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, VertExprArray, m
from proveit.logic import Equals
from proveit.numbers import Add, Interval, one
from proveit.physics.quantum import NumKet, Z
from proveit.physics.quantum.QPE import _Psi_ket, _alpha_m_sqrd, _ket_u, _s, _s_wire, _t
from proveit.physics.quantum.circuits import Input, Measure, MultiQubitElem, Output, Qcircuit
from proveit.statistics import Prob

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Interval(one, _t)
sub_expr3 = Interval(Add(_t, one), Add(_t, _s))
expr = Equals(Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _Psi_ket, part = sub_expr1), targets = sub_expr2), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr3), one, _s)], [ExprRange(sub_expr1, Measure(basis = Z), one, _t), _s_wire], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(m, _t), part = sub_expr1), targets = sub_expr2), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr3), one, _s)]))), _alpha_m_sqrd)

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())

\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert \Psi \rangle} & \meter & \multiqout{3}{\lvert m \rangle_{t}} \\
\ghostqin{\lvert \Psi \rangle} & \meter & \ghostqout{\lvert m \rangle_{t}} \\
\ghostqin{\lvert \Psi \rangle} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert m \rangle_{t}} \\
\ghostqin{\lvert \Psi \rangle} & \meter & \ghostqout{\lvert m \rangle_{t}} \\
\qin{\lvert u \rangle} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) = \left|\alpha_{m}\right|^{2}

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, 11
9	Operation	operator: 12 operands: 13
10	Operation	operator: 14 operand: 19
11	Literal
12	Literal
13	ExprTuple	16, 17, 18
14	Literal
15	ExprTuple	19
16	ExprTuple	20, 21
17	ExprTuple	22, 23
18	ExprTuple	24, 25
19	Operation	operator: 26 operand: 77
20	ExprRange	lambda_map: 28 start_index: 78 end_index: 79
21	ExprRange	lambda_map: 29 start_index: 78 end_index: 80
22	ExprRange	lambda_map: 30 start_index: 78 end_index: 79
23	ExprRange	lambda_map: 31 start_index: 78 end_index: 80
24	ExprRange	lambda_map: 32 start_index: 78 end_index: 79
25	ExprRange	lambda_map: 33 start_index: 78 end_index: 80
26	Literal
27	ExprTuple	77
28	Lambda	parameter: 69 body: 34
29	Lambda	parameter: 69 body: 35
30	Lambda	parameter: 69 body: 36
31	Lambda	parameter: 69 body: 37
32	Lambda	parameter: 69 body: 38
33	Lambda	parameter: 69 body: 40
34	Operation	operator: 48 operands: 41
35	Operation	operator: 48 operands: 42
36	Operation	operator: 43 operands: 44
37	Operation	operator: 45 operands: 46
38	Operation	operator: 48 operands: 47
39	ExprTuple	69
40	Operation	operator: 48 operands: 49
41	NamedExprs	element: 50 targets: 55
42	NamedExprs	element: 51 targets: 57
43	Literal
44	NamedExprs	basis: 52
45	Literal
46	NamedExprs	operation: 53
47	NamedExprs	element: 54 targets: 55
48	Literal
49	NamedExprs	element: 56 targets: 57
50	Operation	operator: 59 operands: 58
51	Operation	operator: 59 operands: 63
52	Literal
53	Literal
54	Operation	operator: 62 operands: 60
55	Operation	operator: 64 operands: 61
56	Operation	operator: 62 operands: 63
57	Operation	operator: 64 operands: 65
58	NamedExprs	state: 66 part: 69
59	Literal
60	NamedExprs	state: 67 part: 69
61	ExprTuple	78, 79
62	Literal
63	NamedExprs	state: 68 part: 69
64	Literal
65	ExprTuple	70, 71
66	Literal
67	Operation	operator: 72 operands: 73
68	Literal
69	Variable
70	Operation	operator: 75 operands: 74
71	Operation	operator: 75 operands: 76
72	Literal
73	ExprTuple	77, 79
74	ExprTuple	79, 78
75	Literal
76	ExprTuple	79, 80
77	Variable
78	Literal
79	Literal
80	Literal