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, Mult, Neg, e, i, one, pi, subtract, two
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _phase

# build up the expression from sub-expressions
expr = Equals(VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Add(Neg(t), one))), _phase)), ket1)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, subtract(t, one)), _phase)), ket1)))

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())

\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) = \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)

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	ExprTuple	9, 8
6	Literal
7	ExprTuple	9, 10
8	Operation	operator: 13 operands: 11
9	Operation	operator: 21 operand: 16
10	Operation	operator: 13 operands: 14
11	ExprTuple	15, 18
12	ExprTuple	16
13	Literal
14	ExprTuple	17, 18
15	Operation	operator: 34 operands: 19
16	Literal
17	Operation	operator: 34 operands: 20
18	Operation	operator: 21 operand: 47
19	ExprTuple	23, 22
20	ExprTuple	23, 24
21	Literal
22	Operation	operator: 26 operands: 25
23	Literal
24	Operation	operator: 26 operands: 27
25	ExprTuple	37, 29, 30, 28, 32
26	Literal
27	ExprTuple	37, 29, 30, 31, 32
28	Operation	operator: 34 operands: 33
29	Literal
30	Literal
31	Operation	operator: 34 operands: 35
32	Literal
33	ExprTuple	37, 36
34	Literal
35	ExprTuple	37, 38
36	Operation	operator: 48 operand: 41
37	Literal
38	Operation	operator: 43 operands: 40
39	ExprTuple	41
40	ExprTuple	50, 42
41	Operation	operator: 43 operands: 44
42	Operation	operator: 48 operand: 47
43	Literal
44	ExprTuple	46, 47
45	ExprTuple	47
46	Operation	operator: 48 operand: 50
47	Literal
48	Literal
49	ExprTuple	50
50	Variable