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, Variable, VertExprArray, t
from proveit.logic import Equals, InSet
from proveit.numbers import Add, Interval, NaturalPos, one
from proveit.physics.quantum import ket_plus
from proveit.physics.quantum.QPE import QPE1, _U, _ket_u, _psi_t_ket, _s
from proveit.physics.quantum.circuits import Gate, Input, MultiQubitElem, Output, Qcircuit
from proveit.statistics import Prob

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(t, one)
sub_expr3 = Add(t, _s)
sub_expr4 = Interval(sub_expr2, sub_expr3)
sub_expr5 = MultiQubitElem(element = Gate(operation = QPE1(_U, t), part = sub_expr1), targets = Interval(one, sub_expr3))
expr = Lambda(t, Conditional(Equals(Prob(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_expr4), one, _s)], [ExprRange(sub_expr1, sub_expr5, one, t), ExprRange(sub_expr1, sub_expr5, sub_expr2, sub_expr3)], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _psi_t_ket, part = sub_expr1), targets = Interval(one, t)), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)]))), one), 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())

t \mapsto \left\{\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}_1\left(U, t\right)} & \multiqout{3}{\lvert \psi_{t} \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\lvert u \rangle}
} \end{array}\right) = 1 \textrm{ if } 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	parameter: 77 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operands: 5
3	Operation	operator: 6 operands: 7
4	Literal
5	ExprTuple	8, 76
6	Literal
7	ExprTuple	77, 9
8	Operation	operator: 10 operand: 12
9	Literal
10	Literal
11	ExprTuple	12
12	Operation	operator: 13 operands: 14
13	Literal
14	ExprTuple	15, 16, 17
15	ExprTuple	18, 19
16	ExprTuple	20, 21
17	ExprTuple	22, 23
18	ExprRange	lambda_map: 24 start_index: 76 end_index: 77
19	ExprRange	lambda_map: 25 start_index: 76 end_index: 78
20	ExprRange	lambda_map: 26 start_index: 76 end_index: 77
21	ExprRange	lambda_map: 26 start_index: 66 end_index: 67
22	ExprRange	lambda_map: 27 start_index: 76 end_index: 77
23	ExprRange	lambda_map: 28 start_index: 76 end_index: 78
24	Lambda	parameter: 65 body: 29
25	Lambda	parameter: 65 body: 30
26	Lambda	parameter: 65 body: 31
27	Lambda	parameter: 65 body: 32
28	Lambda	parameter: 65 body: 34
29	Operation	operator: 51 operands: 35
30	Operation	operator: 39 operands: 36
31	Operation	operator: 39 operands: 37
32	Operation	operator: 39 operands: 38
33	ExprTuple	65
34	Operation	operator: 39 operands: 40
35	NamedExprs	state: 41
36	NamedExprs	element: 42 targets: 48
37	NamedExprs	element: 43 targets: 44
38	NamedExprs	element: 45 targets: 46
39	Literal
40	NamedExprs	element: 47 targets: 48
41	Operation	operator: 49 operand: 61
42	Operation	operator: 51 operands: 58
43	Operation	operator: 52 operands: 53
44	Operation	operator: 59 operands: 54
45	Operation	operator: 57 operands: 55
46	Operation	operator: 59 operands: 56
47	Operation	operator: 57 operands: 58
48	Operation	operator: 59 operands: 60
49	Literal
50	ExprTuple	61
51	Literal
52	Literal
53	NamedExprs	operation: 62 part: 65
54	ExprTuple	76, 67
55	NamedExprs	state: 63 part: 65
56	ExprTuple	76, 77
57	Literal
58	NamedExprs	state: 64 part: 65
59	Literal
60	ExprTuple	66, 67
61	Literal
62	Operation	operator: 68 operands: 69
63	Operation	operator: 70 operand: 77
64	Literal
65	Variable
66	Operation	operator: 73 operands: 72
67	Operation	operator: 73 operands: 74
68	Literal
69	ExprTuple	75, 77
70	Literal
71	ExprTuple	77
72	ExprTuple	77, 76
73	Literal
74	ExprTuple	77, 78
75	Literal
76	Literal
77	Variable
78	Literal