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 SubIndexed, _phase, _psi

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Equals(SubIndexed(_psi, [Add(t, one)]), TensorProd(ExprRange(sub_expr1, ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1))), Neg(t), zero).with_decreasing_order()))

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())

\lvert \psi_{t + 1} \rangle = \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}  \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)

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 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10
9	Operation	operator: 11 operands: 12
10	ExprRange	lambda_map: 13 start_index: 14 end_index: 32
11	Literal
12	ExprTuple	18, 42
13	Lambda	parameter: 55 body: 15
14	Operation	operator: 53 operand: 18
15	Operation	operator: 29 operands: 17
16	ExprTuple	18
17	ExprTuple	19, 20
18	Variable
19	Operation	operator: 35 operands: 21
20	Operation	operator: 22 operands: 23
21	ExprTuple	42, 24
22	Literal
23	ExprTuple	25, 26
24	Operation	operator: 49 operands: 27
25	Operation	operator: 38 operand: 32
26	Operation	operator: 29 operands: 30
27	ExprTuple	51, 31
28	ExprTuple	32
29	Literal
30	ExprTuple	33, 34
31	Operation	operator: 35 operands: 36
32	Literal
33	Operation	operator: 49 operands: 37
34	Operation	operator: 38 operand: 42
35	Literal
36	ExprTuple	42, 51
37	ExprTuple	40, 41
38	Literal
39	ExprTuple	42
40	Literal
41	Operation	operator: 43 operands: 44
42	Literal
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: 53 operand: 55
53	Literal
54	ExprTuple	55
55	Variable