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
from proveit.logic import Equals, Implies
from proveit.numbers import Add, Interval, one
from proveit.physics.quantum import I, ket_plus
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import QPE1, _Psi_ket, _U, _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, one)
sub_expr3 = Add(_t, _s)
sub_expr4 = Interval(one, _t)
sub_expr5 = Interval(sub_expr2, sub_expr3)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE1(_U, _t), part = sub_expr1), targets = Interval(one, sub_expr3))
sub_expr7 = ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)
sub_expr8 = ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)
sub_expr9 = [ExprRange(sub_expr1, Input(state = ket_plus), one, _t), sub_expr7]
sub_expr10 = [ExprRange(sub_expr1, sub_expr6, one, _t), ExprRange(sub_expr1, sub_expr6, sub_expr2, sub_expr3)]
sub_expr11 = [ExprRange(sub_expr1, MultiQubitElem(element = Gate(operation = InverseFourierTransform(_t), part = sub_expr1), targets = sub_expr4), one, _t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, _s)]
sub_expr12 = [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _Psi_ket, part = sub_expr1), targets = sub_expr4), one, _t), sub_expr8]
expr = Implies(Equals(Prob(Qcircuit(vert_expr_array = VertExprArray(sub_expr9, sub_expr10, [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _psi__t_ket, part = sub_expr1), targets = sub_expr4), one, _t), sub_expr8]))), one), Equals(Prob(Qcircuit(vert_expr_array = VertExprArray(sub_expr9, sub_expr10, sub_expr11, sub_expr12))), Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _psi__t_ket, part = sub_expr1), targets = sub_expr4), one, _t), sub_expr7], sub_expr11, sub_expr12)))).with_wrapping_at(2)).with_wrapping_at(2)

# 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}_1\left(U, t\right)} & \multiqout{3}{\lvert \psi_{t} \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\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}_1\left(U, t\right)} & \multigate{3}{{\mathrm {FT}}^{\dag}_{t}} & \multiqout{3}{\lvert \Psi \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghost{{\mathrm {FT}}^{\dag}_{t}} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghost{{\mathrm {FT}}^{\dag}_{t}} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghost{{\mathrm {FT}}^{\dag}_{t}} & \ghostqout{\lvert \Psi \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\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) \end{array} \end{array}\right) \end{array} \end{array}

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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')

	core type	sub-expressions
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, 111
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: 111 end_index: 112
29	ExprRange	lambda_map: 39 start_index: 111 end_index: 112
30	ExprRange	lambda_map: 40 start_index: 111 end_index: 112
31	ExprRange	lambda_map: 40 start_index: 100 end_index: 101
32	ExprRange	lambda_map: 41 start_index: 111 end_index: 112
33	ExprRange	lambda_map: 42 start_index: 111 end_index: 113
34	ExprRange	lambda_map: 43 start_index: 111 end_index: 112
35	ExprRange	lambda_map: 44 start_index: 111 end_index: 113
36	ExprRange	lambda_map: 45 start_index: 111 end_index: 112
37	ExprRange	lambda_map: 46 start_index: 111 end_index: 113
38	Lambda	parameter: 99 body: 47
39	Lambda	parameter: 99 body: 48
40	Lambda	parameter: 99 body: 49
41	Lambda	parameter: 99 body: 50
42	Lambda	parameter: 99 body: 51
43	Lambda	parameter: 99 body: 52
44	Lambda	parameter: 99 body: 53
45	Lambda	parameter: 99 body: 54
46	Lambda	parameter: 99 body: 56
47	Operation	operator: 65 operands: 57
48	Operation	operator: 84 operands: 58
49	Operation	operator: 65 operands: 59
50	Operation	operator: 65 operands: 60
51	Operation	operator: 65 operands: 61
52	Operation	operator: 65 operands: 62
53	Operation	operator: 85 operands: 63
54	Operation	operator: 65 operands: 64
55	ExprTuple	99
56	Operation	operator: 65 operands: 66
57	NamedExprs	element: 67 targets: 76
58	NamedExprs	state: 68
59	NamedExprs	element: 69 targets: 70
60	NamedExprs	element: 71 targets: 76
61	NamedExprs	element: 72 targets: 78
62	NamedExprs	element: 73 targets: 76
63	NamedExprs	operation: 74
64	NamedExprs	element: 75 targets: 76
65	Literal
66	NamedExprs	element: 77 targets: 78
67	Operation	operator: 89 operands: 83
68	Operation	operator: 79 operand: 93
69	Operation	operator: 85 operands: 81
70	Operation	operator: 91 operands: 82
71	Operation	operator: 84 operands: 83
72	Operation	operator: 84 operands: 90
73	Operation	operator: 85 operands: 86
74	Literal
75	Operation	operator: 89 operands: 87
76	Operation	operator: 91 operands: 88
77	Operation	operator: 89 operands: 90
78	Operation	operator: 91 operands: 92
79	Literal
80	ExprTuple	93
81	NamedExprs	operation: 94 part: 99
82	ExprTuple	111, 101
83	NamedExprs	state: 95 part: 99
84	Literal
85	Literal
86	NamedExprs	operation: 96 part: 99
87	NamedExprs	state: 97 part: 99
88	ExprTuple	111, 112
89	Literal
90	NamedExprs	state: 98 part: 99
91	Literal
92	ExprTuple	100, 101
93	Literal
94	Operation	operator: 102 operands: 103
95	Operation	operator: 104 operand: 112
96	Operation	operator: 105 operand: 112
97	Literal
98	Literal
99	Variable
100	Operation	operator: 108 operands: 107
101	Operation	operator: 108 operands: 109
102	Literal
103	ExprTuple	110, 112
104	Literal
105	Literal
106	ExprTuple	112
107	ExprTuple	112, 111
108	Literal
109	ExprTuple	112, 113
110	Literal
111	Literal
112	Literal
113	Literal

Expression of type Implies¶

from the theory of proveit.physics.quantum.QPE¶

Expression of type Implies ¶

from the theory of proveit.physics.quantum.QPE ¶