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 Variable, t
from proveit.linear_algebra import ScalarMult, VecAdd
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, frac, i, one, pi, sqrt, two
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase
from proveit.physics.quantum.circuits import MultiQubitElem, Output

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))
expr = Equals(MultiQubitElem(element = Output(state = sub_expr2, part = one), targets = Interval(Add(Add(sub_expr1, Neg(one), t), one), Add(sub_expr1, t))), Output(state = sub_expr2))

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} \Qcircuit@C=1em @R=.7em{
& & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-{_{-}a}} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)~\mbox{part}~1~\mbox{on}~\{\left({_{-}a} - 1 + t\right) + 1~\ldotp \ldotp~{_{-}a} + t\}} 
} \end{array} = \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-{_{-}a}} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} 
} \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.	()	()	('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: 5 operands: 6
4	Operation	operator: 10 operands: 7
5	Literal
6	NamedExprs	element: 8 targets: 9
7	NamedExprs	state: 14
8	Operation	operator: 10 operands: 11
9	Operation	operator: 12 operands: 13
10	Literal
11	NamedExprs	state: 14 part: 48
12	Literal
13	ExprTuple	15, 16
14	Operation	operator: 35 operands: 17
15	Operation	operator: 26 operands: 18
16	Operation	operator: 26 operands: 19
17	ExprTuple	20, 21
18	ExprTuple	22, 48
19	ExprTuple	61, 32
20	Operation	operator: 41 operands: 23
21	Operation	operator: 24 operands: 25
22	Operation	operator: 26 operands: 27
23	ExprTuple	48, 28
24	Literal
25	ExprTuple	29, 30
26	Literal
27	ExprTuple	61, 31, 32
28	Operation	operator: 55 operands: 33
29	Operation	operator: 44 operand: 38
30	Operation	operator: 35 operands: 36
31	Operation	operator: 59 operand: 48
32	Variable
33	ExprTuple	57, 37
34	ExprTuple	38
35	Literal
36	ExprTuple	39, 40
37	Operation	operator: 41 operands: 42
38	Literal
39	Operation	operator: 55 operands: 43
40	Operation	operator: 44 operand: 48
41	Literal
42	ExprTuple	48, 57
43	ExprTuple	46, 47
44	Literal
45	ExprTuple	48
46	Literal
47	Operation	operator: 49 operands: 50
48	Literal
49	Literal
50	ExprTuple	57, 51, 52, 53, 54
51	Literal
52	Literal
53	Operation	operator: 55 operands: 56
54	Literal
55	Literal
56	ExprTuple	57, 58
57	Literal
58	Operation	operator: 59 operand: 61
59	Literal
60	ExprTuple	61
61	Variable