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.linear_algebra import TensorProd
from proveit.logic import Implies
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, QcircuitEquiv

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Interval(one, _t)
sub_expr3 = Add(_t, one)
sub_expr4 = Add(_t, _s)
sub_expr5 = Interval(sub_expr3, sub_expr4)
sub_expr6 = MultiQubitElem(element = Input(state = TensorProd(_psi__t_ket, _ket_u), part = sub_expr1), targets = Interval(one, sub_expr4))
sub_expr7 = [ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _psi__t_ket, part = sub_expr1), targets = sub_expr2), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)]
sub_expr8 = [ExprRange(sub_expr1, MultiQubitElem(element = Gate(operation = InverseFourierTransform(_t), part = sub_expr1), targets = sub_expr2), one, _t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, _s)]
sub_expr9 = [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _Psi_ket, part = sub_expr1), targets = sub_expr2), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)]
sub_expr10 = [ExprRange(sub_expr1, sub_expr6, one, _t).with_wrapping_at(2,6), ExprRange(sub_expr1, sub_expr6, sub_expr3, sub_expr4).with_wrapping_at(2,6)]
expr = Implies(QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray(sub_expr7)), Qcircuit(vert_expr_array = VertExprArray(sub_expr10))), QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray(sub_expr7, sub_expr8, sub_expr9)), Qcircuit(vert_expr_array = VertExprArray(sub_expr10, sub_expr8, sub_expr9))).with_wrapping_at(1)).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(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert \psi_{t} \rangle} & { /^{t} } \qw \\
\qin{\lvert u \rangle} & { /^{s} } \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{1}{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} & { /^{t} } \qw \\
\ghostqin{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} & { /^{s} } \qw
} \end{array}\right)\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \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) \\  \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{1}{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} & \gate{{\mathrm {FT}}^{\dag}_{t}} & \qout{\lvert \Psi \rangle} \\
\ghostqin{\lvert \psi_{t} \rangle {\otimes} \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, 9
6	Literal
7	ExprTuple	10, 11
8	Operation	operator: 15 operand: 17
9	Operation	operator: 15 operand: 18
10	Operation	operator: 15 operands: 14
11	Operation	operator: 15 operands: 16
12	ExprTuple	17
13	ExprTuple	18
14	ExprTuple	17, 19, 20
15	Literal
16	ExprTuple	18, 19, 20
17	ExprTuple	21, 22
18	ExprTuple	23, 24
19	ExprTuple	25, 26
20	ExprTuple	27, 28
21	ExprRange	lambda_map: 29 start_index: 88 end_index: 92
22	ExprRange	lambda_map: 30 start_index: 88 end_index: 89
23	ExprRange	lambda_map: 31 start_index: 88 end_index: 92
24	ExprRange	lambda_map: 31 start_index: 78 end_index: 79
25	ExprRange	lambda_map: 32 start_index: 88 end_index: 92
26	ExprRange	lambda_map: 33 start_index: 88 end_index: 89
27	ExprRange	lambda_map: 34 start_index: 88 end_index: 92
28	ExprRange	lambda_map: 35 start_index: 88 end_index: 89
29	Lambda	parameter: 77 body: 36
30	Lambda	parameter: 77 body: 37
31	Lambda	parameter: 77 body: 38
32	Lambda	parameter: 77 body: 39
33	Lambda	parameter: 77 body: 40
34	Lambda	parameter: 77 body: 41
35	Lambda	parameter: 77 body: 43
36	Operation	operator: 50 operands: 44
37	Operation	operator: 50 operands: 45
38	Operation	operator: 50 operands: 46
39	Operation	operator: 50 operands: 47
40	Operation	operator: 66 operands: 48
41	Operation	operator: 50 operands: 49
42	ExprTuple	77
43	Operation	operator: 50 operands: 51
44	NamedExprs	element: 52 targets: 59
45	NamedExprs	element: 53 targets: 61
46	NamedExprs	element: 54 targets: 55
47	NamedExprs	element: 56 targets: 59
48	NamedExprs	operation: 57
49	NamedExprs	element: 58 targets: 59
50	Literal
51	NamedExprs	element: 60 targets: 61
52	Operation	operator: 63 operands: 62
53	Operation	operator: 63 operands: 71
54	Operation	operator: 63 operands: 64
55	Operation	operator: 72 operands: 65
56	Operation	operator: 66 operands: 67
57	Literal
58	Operation	operator: 70 operands: 68
59	Operation	operator: 72 operands: 69
60	Operation	operator: 70 operands: 71
61	Operation	operator: 72 operands: 73
62	NamedExprs	state: 86 part: 77
63	Literal
64	NamedExprs	state: 74 part: 77
65	ExprTuple	88, 79
66	Literal
67	NamedExprs	operation: 75 part: 77
68	NamedExprs	state: 76 part: 77
69	ExprTuple	88, 92
70	Literal
71	NamedExprs	state: 87 part: 77
72	Literal
73	ExprTuple	78, 79
74	Operation	operator: 80 operands: 81
75	Operation	operator: 82 operand: 92
76	Literal
77	Variable
78	Operation	operator: 84 operands: 83
79	Operation	operator: 84 operands: 85
80	Literal
81	ExprTuple	86, 87
82	Literal
83	ExprTuple	92, 88
84	Literal
85	ExprTuple	92, 89
86	Operation	operator: 90 operand: 92
87	Literal
88	Literal
89	Literal
90	Literal
91	ExprTuple	92
92	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 ¶