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.core_expr_types import Len
from proveit.linear_algebra import ScalarMult, 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 _phase

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

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(\left(\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)\right), \left(\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-\left(-t + 2\right)} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)\right), \ldots, \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)| = {_{-}b}

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	Variable
5	Literal
6	ExprTuple	7
7	ExprRange	lambda_map: 8 start_index: 9 end_index: 30
8	Lambda	parameter: 53 body: 10
9	Operation	operator: 11 operands: 12
10	Operation	operator: 27 operands: 13
11	Literal
12	ExprTuple	14, 40
13	ExprTuple	15, 16
14	Operation	operator: 51 operand: 21
15	Operation	operator: 33 operands: 18
16	Operation	operator: 19 operands: 20
17	ExprTuple	21
18	ExprTuple	40, 22
19	Literal
20	ExprTuple	23, 24
21	Variable
22	Operation	operator: 47 operands: 25
23	Operation	operator: 36 operand: 30
24	Operation	operator: 27 operands: 28
25	ExprTuple	49, 29
26	ExprTuple	30
27	Literal
28	ExprTuple	31, 32
29	Operation	operator: 33 operands: 34
30	Literal
31	Operation	operator: 47 operands: 35
32	Operation	operator: 36 operand: 40
33	Literal
34	ExprTuple	40, 49
35	ExprTuple	38, 39
36	Literal
37	ExprTuple	40
38	Literal
39	Operation	operator: 41 operands: 42
40	Literal
41	Literal
42	ExprTuple	49, 43, 44, 45, 46
43	Literal
44	Literal
45	Operation	operator: 47 operands: 48
46	Literal
47	Literal
48	ExprTuple	49, 50
49	Literal
50	Operation	operator: 51 operand: 53
51	Literal
52	ExprTuple	53
53	Variable