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 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 QPE, QPE1, _Psi_circuit, _Psi_ket, _U, _ket_u, _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 = Add(_t, _s)
sub_expr3 = Add(_t, one)
sub_expr4 = Interval(one, _t)
sub_expr5 = Interval(one, sub_expr2)
sub_expr6 = Interval(sub_expr3, sub_expr2)
sub_expr7 = MultiQubitElem(element = Gate(operation = QPE1(_U, _t), part = sub_expr1), targets = sub_expr5)
sub_expr8 = MultiQubitElem(element = Gate(operation = QPE(_U, _t), part = sub_expr1), targets = sub_expr5)
sub_expr9 = [ExprRange(sub_expr1, sub_expr7, one, _t), ExprRange(sub_expr1, sub_expr7, sub_expr3, sub_expr2)]
sub_expr10 = [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)]
expr = Implies(QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray(sub_expr9, sub_expr10)), Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, sub_expr8, one, _t), ExprRange(sub_expr1, sub_expr8, sub_expr3, sub_expr2)]))), QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Input(state = ket_plus), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr6), one, _s)], sub_expr9, sub_expr10, [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _Psi_ket, part = sub_expr1), targets = sub_expr4), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr6), one, _s)])), _Psi_circuit).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{
& \multigate{1}{\textrm{QPE}_1\left(U, t\right)} & \gate{{\mathrm {FT}}^{\dag}_{t}} & { /^{t} } \qw \\
& \ghost{\textrm{QPE}_1\left(U, t\right)} & { /^{s} } \qw & { /^{s} } \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \multigate{1}{\textrm{QPE}\left(U, t\right)} & { /^{t} } \qw \\
& \ghost{\textrm{QPE}\left(U, t\right)} & { /^{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 + \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) \\  \cong \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) \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 operands: 12
9	Operation	operator: 15 operand: 20
10	Operation	operator: 15 operands: 14
11	Operation	operator: 15 operands: 16
12	ExprTuple	17, 18
13	ExprTuple	20
14	ExprTuple	19, 17, 18, 21
15	Literal
16	ExprTuple	19, 20, 21
17	ExprTuple	22, 23
18	ExprTuple	24, 25
19	ExprTuple	26, 27
20	ExprTuple	28, 29
21	ExprTuple	30, 31
22	ExprRange	lambda_map: 32 start_index: 101 end_index: 102
23	ExprRange	lambda_map: 32 start_index: 90 end_index: 91
24	ExprRange	lambda_map: 33 start_index: 101 end_index: 102
25	ExprRange	lambda_map: 34 start_index: 101 end_index: 103
26	ExprRange	lambda_map: 35 start_index: 101 end_index: 102
27	ExprRange	lambda_map: 36 start_index: 101 end_index: 103
28	ExprRange	lambda_map: 37 start_index: 101 end_index: 102
29	ExprRange	lambda_map: 37 start_index: 90 end_index: 91
30	ExprRange	lambda_map: 38 start_index: 101 end_index: 102
31	ExprRange	lambda_map: 39 start_index: 101 end_index: 103
32	Lambda	parameter: 89 body: 40
33	Lambda	parameter: 89 body: 41
34	Lambda	parameter: 89 body: 42
35	Lambda	parameter: 89 body: 43
36	Lambda	parameter: 89 body: 44
37	Lambda	parameter: 89 body: 45
38	Lambda	parameter: 89 body: 46
39	Lambda	parameter: 89 body: 48
40	Operation	operator: 56 operands: 49
41	Operation	operator: 56 operands: 50
42	Operation	operator: 74 operands: 51
43	Operation	operator: 73 operands: 52
44	Operation	operator: 56 operands: 53
45	Operation	operator: 56 operands: 54
46	Operation	operator: 56 operands: 55
47	ExprTuple	89
48	Operation	operator: 56 operands: 57
49	NamedExprs	element: 58 targets: 64
50	NamedExprs	element: 59 targets: 66
51	NamedExprs	operation: 60
52	NamedExprs	state: 61
53	NamedExprs	element: 62 targets: 68
54	NamedExprs	element: 63 targets: 64
55	NamedExprs	element: 65 targets: 66
56	Literal
57	NamedExprs	element: 67 targets: 68
58	Operation	operator: 74 operands: 69
59	Operation	operator: 74 operands: 70
60	Literal
61	Operation	operator: 71 operand: 85
62	Operation	operator: 73 operands: 80
63	Operation	operator: 74 operands: 75
64	Operation	operator: 81 operands: 76
65	Operation	operator: 79 operands: 77
66	Operation	operator: 81 operands: 78
67	Operation	operator: 79 operands: 80
68	Operation	operator: 81 operands: 82
69	NamedExprs	operation: 83 part: 89
70	NamedExprs	operation: 84 part: 89
71	Literal
72	ExprTuple	85
73	Literal
74	Literal
75	NamedExprs	operation: 86 part: 89
76	ExprTuple	101, 91
77	NamedExprs	state: 87 part: 89
78	ExprTuple	101, 102
79	Literal
80	NamedExprs	state: 88 part: 89
81	Literal
82	ExprTuple	90, 91
83	Operation	operator: 92 operands: 96
84	Operation	operator: 93 operand: 102
85	Literal
86	Operation	operator: 95 operands: 96
87	Literal
88	Literal
89	Variable
90	Operation	operator: 98 operands: 97
91	Operation	operator: 98 operands: 99
92	Literal
93	Literal
94	ExprTuple	102
95	Literal
96	ExprTuple	100, 102
97	ExprTuple	102, 101
98	Literal
99	ExprTuple	102, 103
100	Literal
101	Literal
102	Literal
103	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 ¶