Expression of type ExprTuple

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, ExprTuple, Lambda, U, Variable, VertExprArray, s, t
from proveit.linear_algebra import MatrixMult, ScalarMult
from proveit.logic import And, Equals, InSet
from proveit.numbers import Add, Exp, Interval, Mult, Real, e, i, one, pi, subtract, two, zero
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 = Mult(two_pow_t, phase)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE(U, t), part = sub_expr1), targets = Interval(one, sub_expr3))
expr = ExprTuple(Lambda(phase, Conditional(Equals(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_expr6, one, t), ExprRange(sub_expr1, sub_expr6, 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(sub_expr5, 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)]))), one), And(InSet(phase, Real), InSet(sub_expr5, Interval(zero, subtract(two_pow_t, one))), Equals(MatrixMult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, phase)), var_ket_u))))))

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())

\left(\varphi \mapsto \left\{\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 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \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 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) = 1 \textrm{ if } \varphi \in \mathbb{R} ,  \left(2^{t} \cdot \varphi\right) \in \{0~\ldotp \ldotp~2^{t} - 1\} ,  \left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right)\right..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameter: 124 body: 3
2	ExprTuple	124
3	Conditional	value: 4 condition: 5
4	Operation	operator: 18 operands: 6
5	Operation	operator: 7 operands: 8
6	ExprTuple	9, 119
7	Literal
8	ExprTuple	10, 11, 12
9	Operation	operator: 13 operand: 20
10	Operation	operator: 16 operands: 15
11	Operation	operator: 16 operands: 17
12	Operation	operator: 18 operands: 19
13	Literal
14	ExprTuple	20
15	ExprTuple	124, 21
16	Literal
17	ExprTuple	118, 22
18	Literal
19	ExprTuple	23, 24
20	Operation	operator: 25 operands: 26
21	Literal
22	Operation	operator: 101 operands: 27
23	Operation	operator: 28 operands: 29
24	Operation	operator: 30 operands: 31
25	Literal
26	ExprTuple	32, 33, 34, 35
27	ExprTuple	36, 37
28	Literal
29	ExprTuple	117, 106
30	Literal
31	ExprTuple	38, 106
32	ExprTuple	39, 40
33	ExprTuple	41, 42
34	ExprTuple	43, 44
35	ExprTuple	45, 46
36	Literal
37	Operation	operator: 115 operands: 47
38	Operation	operator: 125 operands: 48
39	ExprRange	lambda_map: 49 start_index: 119 end_index: 128
40	ExprRange	lambda_map: 50 start_index: 119 end_index: 120
41	ExprRange	lambda_map: 51 start_index: 119 end_index: 128
42	ExprRange	lambda_map: 51 start_index: 108 end_index: 109
43	ExprRange	lambda_map: 52 start_index: 119 end_index: 128
44	ExprRange	lambda_map: 53 start_index: 119 end_index: 120
45	ExprRange	lambda_map: 54 start_index: 119 end_index: 128
46	ExprRange	lambda_map: 55 start_index: 119 end_index: 120
47	ExprTuple	123, 56
48	ExprTuple	57, 58
49	Lambda	parameter: 107 body: 59
50	Lambda	parameter: 107 body: 60
51	Lambda	parameter: 107 body: 61
52	Lambda	parameter: 107 body: 62
53	Lambda	parameter: 107 body: 63
54	Lambda	parameter: 107 body: 64
55	Lambda	parameter: 107 body: 66
56	Operation	operator: 67 operand: 119
57	Literal
58	Operation	operator: 121 operands: 69
59	Operation	operator: 93 operands: 70
60	Operation	operator: 77 operands: 71
61	Operation	operator: 77 operands: 72
62	Operation	operator: 73 operands: 74
63	Operation	operator: 94 operands: 75
64	Operation	operator: 77 operands: 76
65	ExprTuple	107
66	Operation	operator: 77 operands: 78
67	Literal
68	ExprTuple	119
69	ExprTuple	127, 79, 80, 124
70	NamedExprs	state: 81
71	NamedExprs	element: 82 targets: 90
72	NamedExprs	element: 83 targets: 84
73	Literal
74	NamedExprs	basis: 85
75	NamedExprs	operation: 86
76	NamedExprs	element: 87 targets: 88
77	Literal
78	NamedExprs	element: 89 targets: 90
79	Literal
80	Literal
81	Operation	operator: 91 operand: 103
82	Operation	operator: 93 operands: 100
83	Operation	operator: 94 operands: 95
84	Operation	operator: 101 operands: 96
85	Literal
86	Literal
87	Operation	operator: 99 operands: 97
88	Operation	operator: 101 operands: 98
89	Operation	operator: 99 operands: 100
90	Operation	operator: 101 operands: 102
91	Literal
92	ExprTuple	103
93	Literal
94	Literal
95	NamedExprs	operation: 104 part: 107
96	ExprTuple	119, 109
97	NamedExprs	state: 105 part: 107
98	ExprTuple	119, 128
99	Literal
100	NamedExprs	state: 106 part: 107
101	Literal
102	ExprTuple	108, 109
103	Literal
104	Operation	operator: 110 operands: 111
105	Operation	operator: 112 operands: 113
106	Variable
107	Variable
108	Operation	operator: 115 operands: 114
109	Operation	operator: 115 operands: 116
110	Literal
111	ExprTuple	117, 128
112	Literal
113	ExprTuple	118, 128
114	ExprTuple	128, 119
115	Literal
116	ExprTuple	128, 120
117	Variable
118	Operation	operator: 121 operands: 122
119	Literal
120	Variable
121	Literal
122	ExprTuple	123, 124
123	Operation	operator: 125 operands: 126
124	Variable
125	Literal
126	ExprTuple	127, 128
127	Literal
128	Variable