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 = VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), ket1))
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(TensorProd(ScalarMult(frac(one, Exp(two, subtract(sub_expr2, sub_expr1))), sub_expr3), ScalarMult(frac(one, Exp(two, sub_expr1)), sub_expr4)), ScalarMult(frac(one, Exp(two, sub_expr2)), TensorProd(sub_expr3, 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(\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) = \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: 14 operands: 5
4	Operation	operator: 47 operands: 6
5	ExprTuple	7, 8
6	ExprTuple	9, 10
7	Operation	operator: 47 operands: 11
8	Operation	operator: 47 operands: 12
9	Operation	operator: 72 operands: 13
10	Operation	operator: 14 operands: 15
11	ExprTuple	16, 19
12	ExprTuple	17, 20
13	ExprTuple	90, 18
14	Literal
15	ExprTuple	19, 20
16	Operation	operator: 72 operands: 21
17	Operation	operator: 72 operands: 22
18	Operation	operator: 84 operands: 23
19	Operation	operator: 24 operands: 25
20	Operation	operator: 26 operand: 32
21	ExprTuple	90, 28
22	ExprTuple	90, 29
23	ExprTuple	88, 51
24	Literal
25	ExprTuple	30, 31
26	Literal
27	ExprTuple	32
28	Operation	operator: 84 operands: 33
29	Operation	operator: 84 operands: 34
30	Operation	operator: 46 operand: 69
31	Operation	operator: 47 operands: 36
32	Lambda	parameter: 81 body: 38
33	ExprTuple	88, 39
34	ExprTuple	88, 66
35	ExprTuple	69
36	ExprTuple	40, 41
37	ExprTuple	81
38	Conditional	value: 42 condition: 43
39	Operation	operator: 76 operands: 44
40	Operation	operator: 84 operands: 45
41	Operation	operator: 46 operand: 90
42	Operation	operator: 47 operands: 48
43	Operation	operator: 49 operands: 50
44	ExprTuple	51, 52
45	ExprTuple	67, 53
46	Literal
47	Literal
48	ExprTuple	54, 55
49	Literal
50	ExprTuple	81, 56
51	Operation	operator: 72 operands: 57
52	Operation	operator: 86 operand: 66
53	Operation	operator: 74 operands: 59
54	Operation	operator: 84 operands: 60
55	Operation	operator: 61 operands: 62
56	Operation	operator: 63 operands: 64
57	ExprTuple	65, 88
58	ExprTuple	66
59	ExprTuple	88, 78, 79, 80, 82
60	ExprTuple	67, 68
61	Literal
62	ExprTuple	81, 89
63	Literal
64	ExprTuple	69, 70
65	Operation	operator: 76 operands: 71
66	Operation	operator: 72 operands: 73
67	Literal
68	Operation	operator: 74 operands: 75
69	Literal
70	Operation	operator: 76 operands: 77
71	ExprTuple	89, 90
72	Literal
73	ExprTuple	89, 88
74	Literal
75	ExprTuple	88, 78, 79, 80, 81
76	Literal
77	ExprTuple	82, 83
78	Literal
79	Literal
80	Literal
81	Variable
82	Operation	operator: 84 operands: 85
83	Operation	operator: 86 operand: 90
84	Literal
85	ExprTuple	88, 89
86	Literal
87	ExprTuple	90
88	Literal
89	Variable
90	Literal