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 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 = frac(one, Exp(two, frac(t, two)))
sub_expr2 = VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1))
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(TensorProd(sub_expr2, ScalarMult(sub_expr1, sub_expr3)), ScalarMult(sub_expr1, TensorProd(sub_expr2, sub_expr3))).with_wrapping_at(1)

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())

\begin{array}{c} \begin{array}{l} \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}{2^{\frac{t}{2}}} \cdot \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) \\  = \left(\frac{1}{2^{\frac{t}{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) \end{array} \end{array}

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.	()	(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')

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