Expression of type Lambda

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, Lambda, 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 = Lambda([s, t], 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))))

expr:

# 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(s, t\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	87, 88
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 86 body: 13
10	Operation	operator: 21 operands: 14
11	Operation	operator: 21 operands: 15
12	ExprTuple	86
13	Conditional	value: 16 condition: 17
14	ExprTuple	87, 18
15	ExprTuple	88, 18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 21 operands: 22
18	Literal
19	Literal
20	ExprTuple	23, 24
21	Literal
22	ExprTuple	86, 25
23	Operation	operator: 27 operand: 31
24	Operation	operator: 27 operands: 28
25	Operation	operator: 29 operand: 34
26	ExprTuple	31
27	Literal
28	ExprTuple	32, 33
29	Literal
30	ExprTuple	34
31	ExprTuple	35, 36
32	ExprTuple	37, 38
33	ExprTuple	39, 40
34	Operation	operator: 41 operands: 42
35	ExprRange	lambda_map: 43 start_index: 78 end_index: 88
36	ExprRange	lambda_map: 43 start_index: 45 end_index: 75
37	ExprRange	lambda_map: 44 start_index: 78 end_index: 88
38	ExprRange	lambda_map: 44 start_index: 45 end_index: 75
39	ExprRange	lambda_map: 46 start_index: 78 end_index: 88
40	ExprRange	lambda_map: 47 start_index: 78 end_index: 87
41	Literal
42	ExprTuple	48, 87
43	Lambda	parameter: 77 body: 49
44	Lambda	parameter: 77 body: 50
45	Operation	operator: 82 operands: 51
46	Lambda	parameter: 77 body: 52
47	Lambda	parameter: 77 body: 54
48	Literal
49	Operation	operator: 57 operands: 55
50	Operation	operator: 57 operands: 56
51	ExprTuple	88, 78
52	Operation	operator: 57 operands: 58
53	ExprTuple	77
54	Operation	operator: 69 operands: 59
55	NamedExprs	element: 60 targets: 62
56	NamedExprs	element: 61 targets: 62
57	Literal
58	NamedExprs	element: 63 targets: 64
59	NamedExprs	operation: 65
60	Operation	operator: 69 operands: 66
61	Operation	operator: 69 operands: 67
62	Operation	operator: 71 operands: 68
63	Operation	operator: 69 operands: 70
64	Operation	operator: 71 operands: 72
65	Literal
66	NamedExprs	operation: 73 part: 77
67	NamedExprs	operation: 74 part: 77
68	ExprTuple	78, 75
69	Literal
70	NamedExprs	operation: 76 part: 77
71	Literal
72	ExprTuple	78, 88
73	Operation	operator: 79 operands: 81
74	Operation	operator: 80 operands: 81
75	Operation	operator: 82 operands: 83
76	Operation	operator: 84 operand: 88
77	Variable
78	Literal
79	Literal
80	Literal
81	ExprTuple	86, 88
82	Literal
83	ExprTuple	88, 87
84	Literal
85	ExprTuple	88
86	Variable
87	Variable
88	Variable