Expression of type Prob

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.numbers import Add, Interval, Mod, Mult, Round, one
from proveit.physics.quantum import I, NumKet, Z, ket_plus, var_ket_u
from proveit.physics.quantum.QPE import QPE, phase, two_pow_t
from proveit.physics.quantum.circuits import Gate, Input, Measure, 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 = QPE(U, t), part = sub_expr1), targets = Interval(one, sub_expr3))
expr = Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Input(state = ket_plus), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = var_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, Measure(basis = Z), one, t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, s)], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(Mod(Round(Mult(two_pow_t, phase)), two_pow_t), t), part = sub_expr1), targets = Interval(one, t)), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = var_ket_u, part = sub_expr1), targets = sub_expr4), one, 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())

\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right)

stored_expr.style_options()

no style options

# 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	Operation	operator: 4 operands: 5
4	Literal
5	ExprTuple	6, 7, 8, 9
6	ExprTuple	10, 11
7	ExprTuple	12, 13
8	ExprTuple	14, 15
9	ExprTuple	16, 17
10	ExprRange	lambda_map: 18 start_index: 80 end_index: 95
11	ExprRange	lambda_map: 19 start_index: 80 end_index: 81
12	ExprRange	lambda_map: 20 start_index: 80 end_index: 95
13	ExprRange	lambda_map: 20 start_index: 69 end_index: 70
14	ExprRange	lambda_map: 21 start_index: 80 end_index: 95
15	ExprRange	lambda_map: 22 start_index: 80 end_index: 81
16	ExprRange	lambda_map: 23 start_index: 80 end_index: 95
17	ExprRange	lambda_map: 24 start_index: 80 end_index: 81
18	Lambda	parameter: 68 body: 25
19	Lambda	parameter: 68 body: 26
20	Lambda	parameter: 68 body: 27
21	Lambda	parameter: 68 body: 28
22	Lambda	parameter: 68 body: 29
23	Lambda	parameter: 68 body: 30
24	Lambda	parameter: 68 body: 32
25	Operation	operator: 54 operands: 33
26	Operation	operator: 40 operands: 34
27	Operation	operator: 40 operands: 35
28	Operation	operator: 36 operands: 37
29	Operation	operator: 55 operands: 38
30	Operation	operator: 40 operands: 39
31	ExprTuple	68
32	Operation	operator: 40 operands: 41
33	NamedExprs	state: 42
34	NamedExprs	element: 43 targets: 51
35	NamedExprs	element: 44 targets: 45
36	Literal
37	NamedExprs	basis: 46
38	NamedExprs	operation: 47
39	NamedExprs	element: 48 targets: 49
40	Literal
41	NamedExprs	element: 50 targets: 51
42	Operation	operator: 52 operand: 64
43	Operation	operator: 54 operands: 61
44	Operation	operator: 55 operands: 56
45	Operation	operator: 62 operands: 57
46	Literal
47	Literal
48	Operation	operator: 60 operands: 58
49	Operation	operator: 62 operands: 59
50	Operation	operator: 60 operands: 61
51	Operation	operator: 62 operands: 63
52	Literal
53	ExprTuple	64
54	Literal
55	Literal
56	NamedExprs	operation: 65 part: 68
57	ExprTuple	80, 70
58	NamedExprs	state: 66 part: 68
59	ExprTuple	80, 95
60	Literal
61	NamedExprs	state: 67 part: 68
62	Literal
63	ExprTuple	69, 70
64	Literal
65	Operation	operator: 71 operands: 72
66	Operation	operator: 73 operands: 74
67	Variable
68	Variable
69	Operation	operator: 76 operands: 75
70	Operation	operator: 76 operands: 77
71	Literal
72	ExprTuple	78, 95
73	Literal
74	ExprTuple	79, 95
75	ExprTuple	95, 80
76	Literal
77	ExprTuple	95, 81
78	Variable
79	Operation	operator: 82 operands: 83
80	Literal
81	Variable
82	Literal
83	ExprTuple	84, 90
84	Operation	operator: 85 operand: 87
85	Literal
86	ExprTuple	87
87	Operation	operator: 88 operands: 89
88	Literal
89	ExprTuple	90, 91
90	Operation	operator: 92 operands: 93
91	Variable
92	Literal
93	ExprTuple	94, 95
94	Literal
95	Variable