Expression of type Forall

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, U, Variable, VertExprArray, s, t
from proveit.linear_algebra import Unitary
from proveit.logic import Forall
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 = 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)))

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())

\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)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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