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
from proveit.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import And, Equals, Forall, InSet
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([var_ket_u, var_ket_v], Conditional(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), And(InSet(var_ket_u, l_ket_domain), InSet(var_ket_v, l_ket_domain))))

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