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