Expression of type Lambda

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, Lambda, U
from proveit.linear_algebra import ScalarMult
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Exp, Mult, 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 = Lambda(var_ket_u, Conditional(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))), And(InSet(var_ket_u, m_ket_domain), normalized_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())

\lvert u \rangle \mapsto \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) \textrm{ if } \lvert u \rangle \in \mathbb{C}^{2^{m}} ,  \left \|\lvert u \rangle\right \| = 1\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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