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, 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 = frac(one, sqrt(two))
sub_expr3 = VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, two_pow_t, _phase)), ket1))
sub_expr4 = ExprRange(sub_expr1, ScalarMult(sub_expr2, 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(ScalarMult(sub_expr2, sub_expr3), sub_expr4), ScalarMult(sub_expr2, TensorProd(sub_expr3, sub_expr4))).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}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t} \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 - 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) \\  = \left(\frac{1}{\sqrt{2}} \cdot \left(\left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t} \cdot \varphi} \cdot \lvert 1 \rangle\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)\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: 39 operands: 6
5	ExprTuple	7, 13
6	ExprTuple	25, 8
7	Operation	operator: 39 operands: 9
8	Operation	operator: 10 operands: 11
9	ExprTuple	25, 12
10	Literal
11	ExprTuple	12, 13
12	Operation	operator: 30 operands: 14
13	ExprRange	lambda_map: 15 start_index: 16 end_index: 43
14	ExprTuple	34, 17
15	Lambda	parameter: 68 body: 18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 39 operands: 21
18	Operation	operator: 39 operands: 22
19	Literal
20	ExprTuple	23, 55
21	ExprTuple	24, 45
22	ExprTuple	25, 26
23	Operation	operator: 66 operand: 52
24	Operation	operator: 62 operands: 28
25	Operation	operator: 47 operands: 29
26	Operation	operator: 30 operands: 31
27	ExprTuple	52
28	ExprTuple	53, 32
29	ExprTuple	55, 33
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 56 operands: 36
33	Operation	operator: 62 operands: 37
34	Operation	operator: 50 operand: 43
35	Operation	operator: 39 operands: 40
36	ExprTuple	64, 58, 59, 41, 61
37	ExprTuple	64, 42
38	ExprTuple	43
39	Literal
40	ExprTuple	44, 45
41	Operation	operator: 62 operands: 46
42	Operation	operator: 47 operands: 48
43	Literal
44	Operation	operator: 62 operands: 49
45	Operation	operator: 50 operand: 55
46	ExprTuple	64, 52
47	Literal
48	ExprTuple	55, 64
49	ExprTuple	53, 54
50	Literal
51	ExprTuple	55
52	Variable
53	Literal
54	Operation	operator: 56 operands: 57
55	Literal
56	Literal
57	ExprTuple	64, 58, 59, 60, 61
58	Literal
59	Literal
60	Operation	operator: 62 operands: 63
61	Literal
62	Literal
63	ExprTuple	64, 65
64	Literal
65	Operation	operator: 66 operand: 68
66	Literal
67	ExprTuple	68
68	Variable