24	Operation	operator: 40 operand: 62
25	Operation	operator: 48 operands: 31
26	Literal
27	ExprTuple	35, 32
28	Literal
29	ExprTuple	33, 78
30	ExprTuple	62
31	ExprTuple	34, 35
32	Operation	operator: 36 operand: 41
33	Operation	operator: 66 operands: 38
34	Operation	operator: 74 operands: 39
35	Operation	operator: 40 operand: 80
36	Literal
37	ExprTuple	41
38	ExprTuple	79, 80
39	ExprTuple	60, 42
40	Literal
41	Lambda	parameter: 71 body: 44
42	Operation	operator: 64 operands: 45
43	ExprTuple	71
44	Conditional	value: 46 condition: 47
45	ExprTuple	78, 68, 69, 70, 72
46	Operation	operator: 48 operands: 49
47	Operation	operator: 50 operands: 51
48	Literal
49	ExprTuple	52, 53
50	Literal
51	ExprTuple	71, 54
52	Operation	operator: 74 operands: 55
53	Operation	operator: 56 operands: 57
54	Operation	operator: 58 operands: 59
55	ExprTuple	60, 61
56	Literal
57	ExprTuple	71, 79
58	Literal
59	ExprTuple	62, 63
60	Literal
61	Operation	operator: 64 operands: 65
62	Literal
63	Operation	operator: 66 operands: 67
64	Literal
65	ExprTuple	78, 68, 69, 70, 71
66	Literal
67	ExprTuple	72, 73
68	Literal
69	Literal
70	Literal
71	Variable
72	Operation	operator: 74 operands: 75
73	Operation	operator: 76 operand: 80
74	Literal
75	ExprTuple	78, 79
76	Literal
77	ExprTuple	80
78	Literal
79	Variable
80	Literal