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 = Add(sub_expr1, t)
sub_expr3 = Output(state = ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1))), part = one)
expr = Equals(MultiQubitElem(element = sub_expr3, targets = Interval(Add(Add(sub_expr1, Neg(one), t), one), sub_expr2)), MultiQubitElem(element = sub_expr3, targets = Interval(sub_expr2, 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)~\mbox{part}~1~\mbox{on}~\{{_{-}a} + t~\ldotp \ldotp~{_{-}a} + t\}} 
} \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: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	NamedExprs	element: 9 targets: 8
6	Literal
7	NamedExprs	element: 9 targets: 10
8	Operation	operator: 14 operands: 11
9	Operation	operator: 12 operands: 13
10	Operation	operator: 14 operands: 15
11	ExprTuple	16, 18
12	Literal
13	NamedExprs	state: 17 part: 50
14	Literal
15	ExprTuple	18, 18
16	Operation	operator: 25 operands: 19
17	Operation	operator: 37 operands: 20
18	Operation	operator: 25 operands: 21
19	ExprTuple	22, 50
20	ExprTuple	23, 24
21	ExprTuple	63, 31
22	Operation	operator: 25 operands: 26
23	Operation	operator: 43 operands: 27
24	Operation	operator: 28 operands: 29
25	Literal
26	ExprTuple	63, 30, 31
27	ExprTuple	50, 32
28	Literal
29	ExprTuple	33, 34
30	Operation	operator: 61 operand: 50
31	Variable
32	Operation	operator: 57 operands: 35
33	Operation	operator: 46 operand: 40
34	Operation	operator: 37 operands: 38
35	ExprTuple	59, 39
36	ExprTuple	40
37	Literal
38	ExprTuple	41, 42
39	Operation	operator: 43 operands: 44
40	Literal
41	Operation	operator: 57 operands: 45
42	Operation	operator: 46 operand: 50
43	Literal
44	ExprTuple	50, 59
45	ExprTuple	48, 49
46	Literal
47	ExprTuple	50
48	Literal
49	Operation	operator: 51 operands: 52
50	Literal
51	Literal
52	ExprTuple	59, 53, 54, 55, 56
53	Literal
54	Literal
55	Operation	operator: 57 operands: 58
56	Literal
57	Literal
58	ExprTuple	59, 60
59	Literal
60	Operation	operator: 61 operand: 63
61	Literal
62	ExprTuple	63
63	Variable