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, k, l, m
from proveit.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import NaturalPos
from proveit.physics.quantum import l_ket_domain, m_ket_domain, var_ket_psi, var_ket_u, var_ket_v
from proveit.physics.quantum.circuits import circuit__u_Akl_v, circuit__u_Akl_v__psi_m
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Lambda([k, l, m], Conditional(Forall(instance_param_or_params = [var_ket_u, var_ket_v], instance_expr = Forall(instance_param_or_params = [var_ket_psi], instance_expr = Forall(instance_param_or_params = [A11_to_Akl], instance_expr = Equals(Prob(circuit__u_Akl_v), Prob(circuit__u_Akl_v__psi_m)).with_wrapping_at(2)).with_wrapping(), domain = m_ket_domain), domain = l_ket_domain), And(InSet(k, NaturalPos), InSet(l, NaturalPos), InSet(m, NaturalPos))))

# 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(k, l, m\right) \mapsto \left\{\forall_{\lvert u \rangle, \lvert v \rangle \in \mathbb{C}^{2^{l}}}~\left[\forall_{\lvert \psi \rangle \in \mathbb{C}^{2^{m}}}~\left[\begin{array}{l}\forall_{A_{1, 1}, A_{1, 2}, \ldots, A_{1, l}, A_{2, 1}, A_{2, 2}, \ldots, A_{2, l}, \ldots\ldots, A_{k, 1}, A_{k, 2}, \ldots, A_{k, l}}~\\
\left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle}
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle} \\
\qin{\lvert \psi \rangle} & { /^{m} } \qw & { /^{m} } \qw & \gate{\cdots} & { /^{m} } \qw & \qout{\lvert \psi \rangle}
} \end{array}\right) \end{array} \end{array}\right)\end{array}\right]\right] \textrm{ if } k \in \mathbb{N}^+ ,  l \in \mathbb{N}^+ ,  m \in \mathbb{N}^+\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Lambda	parameters: 1 body: 2
1	ExprTuple	74, 130, 131
2	Conditional	value: 3 condition: 4
3	Operation	operator: 32 operand: 7
4	Operation	operator: 20 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9, 10
7	Lambda	parameters: 11 body: 12
8	Operation	operator: 34 operands: 13
9	Operation	operator: 34 operands: 14
10	Operation	operator: 34 operands: 15
11	ExprTuple	118, 121
12	Conditional	value: 16 condition: 17
13	ExprTuple	74, 18
14	ExprTuple	130, 18
15	ExprTuple	131, 18
16	Operation	operator: 32 operand: 22
17	Operation	operator: 20 operands: 21
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	23, 24
22	Lambda	parameter: 122 body: 26
23	Operation	operator: 34 operands: 27
24	Operation	operator: 34 operands: 28
25	ExprTuple	122
26	Conditional	value: 29 condition: 30
27	ExprTuple	118, 31
28	ExprTuple	121, 31
29	Operation	operator: 32 operand: 37
30	Operation	operator: 34 operands: 35
31	Operation	operator: 42 operands: 36
32	Literal
33	ExprTuple	37
34	Literal
35	ExprTuple	122, 38
36	ExprTuple	48, 39
37	Lambda	parameters: 40 body: 41
38	Operation	operator: 42 operands: 43
39	Operation	operator: 53 operands: 44
40	ExprTuple	45
41	Operation	operator: 46 operands: 47
42	Literal
43	ExprTuple	48, 49
44	ExprTuple	58, 130
45	ExprRange	lambda_map: 50 start_index: 129 end_index: 74
46	Literal
47	ExprTuple	51, 52
48	Literal
49	Operation	operator: 53 operands: 54
50	Lambda	parameter: 119 body: 86
51	Operation	operator: 56 operand: 59
52	Operation	operator: 56 operand: 60
53	Literal
54	ExprTuple	58, 131
55	ExprTuple	59
56	Literal
57	ExprTuple	60
58	Literal
59	Operation	operator: 62 operands: 61
60	Operation	operator: 62 operands: 63
61	ExprTuple	64, 65, 66
62	Literal
63	ExprTuple	67, 68, 69
64	ExprTuple	71
65	ExprRange	lambda_map: 70 start_index: 129 end_index: 74
66	ExprTuple	75
67	ExprTuple	71, 72
68	ExprRange	lambda_map: 73 start_index: 129 end_index: 74
69	ExprTuple	75, 76
70	Lambda	parameter: 119 body: 77
71	ExprRange	lambda_map: 78 start_index: 129 end_index: 130
72	ExprRange	lambda_map: 79 start_index: 129 end_index: 131
73	Lambda	parameter: 119 body: 81
74	Variable
75	ExprRange	lambda_map: 82 start_index: 129 end_index: 130
76	ExprRange	lambda_map: 83 start_index: 129 end_index: 131
77	ExprTuple	86
78	Lambda	parameter: 123 body: 84
79	Lambda	parameter: 123 body: 85
80	ExprTuple	119
81	ExprTuple	86, 87
82	Lambda	parameter: 123 body: 88
83	Lambda	parameter: 123 body: 89
84	Operation	operator: 95 operands: 90
85	Operation	operator: 95 operands: 91
86	ExprRange	lambda_map: 92 start_index: 129 end_index: 130
87	ExprRange	lambda_map: 93 start_index: 129 end_index: 131
88	Operation	operator: 95 operands: 94
89	Operation	operator: 95 operands: 96
90	NamedExprs	element: 97 targets: 103
91	NamedExprs	element: 98 targets: 105
92	Lambda	parameter: 123 body: 99
93	Lambda	parameter: 123 body: 101
94	NamedExprs	element: 102 targets: 103
95	Literal
96	NamedExprs	element: 104 targets: 105
97	Operation	operator: 107 operands: 106
98	Operation	operator: 107 operands: 115
99	IndexedVar	variable: 108 indices: 109
100	ExprTuple	123
101	Operation	operator: 110 operands: 111
102	Operation	operator: 114 operands: 112
103	Operation	operator: 116 operands: 113
104	Operation	operator: 114 operands: 115
105	Operation	operator: 116 operands: 117
106	NamedExprs	state: 118 part: 123
107	Literal
108	Variable
109	ExprTuple	119, 123
110	Literal
111	NamedExprs	operation: 120
112	NamedExprs	state: 121 part: 123
113	ExprTuple	129, 130
114	Literal
115	NamedExprs	state: 122 part: 123
116	Literal
117	ExprTuple	124, 125
118	Variable
119	Variable
120	Literal
121	Variable
122	Variable
123	Variable
124	Operation	operator: 127 operands: 126
125	Operation	operator: 127 operands: 128
126	ExprTuple	130, 129
127	Literal
128	ExprTuple	130, 131
129	Literal
130	Variable
131	Variable

Expression of type Lambda¶

from the theory of proveit.physics.quantum.circuits¶

Expression of type Lambda ¶

from the theory of proveit.physics.quantum.circuits ¶