Expression of type Conditional ¶

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 Conditional, ExprRange, U, Variable, VertExprArray, s, t
from proveit.linear_algebra import Unitary
from proveit.logic import And, Forall, InSet
from proveit.numbers import Add, Exp, Interval, NaturalPos, one, two
from proveit.physics.quantum import I
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import QPE, QPE1
from proveit.physics.quantum.circuits import Gate, MultiQubitElem, 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, sub_expr2)
sub_expr5 = MultiQubitElem(element = Gate(operation = QPE(U, t), part = sub_expr1), targets = sub_expr4)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE1(U, t), part = sub_expr1), targets = sub_expr4)
expr = Conditional(Forall(instance_param_or_params = [U], instance_expr = QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, sub_expr5, one, t), ExprRange(sub_expr1, sub_expr5, sub_expr3, sub_expr2)])), Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, sub_expr6, one, t), ExprRange(sub_expr1, sub_expr6, sub_expr3, sub_expr2)], [ExprRange(sub_expr1, MultiQubitElem(element = Gate(operation = InverseFourierTransform(t), part = sub_expr1), targets = Interval(one, t)), one, t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, s)]))), domain = Unitary(Exp(two, s))), And(InSet(s, NaturalPos), InSet(t, NaturalPos)))

# 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())

\left\{\forall_{U \in \textrm{U}\left(2^{s}\right)}~\left(\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) \cong \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)\right) \textrm{ if } s \in \mathbb{N}^+ ,  t \in \mathbb{N}^+\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	8, 9
7	Lambda	parameter: 84 body: 11
8	Operation	operator: 19 operands: 12
9	Operation	operator: 19 operands: 13
10	ExprTuple	84
11	Conditional	value: 14 condition: 15
12	ExprTuple	85, 16
13	ExprTuple	86, 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	Literal
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	84, 23
21	Operation	operator: 25 operand: 29
22	Operation	operator: 25 operands: 26
23	Operation	operator: 27 operand: 32
24	ExprTuple	29
25	Literal
26	ExprTuple	30, 31
27	Literal
28	ExprTuple	32
29	ExprTuple	33, 34
30	ExprTuple	35, 36
31	ExprTuple	37, 38
32	Operation	operator: 39 operands: 40
33	ExprRange	lambda_map: 41 start_index: 76 end_index: 86
34	ExprRange	lambda_map: 41 start_index: 43 end_index: 73
35	ExprRange	lambda_map: 42 start_index: 76 end_index: 86
36	ExprRange	lambda_map: 42 start_index: 43 end_index: 73
37	ExprRange	lambda_map: 44 start_index: 76 end_index: 86
38	ExprRange	lambda_map: 45 start_index: 76 end_index: 85
39	Literal
40	ExprTuple	46, 85
41	Lambda	parameter: 75 body: 47
42	Lambda	parameter: 75 body: 48
43	Operation	operator: 80 operands: 49
44	Lambda	parameter: 75 body: 50
45	Lambda	parameter: 75 body: 52
46	Literal
47	Operation	operator: 55 operands: 53
48	Operation	operator: 55 operands: 54
49	ExprTuple	86, 76
50	Operation	operator: 55 operands: 56
51	ExprTuple	75
52	Operation	operator: 67 operands: 57
53	NamedExprs	element: 58 targets: 60
54	NamedExprs	element: 59 targets: 60
55	Literal
56	NamedExprs	element: 61 targets: 62
57	NamedExprs	operation: 63
58	Operation	operator: 67 operands: 64
59	Operation	operator: 67 operands: 65
60	Operation	operator: 69 operands: 66
61	Operation	operator: 67 operands: 68
62	Operation	operator: 69 operands: 70
63	Literal
64	NamedExprs	operation: 71 part: 75
65	NamedExprs	operation: 72 part: 75
66	ExprTuple	76, 73
67	Literal
68	NamedExprs	operation: 74 part: 75
69	Literal
70	ExprTuple	76, 86
71	Operation	operator: 77 operands: 79
72	Operation	operator: 78 operands: 79
73	Operation	operator: 80 operands: 81
74	Operation	operator: 82 operand: 86
75	Variable
76	Literal
77	Literal
78	Literal
79	ExprTuple	84, 86
80	Literal
81	ExprTuple	86, 85
82	Literal
83	ExprTuple	86
84	Variable
85	Variable
86	Variable

Expression of type Conditional¶

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

Expression of type Conditional ¶

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