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 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, zero
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 = Output(state = ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Add(Neg(t), one))), _phase)), ket1))), part = one)
expr = Equals(MultiQubitElem(element = sub_expr1, targets = Interval(Add(zero, one), one)), MultiQubitElem(element = sub_expr1, targets = Interval(one, one)))

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^{-\left(-t + 1\right)} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)~\mbox{part}~1~\mbox{on}~\{0 + 1~\ldotp \ldotp~1\}} 
} \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^{-\left(-t + 1\right)} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~1\}} 
} \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, 58
12	Literal
13	NamedExprs	state: 17 part: 58
14	Literal
15	ExprTuple	58, 58
16	Operation	operator: 55 operands: 18
17	Operation	operator: 30 operands: 19
18	ExprTuple	33, 58
19	ExprTuple	20, 21
20	Operation	operator: 36 operands: 22
21	Operation	operator: 23 operands: 24
22	ExprTuple	58, 25
23	Literal
24	ExprTuple	26, 27
25	Operation	operator: 49 operands: 28
26	Operation	operator: 39 operand: 33
27	Operation	operator: 30 operands: 31
28	ExprTuple	51, 32
29	ExprTuple	33
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 36 operands: 37
33	Literal
34	Operation	operator: 49 operands: 38
35	Operation	operator: 39 operand: 58
36	Literal
37	ExprTuple	58, 51
38	ExprTuple	41, 42
39	Literal
40	ExprTuple	58
41	Literal
42	Operation	operator: 43 operands: 44
43	Literal
44	ExprTuple	51, 45, 46, 47, 48
45	Literal
46	Literal
47	Operation	operator: 49 operands: 50
48	Literal
49	Literal
50	ExprTuple	51, 52
51	Literal
52	Operation	operator: 59 operand: 54
53	ExprTuple	54
54	Operation	operator: 55 operands: 56
55	Literal
56	ExprTuple	57, 58
57	Operation	operator: 59 operand: 61
58	Literal
59	Literal
60	ExprTuple	61
61	Variable