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 ket_plus
from proveit.physics.quantum.QPE import QPE1, _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(sub_expr2, sub_expr3)
sub_expr5 = Interval(one, sub_expr3)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE1(_U, _t), part = sub_expr1), targets = sub_expr5)
sub_expr7 = MultiQubitElem(element = Output(state = TensorProd(_psi__t_ket, _ket_u), part = sub_expr1), targets = sub_expr5)
sub_expr8 = [ExprRange(sub_expr1, Input(state = ket_plus), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)]
sub_expr9 = [ExprRange(sub_expr1, sub_expr6, one, _t), ExprRange(sub_expr1, sub_expr6, sub_expr2, sub_expr3)]
expr = Equals(Prob(Qcircuit(vert_expr_array = VertExprArray(sub_expr8, sub_expr9, [ExprRange(sub_expr1, sub_expr7, one, _t).with_wrapping_at(2,6), ExprRange(sub_expr1, sub_expr7, sub_expr2, sub_expr3).with_wrapping_at(2,6)]))), Prob(Qcircuit(vert_expr_array = VertExprArray(sub_expr8, sub_expr9, [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _psi__t_ket, part = sub_expr1), targets = Interval(one, _t)), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)]))))

# 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{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}_1\left(U, t\right)} & \multiqout{4}{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \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 {\otimes} \lvert u \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle}
} \end{array}\right) = \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)

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.	()	()	('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 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	14, 15, 13
11	Literal
12	ExprTuple	14, 15, 16
13	ExprTuple	17, 18
14	ExprTuple	19, 20
15	ExprTuple	21, 22
16	ExprTuple	23, 24
17	ExprRange	lambda_map: 25 start_index: 83 end_index: 87
18	ExprRange	lambda_map: 25 start_index: 71 end_index: 72
19	ExprRange	lambda_map: 26 start_index: 83 end_index: 87
20	ExprRange	lambda_map: 27 start_index: 83 end_index: 84
21	ExprRange	lambda_map: 28 start_index: 83 end_index: 87
22	ExprRange	lambda_map: 28 start_index: 71 end_index: 72
23	ExprRange	lambda_map: 29 start_index: 83 end_index: 87
24	ExprRange	lambda_map: 30 start_index: 83 end_index: 84
25	Lambda	parameter: 70 body: 31
26	Lambda	parameter: 70 body: 32
27	Lambda	parameter: 70 body: 33
28	Lambda	parameter: 70 body: 34
29	Lambda	parameter: 70 body: 35
30	Lambda	parameter: 70 body: 37
31	Operation	operator: 43 operands: 38
32	Operation	operator: 57 operands: 39
33	Operation	operator: 43 operands: 40
34	Operation	operator: 43 operands: 41
35	Operation	operator: 43 operands: 42
36	ExprTuple	70
37	Operation	operator: 43 operands: 44
38	NamedExprs	element: 45 targets: 49
39	NamedExprs	state: 46
40	NamedExprs	element: 47 targets: 53
41	NamedExprs	element: 48 targets: 49
42	NamedExprs	element: 50 targets: 51
43	Literal
44	NamedExprs	element: 52 targets: 53
45	Operation	operator: 63 operands: 54
46	Operation	operator: 55 operand: 68
47	Operation	operator: 57 operands: 64
48	Operation	operator: 58 operands: 59
49	Operation	operator: 65 operands: 60
50	Operation	operator: 63 operands: 61
51	Operation	operator: 65 operands: 62
52	Operation	operator: 63 operands: 64
53	Operation	operator: 65 operands: 66
54	NamedExprs	state: 67 part: 70
55	Literal
56	ExprTuple	68
57	Literal
58	Literal
59	NamedExprs	operation: 69 part: 70
60	ExprTuple	83, 72
61	NamedExprs	state: 80 part: 70
62	ExprTuple	83, 87
63	Literal
64	NamedExprs	state: 81 part: 70
65	Literal
66	ExprTuple	71, 72
67	Operation	operator: 73 operands: 74
68	Literal
69	Operation	operator: 75 operands: 76
70	Variable
71	Operation	operator: 78 operands: 77
72	Operation	operator: 78 operands: 79
73	Literal
74	ExprTuple	80, 81
75	Literal
76	ExprTuple	82, 87
77	ExprTuple	87, 83
78	Literal
79	ExprTuple	87, 84
80	Operation	operator: 85 operand: 87
81	Literal
82	Literal
83	Literal
84	Literal
85	Literal
86	ExprTuple	87
87	Literal

Expression of type Equals¶

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

Expression of type Equals ¶

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