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 = frac(t, two)
sub_expr2 = frac(Add(t, one), two)
sub_expr3 = Exp(e, Mult(two, pi, i, _phase, two_pow_t))
sub_expr4 = 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(frac(one, Exp(two, sub_expr2)), VecAdd(TensorProd(ket0, sub_expr4), ScalarMult(sub_expr3, TensorProd(ket1, sub_expr4)))), TensorProd(ScalarMult(frac(one, Exp(two, subtract(sub_expr2, sub_expr1))), VecAdd(ket0, ScalarMult(sub_expr3, ket1))), ScalarMult(frac(one, Exp(two, sub_expr1)), sub_expr4)))

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(\left(\frac{1}{2^{\frac{t + 1}{2} - \frac{t}{2}}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \lvert 1 \rangle\right)\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)

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: 61 operands: 5
4	Operation	operator: 33 operands: 6
5	ExprTuple	7, 8
6	ExprTuple	9, 10
7	Operation	operator: 66 operands: 11
8	Operation	operator: 25 operands: 12
9	Operation	operator: 61 operands: 13
10	Operation	operator: 61 operands: 14
11	ExprTuple	96, 15
12	ExprTuple	16, 17
13	ExprTuple	18, 19
14	ExprTuple	20, 39
15	Operation	operator: 90 operands: 21
16	Operation	operator: 33 operands: 22
17	Operation	operator: 61 operands: 23
18	Operation	operator: 66 operands: 24
19	Operation	operator: 25 operands: 26
20	Operation	operator: 66 operands: 27
21	ExprTuple	94, 49
22	ExprTuple	30, 39
23	ExprTuple	41, 28
24	ExprTuple	96, 29
25	Literal
26	ExprTuple	30, 31
27	ExprTuple	96, 32
28	Operation	operator: 33 operands: 34
29	Operation	operator: 90 operands: 35
30	Operation	operator: 47 operand: 78
31	Operation	operator: 61 operands: 37
32	Operation	operator: 90 operands: 38
33	Literal
34	ExprTuple	42, 39
35	ExprTuple	94, 40
36	ExprTuple	78
37	ExprTuple	41, 42
38	ExprTuple	94, 60
39	Operation	operator: 43 operand: 48
40	Operation	operator: 82 operands: 45
41	Operation	operator: 90 operands: 46
42	Operation	operator: 47 operand: 96
43	Literal
44	ExprTuple	48
45	ExprTuple	49, 50
46	ExprTuple	76, 51
47	Literal
48	Lambda	parameter: 87 body: 53
49	Operation	operator: 66 operands: 54
50	Operation	operator: 92 operand: 60
51	Operation	operator: 80 operands: 56
52	ExprTuple	87
53	Conditional	value: 57 condition: 58
54	ExprTuple	59, 94
55	ExprTuple	60
56	ExprTuple	94, 84, 85, 86, 88
57	Operation	operator: 61 operands: 62
58	Operation	operator: 63 operands: 64
59	Operation	operator: 82 operands: 65
60	Operation	operator: 66 operands: 67
61	Literal
62	ExprTuple	68, 69
63	Literal
64	ExprTuple	87, 70
65	ExprTuple	95, 96
66	Literal
67	ExprTuple	95, 94
68	Operation	operator: 90 operands: 71
69	Operation	operator: 72 operands: 73
70	Operation	operator: 74 operands: 75
71	ExprTuple	76, 77
72	Literal
73	ExprTuple	87, 95
74	Literal
75	ExprTuple	78, 79
76	Literal
77	Operation	operator: 80 operands: 81
78	Literal
79	Operation	operator: 82 operands: 83
80	Literal
81	ExprTuple	94, 84, 85, 86, 87
82	Literal
83	ExprTuple	88, 89
84	Literal
85	Literal
86	Literal
87	Variable
88	Operation	operator: 90 operands: 91
89	Operation	operator: 92 operand: 96
90	Literal
91	ExprTuple	94, 95
92	Literal
93	ExprTuple	96
94	Literal
95	Variable
96	Literal