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__psi_m__u_Akl_v, circuit__u_Akl_v
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__psi_m__u_Akl_v)).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{
\qin{\lvert \psi \rangle} & { /^{m} } \qw & { /^{m} } \qw & \gate{\cdots} & { /^{m} } \qw & \qout{\lvert \psi \rangle} \\
\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) \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	76, 141, 140
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	128, 132
12	Conditional	value: 16 condition: 17
13	ExprTuple	76, 18
14	ExprTuple	141, 18
15	ExprTuple	140, 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: 131 body: 26
23	Operation	operator: 34 operands: 27
24	Operation	operator: 34 operands: 28
25	ExprTuple	131
26	Conditional	value: 29 condition: 30
27	ExprTuple	128, 31
28	ExprTuple	132, 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	131, 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, 141
45	ExprRange	lambda_map: 50 start_index: 139 end_index: 76
46	Literal
47	ExprTuple	51, 52
48	Literal
49	Operation	operator: 53 operands: 54
50	Lambda	parameter: 130 body: 93
51	Operation	operator: 56 operand: 59
52	Operation	operator: 56 operand: 60
53	Literal
54	ExprTuple	58, 140
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	70
65	ExprRange	lambda_map: 71 start_index: 139 end_index: 76
66	ExprTuple	72
67	ExprTuple	73, 74
68	ExprRange	lambda_map: 75 start_index: 139 end_index: 76
69	ExprTuple	77, 78
70	ExprRange	lambda_map: 79 start_index: 139 end_index: 141
71	Lambda	parameter: 130 body: 80
72	ExprRange	lambda_map: 81 start_index: 139 end_index: 141
73	ExprRange	lambda_map: 82 start_index: 139 end_index: 140
74	ExprRange	lambda_map: 83 start_index: 139 end_index: 141
75	Lambda	parameter: 130 body: 85
76	Variable
77	ExprRange	lambda_map: 86 start_index: 139 end_index: 140
78	ExprRange	lambda_map: 87 start_index: 139 end_index: 141
79	Lambda	parameter: 133 body: 88
80	ExprTuple	93
81	Lambda	parameter: 133 body: 89
82	Lambda	parameter: 133 body: 90
83	Lambda	parameter: 133 body: 91
84	ExprTuple	130
85	ExprTuple	92, 93
86	Lambda	parameter: 133 body: 94
87	Lambda	parameter: 133 body: 95
88	Operation	operator: 103 operands: 96
89	Operation	operator: 103 operands: 97
90	Operation	operator: 103 operands: 98
91	Operation	operator: 103 operands: 99
92	ExprRange	lambda_map: 100 start_index: 139 end_index: 140
93	ExprRange	lambda_map: 101 start_index: 139 end_index: 141
94	Operation	operator: 103 operands: 102
95	Operation	operator: 103 operands: 104
96	NamedExprs	element: 107 targets: 105
97	NamedExprs	element: 113 targets: 105
98	NamedExprs	element: 106 targets: 112
99	NamedExprs	element: 107 targets: 114
100	Lambda	parameter: 133 body: 108
101	Lambda	parameter: 133 body: 110
102	NamedExprs	element: 111 targets: 112
103	Literal
104	NamedExprs	element: 113 targets: 114
105	Operation	operator: 126 operands: 115
106	Operation	operator: 116 operands: 122
107	Operation	operator: 116 operands: 117
108	Operation	operator: 118 operands: 119
109	ExprTuple	133
110	IndexedVar	variable: 120 indices: 121
111	Operation	operator: 124 operands: 122
112	Operation	operator: 126 operands: 123
113	Operation	operator: 124 operands: 125
114	Operation	operator: 126 operands: 127
115	ExprTuple	139, 141
116	Literal
117	NamedExprs	state: 128 part: 133
118	Literal
119	NamedExprs	operation: 129
120	Variable
121	ExprTuple	130, 133
122	NamedExprs	state: 131 part: 133
123	ExprTuple	139, 140
124	Literal
125	NamedExprs	state: 132 part: 133
126	Literal
127	ExprTuple	134, 135
128	Variable
129	Literal
130	Variable
131	Variable
132	Variable
133	Variable
134	Operation	operator: 137 operands: 136
135	Operation	operator: 137 operands: 138
136	ExprTuple	140, 139
137	Literal
138	ExprTuple	140, 141
139	Literal
140	Variable
141	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 ¶