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, Variable, VertExprArray, t
from proveit.linear_algebra import ScalarMult, VecAdd
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import ket0, ket1, ket_plus
from proveit.physics.quantum.QPE import QPE1, _U, _ket_u, _phase, _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 = 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, Output(state = ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))), Add(Neg(t), one), zero).with_decreasing_order(), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _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}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\vdots} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \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
6	ExprTuple	9, 10
7	ExprTuple	11, 12
8	ExprTuple	13, 14
9	ExprRange	lambda_map: 15 start_index: 88 end_index: 71
10	ExprRange	lambda_map: 16 start_index: 88 end_index: 72
11	ExprRange	lambda_map: 17 start_index: 88 end_index: 71
12	ExprRange	lambda_map: 17 start_index: 57 end_index: 58
13	ExprRange	lambda_map: 18 start_index: 19 end_index: 78
14	ExprRange	lambda_map: 20 start_index: 88 end_index: 72
15	Lambda	parameter: 101 body: 21
16	Lambda	parameter: 101 body: 22
17	Lambda	parameter: 101 body: 23
18	Lambda	parameter: 101 body: 24
19	Operation	operator: 65 operands: 25
20	Lambda	parameter: 101 body: 26
21	Operation	operator: 43 operands: 27
22	Operation	operator: 32 operands: 28
23	Operation	operator: 32 operands: 29
24	Operation	operator: 48 operands: 30
25	ExprTuple	31, 88
26	Operation	operator: 32 operands: 33
27	NamedExprs	state: 34
28	NamedExprs	element: 35 targets: 41
29	NamedExprs	element: 36 targets: 37
30	NamedExprs	state: 38
31	Operation	operator: 99 operand: 71
32	Literal
33	NamedExprs	element: 40 targets: 41
34	Operation	operator: 84 operand: 52
35	Operation	operator: 43 operands: 49
36	Operation	operator: 44 operands: 45
37	Operation	operator: 50 operands: 46
38	Operation	operator: 75 operands: 47
39	ExprTuple	71
40	Operation	operator: 48 operands: 49
41	Operation	operator: 50 operands: 51
42	ExprTuple	52
43	Literal
44	Literal
45	NamedExprs	operation: 53 part: 101
46	ExprTuple	88, 58
47	ExprTuple	54, 55
48	Literal
49	NamedExprs	state: 56 part: 101
50	Literal
51	ExprTuple	57, 58
52	Literal
53	Operation	operator: 59 operands: 60
54	Operation	operator: 81 operands: 61
55	Operation	operator: 62 operands: 63
56	Literal
57	Operation	operator: 65 operands: 64
58	Operation	operator: 65 operands: 66
59	Literal
60	ExprTuple	67, 71
61	ExprTuple	88, 68
62	Literal
63	ExprTuple	69, 70
64	ExprTuple	71, 88
65	Literal
66	ExprTuple	71, 72
67	Literal
68	Operation	operator: 95 operands: 73
69	Operation	operator: 84 operand: 78
70	Operation	operator: 75 operands: 76
71	Variable
72	Literal
73	ExprTuple	97, 77
74	ExprTuple	78
75	Literal
76	ExprTuple	79, 80
77	Operation	operator: 81 operands: 82
78	Literal
79	Operation	operator: 95 operands: 83
80	Operation	operator: 84 operand: 88
81	Literal
82	ExprTuple	88, 97
83	ExprTuple	86, 87
84	Literal
85	ExprTuple	88
86	Literal
87	Operation	operator: 89 operands: 90
88	Literal
89	Literal
90	ExprTuple	97, 91, 92, 93, 94
91	Literal
92	Literal
93	Operation	operator: 95 operands: 96
94	Literal
95	Literal
96	ExprTuple	97, 98
97	Literal
98	Operation	operator: 99 operand: 101
99	Literal
100	ExprTuple	101
101	Variable