Expression of type ExprTuple

from the theory of proveit.physics.quantum.circuits

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, ExprTuple, Lambda, U, m
from proveit.linear_algebra import ScalarMult, Unitary
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Exp, Mult, NaturalPos, e, i, one, pi, two
from proveit.physics.quantum import Qmult, m_ket_domain, normalized_var_ket_u, var_ket_u, varphi
from proveit.physics.quantum.circuits import phase_kickback_circuit
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = ExprTuple(Lambda(m, Conditional(Forall(instance_param_or_params = [U], instance_expr = Forall(instance_param_or_params = [var_ket_u], instance_expr = Forall(instance_param_or_params = [varphi], instance_expr = Equals(Prob(phase_kickback_circuit), one), condition = Equals(Qmult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, varphi)), var_ket_u))), domain = m_ket_domain, condition = normalized_var_ket_u), domain = Unitary(Exp(two, m))), InSet(m, 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())

\left(m \mapsto \left\{\forall_{U \in \textrm{U}\left(2^{m}\right)}~\left[\forall_{\lvert u \rangle \in \mathbb{C}^{2^{m}}~|~\left \|\lvert u \rangle\right \| = 1}~\left[\forall_{\varphi~|~\left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right)}~\left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \control{} \qw \qwx[1] & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert u \rangle} & \gate{U} & \qout{\lvert u \rangle}
} \end{array}\right) = 1\right)\right]\right] \textrm{ if } m \in \mathbb{N}^+\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: 127 body: 3
2	ExprTuple	127
3	Conditional	value: 4 condition: 5
4	Operation	operator: 23 operand: 8
5	Operation	operator: 32 operands: 7
6	ExprTuple	8
7	ExprTuple	127, 9
8	Lambda	parameter: 110 body: 11
9	Literal
10	ExprTuple	110
11	Conditional	value: 12 condition: 13
12	Operation	operator: 23 operand: 16
13	Operation	operator: 32 operands: 15
14	ExprTuple	16
15	ExprTuple	110, 17
16	Lambda	parameter: 114 body: 18
17	Operation	operator: 19 operand: 50
18	Conditional	value: 21 condition: 22
19	Literal
20	ExprTuple	50
21	Operation	operator: 23 operand: 27
22	Operation	operator: 25 operands: 26
23	Literal
24	ExprTuple	27
25	Literal
26	ExprTuple	28, 29
27	Lambda	parameter: 142 body: 31
28	Operation	operator: 32 operands: 33
29	Operation	operator: 40 operands: 34
30	ExprTuple	142
31	Conditional	value: 35 condition: 36
32	Literal
33	ExprTuple	114, 37
34	ExprTuple	38, 136
35	Operation	operator: 40 operands: 39
36	Operation	operator: 40 operands: 41
37	Operation	operator: 42 operands: 43
38	Operation	operator: 44 operand: 114
39	ExprTuple	46, 136
40	Literal
41	ExprTuple	47, 48
42	Literal
43	ExprTuple	49, 50
44	Literal
45	ExprTuple	114
46	Operation	operator: 51 operand: 57
47	Operation	operator: 53 operands: 54
48	Operation	operator: 119 operands: 55
49	Literal
50	Operation	operator: 130 operands: 56
51	Literal
52	ExprTuple	57
53	Literal
54	ExprTuple	110, 114
55	ExprTuple	125, 114
56	ExprTuple	139, 127
57	Operation	operator: 58 operands: 59
58	Literal
59	ExprTuple	60, 61, 62
60	ExprTuple	63, 64
61	ExprTuple	65, 66
62	ExprTuple	67, 68
63	Operation	operator: 100 operands: 69
64	ExprRange	lambda_map: 70 start_index: 136 end_index: 75
65	Operation	operator: 91 operands: 71
66	ExprRange	lambda_map: 72 start_index: 136 end_index: 75
67	Operation	operator: 106 operands: 73
68	ExprRange	lambda_map: 74 start_index: 136 end_index: 75
69	NamedExprs	state: 76
70	Lambda	parameter: 115 body: 77
71	NamedExprs	element: 78 targets: 79
72	Lambda	parameter: 115 body: 80
73	NamedExprs	state: 81
74	Lambda	parameter: 115 body: 83
75	Operation	operator: 121 operands: 84
76	Operation	operator: 132 operand: 93
77	Operation	operator: 91 operands: 86
78	Literal
79	Operation	operator: 87 operand: 139
80	Operation	operator: 91 operands: 89
81	Operation	operator: 119 operands: 90
82	ExprTuple	115
83	Operation	operator: 91 operands: 92
84	ExprTuple	127, 136
85	ExprTuple	93
86	NamedExprs	element: 94 targets: 99
87	Literal
88	ExprTuple	139
89	NamedExprs	element: 95 targets: 99
90	ExprTuple	96, 97
91	Literal
92	NamedExprs	element: 98 targets: 99
93	Literal
94	Operation	operator: 100 operands: 107
95	Operation	operator: 101 operands: 102
96	Operation	operator: 128 operands: 103
97	Operation	operator: 104 operands: 105
98	Operation	operator: 106 operands: 107
99	Operation	operator: 108 operands: 109
100	Literal
101	Literal
102	NamedExprs	operation: 110 part: 115
103	ExprTuple	136, 111
104	Literal
105	ExprTuple	112, 113
106	Literal
107	NamedExprs	state: 114 part: 115
108	Literal
109	ExprTuple	139, 116
110	Variable
111	Operation	operator: 130 operands: 117
112	Operation	operator: 132 operand: 124
113	Operation	operator: 119 operands: 120
114	Variable
115	Variable
116	Operation	operator: 121 operands: 122
117	ExprTuple	139, 123
118	ExprTuple	124
119	Literal
120	ExprTuple	125, 126
121	Literal
122	ExprTuple	127, 139
123	Operation	operator: 128 operands: 129
124	Literal
125	Operation	operator: 130 operands: 131
126	Operation	operator: 132 operand: 136
127	Variable
128	Literal
129	ExprTuple	136, 139
130	Literal
131	ExprTuple	134, 135
132	Literal
133	ExprTuple	136
134	Literal
135	Operation	operator: 137 operands: 138
136	Literal
137	Literal
138	ExprTuple	139, 140, 141, 142
139	Literal
140	Literal
141	Literal
142	Variable