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 InSet
from proveit.numbers import Add, Exp, Interval, 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(U, Conditional(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)]))), InSet(U, Unitary(Exp(two, s)))))

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())

U \mapsto \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) \textrm{ if } U \in \textrm{U}\left(2^{s}\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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