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
from proveit.linear_algebra import TensorProd
from proveit.logic import Equals
from proveit.numbers import Add, Interval, one
from proveit.physics.quantum import I
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _Psi_ket, _ket_u, _psi__t_ket, _s, _t
from proveit.physics.quantum.circuits import Gate, Input, 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 = Add(_t, _s)
sub_expr3 = Interval(one, _t)
sub_expr4 = Interval(Add(_t, one), sub_expr2)
sub_expr5 = [ExprRange(sub_expr1, MultiQubitElem(element = Gate(operation = InverseFourierTransform(_t), part = sub_expr1), targets = sub_expr3), one, _t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, _s)]
sub_expr6 = [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _Psi_ket, part = sub_expr1), targets = sub_expr3), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)]
expr = Equals(Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, MultiQubitElem(element = Input(state = TensorProd(_psi__t_ket, _ket_u), part = sub_expr1), targets = Interval(one, sub_expr2)), one, sub_expr2)], sub_expr5, sub_expr6))), Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _psi__t_ket, part = sub_expr1), targets = sub_expr3), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)], sub_expr5, sub_expr6))))

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(QCIRCUIT\left(VertExprArray\left(\begin{array}{c} \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~t + s~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}\right),  \\ \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{{\mathrm {FT}}^{\dag}_{t}~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{{\mathrm {FT}}^{\dag}_{t}~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{{\mathrm {FT}}^{\dag}_{t}~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t\}} & \qw 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}, ..\left(s - 3\right) \times.., \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}\right),  \\ \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \Psi \rangle~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \Psi \rangle~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert \Psi \rangle~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}\right) \end{array}\right)\right)\right) = \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert \psi_{t} \rangle} & \gate{{\mathrm {FT}}^{\dag}_{t}} & \qout{\lvert \Psi \rangle} \\
\qin{\lvert u \rangle} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \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',)
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 operand: 8
4	Operation	operator: 6 operand: 9
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	Operation	operator: 11 operands: 10
9	Operation	operator: 11 operands: 12
10	ExprTuple	13, 15, 16
11	Literal
12	ExprTuple	14, 15, 16
13	ExprTuple	17
14	ExprTuple	18, 19
15	ExprTuple	20, 21
16	ExprTuple	22, 23
17	ExprRange	lambda_map: 24 start_index: 83 end_index: 74
18	ExprRange	lambda_map: 25 start_index: 83 end_index: 87
19	ExprRange	lambda_map: 26 start_index: 83 end_index: 84
20	ExprRange	lambda_map: 27 start_index: 83 end_index: 87
21	ExprRange	lambda_map: 28 start_index: 83 end_index: 84
22	ExprRange	lambda_map: 29 start_index: 83 end_index: 87
23	ExprRange	lambda_map: 30 start_index: 83 end_index: 84
24	Lambda	parameter: 72 body: 31
25	Lambda	parameter: 72 body: 32
26	Lambda	parameter: 72 body: 33
27	Lambda	parameter: 72 body: 34
28	Lambda	parameter: 72 body: 35
29	Lambda	parameter: 72 body: 36
30	Lambda	parameter: 72 body: 38
31	Operation	operator: 45 operands: 39
32	Operation	operator: 45 operands: 40
33	Operation	operator: 45 operands: 41
34	Operation	operator: 45 operands: 42
35	Operation	operator: 61 operands: 43
36	Operation	operator: 45 operands: 44
37	ExprTuple	72
38	Operation	operator: 45 operands: 46
39	NamedExprs	element: 47 targets: 48
40	NamedExprs	element: 49 targets: 54
41	NamedExprs	element: 50 targets: 56
42	NamedExprs	element: 51 targets: 54
43	NamedExprs	operation: 52
44	NamedExprs	element: 53 targets: 54
45	Literal
46	NamedExprs	element: 55 targets: 56
47	Operation	operator: 60 operands: 57
48	Operation	operator: 67 operands: 58
49	Operation	operator: 60 operands: 59
50	Operation	operator: 60 operands: 66
51	Operation	operator: 61 operands: 62
52	Literal
53	Operation	operator: 65 operands: 63
54	Operation	operator: 67 operands: 64
55	Operation	operator: 65 operands: 66
56	Operation	operator: 67 operands: 68
57	NamedExprs	state: 69 part: 72
58	ExprTuple	83, 74
59	NamedExprs	state: 81 part: 72
60	Literal
61	Literal
62	NamedExprs	operation: 70 part: 72
63	NamedExprs	state: 71 part: 72
64	ExprTuple	83, 87
65	Literal
66	NamedExprs	state: 82 part: 72
67	Literal
68	ExprTuple	73, 74
69	Operation	operator: 75 operands: 76
70	Operation	operator: 77 operand: 87
71	Literal
72	Variable
73	Operation	operator: 79 operands: 78
74	Operation	operator: 79 operands: 80
75	Literal
76	ExprTuple	81, 82
77	Literal
78	ExprTuple	87, 83
79	Literal
80	ExprTuple	87, 84
81	Operation	operator: 85 operand: 87
82	Literal
83	Literal
84	Literal
85	Literal
86	ExprTuple	87
87	Literal