Expression of type Len

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 Lambda, Variable, t
from proveit.core_expr_types import Len
from proveit.linear_algebra import ScalarMult, VecAdd
from proveit.numbers import Add, Exp, 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 Output

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = frac(one, sqrt(two))
expr = Len(operands = [Lambda(Variable("_b", latex_format = r"{_{-}b}"), Output(state = ScalarMult(sub_expr2, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Add(Neg(t), one))), _phase)), ket1))))), Lambda(sub_expr1, Output(state = ScalarMult(sub_expr2, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _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({_{-}b} \mapsto \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)} 
} \end{array}, {_{-}a} \mapsto \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)} 
} \end{array}\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',)

# 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	Lambda	parameter: 8 body: 6
4	Lambda	parameter: 63 body: 7
5	ExprTuple	8
6	Operation	operator: 10 operands: 9
7	Operation	operator: 10 operands: 11
8	Variable
9	NamedExprs	state: 12
10	Literal
11	NamedExprs	state: 13
12	Operation	operator: 30 operands: 14
13	Operation	operator: 30 operands: 15
14	ExprTuple	17, 16
15	ExprTuple	17, 18
16	Operation	operator: 21 operands: 19
17	Operation	operator: 38 operands: 20
18	Operation	operator: 21 operands: 22
19	ExprTuple	25, 23
20	ExprTuple	67, 24
21	Literal
22	ExprTuple	25, 26
23	Operation	operator: 30 operands: 27
24	Operation	operator: 55 operands: 28
25	Operation	operator: 41 operand: 34
26	Operation	operator: 30 operands: 31
27	ExprTuple	32, 36
28	ExprTuple	58, 33
29	ExprTuple	34
30	Literal
31	ExprTuple	35, 36
32	Operation	operator: 55 operands: 37
33	Operation	operator: 38 operands: 39
34	Literal
35	Operation	operator: 55 operands: 40
36	Operation	operator: 41 operand: 67
37	ExprTuple	44, 43
38	Literal
39	ExprTuple	67, 58
40	ExprTuple	44, 45
41	Literal
42	ExprTuple	67
43	Operation	operator: 47 operands: 46
44	Literal
45	Operation	operator: 47 operands: 48
46	ExprTuple	58, 50, 51, 49, 53
47	Literal
48	ExprTuple	58, 50, 51, 52, 53
49	Operation	operator: 55 operands: 54
50	Literal
51	Literal
52	Operation	operator: 55 operands: 56
53	Literal
54	ExprTuple	58, 57
55	Literal
56	ExprTuple	58, 59
57	Operation	operator: 68 operand: 62
58	Literal
59	Operation	operator: 68 operand: 63
60	ExprTuple	62
61	ExprTuple	63
62	Operation	operator: 64 operands: 65
63	Variable
64	Literal
65	ExprTuple	66, 67
66	Operation	operator: 68 operand: 70
67	Literal
68	Literal
69	ExprTuple	70
70	Variable