Expression of type Implies

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, Implies
from proveit.numbers import Add, Interval, one
from proveit.physics.quantum import NumKet, Z
from proveit.physics.quantum.QPE import _Psi_circuit, _Psi_ket, _ket_u, _phase_est_circuit, _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 = Implies(Equals(Prob(_Psi_circuit), one), Equals(Prob(_phase_est_circuit), 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)])))).with_wrapping_at(2)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \multiqout{3}{\lvert \Psi \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \qout{\lvert u \rangle}
} \end{array}\right) = 1\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert m \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert m \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) =  \\ \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) \end{array} \end{array}\right) \end{array} \end{array}

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.	()	(2)	('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 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 110
6	Literal
7	ExprTuple	9, 10
8	Operation	operator: 13 operand: 15
9	Operation	operator: 13 operand: 16
10	Operation	operator: 13 operand: 17
11	ExprTuple	15
12	ExprTuple	16
13	Literal
14	ExprTuple	17
15	Operation	operator: 20 operands: 18
16	Operation	operator: 20 operands: 19
17	Operation	operator: 20 operands: 21
18	ExprTuple	23, 24, 22
19	ExprTuple	23, 24, 26, 27
20	Literal
21	ExprTuple	25, 26, 27
22	ExprTuple	28, 37
23	ExprTuple	29, 33
24	ExprTuple	30, 31
25	ExprTuple	32, 33
26	ExprTuple	34, 35
27	ExprTuple	36, 37
28	ExprRange	lambda_map: 38 start_index: 110 end_index: 111
29	ExprRange	lambda_map: 39 start_index: 110 end_index: 111
30	ExprRange	lambda_map: 40 start_index: 110 end_index: 111
31	ExprRange	lambda_map: 40 start_index: 99 end_index: 100
32	ExprRange	lambda_map: 41 start_index: 110 end_index: 111
33	ExprRange	lambda_map: 42 start_index: 110 end_index: 112
34	ExprRange	lambda_map: 43 start_index: 110 end_index: 111
35	ExprRange	lambda_map: 44 start_index: 110 end_index: 112
36	ExprRange	lambda_map: 45 start_index: 110 end_index: 111
37	ExprRange	lambda_map: 46 start_index: 110 end_index: 112
38	Lambda	parameter: 98 body: 47
39	Lambda	parameter: 98 body: 48
40	Lambda	parameter: 98 body: 49
41	Lambda	parameter: 98 body: 50
42	Lambda	parameter: 98 body: 51
43	Lambda	parameter: 98 body: 52
44	Lambda	parameter: 98 body: 53
45	Lambda	parameter: 98 body: 54
46	Lambda	parameter: 98 body: 56
47	Operation	operator: 66 operands: 57
48	Operation	operator: 86 operands: 58
49	Operation	operator: 66 operands: 59
50	Operation	operator: 66 operands: 60
51	Operation	operator: 66 operands: 61
52	Operation	operator: 62 operands: 63
53	Operation	operator: 82 operands: 64
54	Operation	operator: 66 operands: 65
55	ExprTuple	98
56	Operation	operator: 66 operands: 67
57	NamedExprs	element: 68 targets: 77
58	NamedExprs	state: 69
59	NamedExprs	element: 70 targets: 71
60	NamedExprs	element: 72 targets: 77
61	NamedExprs	element: 73 targets: 79
62	Literal
63	NamedExprs	basis: 74
64	NamedExprs	operation: 75
65	NamedExprs	element: 76 targets: 77
66	Literal
67	NamedExprs	element: 78 targets: 79
68	Operation	operator: 89 operands: 85
69	Operation	operator: 80 operand: 93
70	Operation	operator: 82 operands: 83
71	Operation	operator: 91 operands: 84
72	Operation	operator: 86 operands: 85
73	Operation	operator: 86 operands: 90
74	Literal
75	Literal
76	Operation	operator: 89 operands: 87
77	Operation	operator: 91 operands: 88
78	Operation	operator: 89 operands: 90
79	Operation	operator: 91 operands: 92
80	Literal
81	ExprTuple	93
82	Literal
83	NamedExprs	operation: 94 part: 98
84	ExprTuple	110, 100
85	NamedExprs	state: 95 part: 98
86	Literal
87	NamedExprs	state: 96 part: 98
88	ExprTuple	110, 111
89	Literal
90	NamedExprs	state: 97 part: 98
91	Literal
92	ExprTuple	99, 100
93	Literal
94	Operation	operator: 101 operands: 102
95	Literal
96	Operation	operator: 103 operands: 104
97	Literal
98	Variable
99	Operation	operator: 106 operands: 105
100	Operation	operator: 106 operands: 107
101	Literal
102	ExprTuple	108, 111
103	Literal
104	ExprTuple	109, 111
105	ExprTuple	111, 110
106	Literal
107	ExprTuple	111, 112
108	Literal
109	Variable
110	Literal
111	Literal
112	Literal