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Expression of type Conditional

from the theory of proveit.physics.quantum.circuits

In [1]:
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, Function, Literal, k, p
from proveit.core_expr_types.expr_arrays import A11_to_Akm, B11_to_Bkn, R11_to_Rkm, S11_to_Skn
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Interval, one
from proveit.physics.quantum.circuits import QcircuitEquiv, circuit_Akm, circuit_Bkn, circuit_permuted_Akm, circuit_permuted_Bkn
In [2]:
# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [A11_to_Akm, R11_to_Rkm, B11_to_Bkn, S11_to_Skn], instance_expr = Equals(QcircuitEquiv(circuit_Akm, circuit_Bkn), QcircuitEquiv(circuit_permuted_Akm, circuit_permuted_Bkn)).with_wrapping_at(2)).with_wrapping(), InSet(p, Function(Literal("Perm", latex_format = r"\textrm{Perm}", theory = "proveit.physics.quantum.circuits"), [Interval(one, k)])))
expr:
In [3]:
# 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
In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left\{\begin{array}{l}\forall_{A_{1, 1}, A_{1, 2}, \ldots, A_{1, m}, A_{2, 1}, A_{2, 2}, \ldots, A_{2, m}, \ldots\ldots, A_{k, 1}, A_{k, 2}, \ldots, A_{k, m}, R_{1, 1}, R_{1, 2}, \ldots, R_{1, m}, R_{2, 1}, R_{2, 2}, \ldots, R_{2, m}, \ldots\ldots, R_{k, 1}, R_{k, 2}, \ldots, R_{k, m}, B_{1, 1}, B_{1, 2}, \ldots, B_{1, n}, B_{2, 1}, B_{2, 2}, \ldots, B_{2, n}, \ldots\ldots, B_{k, 1}, B_{k, 2}, \ldots, B_{k, n}, S_{1, 1}, S_{1, 2}, \ldots, S_{1, n}, S_{2, 1}, S_{2, 2}, \ldots, S_{2, n}, \ldots\ldots, S_{k, 1}, S_{k, 2}, \ldots, S_{k, n}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{A_{1, 1}~\mbox{on}~R_{1, 1}} \qwx[1] & \gate{A_{2, 1}~\mbox{on}~R_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{m, 1}~\mbox{on}~R_{m, 1}} \qwx[1] & \qw \\
& \gate{A_{1, 2}~\mbox{on}~R_{1, 2}} \qwx[1] & \gate{A_{2, 2}~\mbox{on}~R_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{m, 2}~\mbox{on}~R_{m, 2}} \qwx[1] & \qw \\
& \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \qw \\
& \gate{A_{1, k}~\mbox{on}~R_{1, k}} & \gate{A_{2, k}~\mbox{on}~R_{2, k}} & \gate{\cdots} & \gate{A_{m, k}~\mbox{on}~R_{m, k}} & \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{B_{1, 1}~\mbox{on}~S_{1, 1}} \qwx[1] & \gate{B_{2, 1}~\mbox{on}~S_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{B_{n, 1}~\mbox{on}~S_{n, 1}} \qwx[1] & \qw \\
& \gate{B_{1, 2}~\mbox{on}~S_{1, 2}} \qwx[1] & \gate{B_{2, 2}~\mbox{on}~S_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{B_{n, 2}~\mbox{on}~S_{n, 2}} \qwx[1] & \qw \\
& \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \qw \\
& \gate{B_{1, k}~\mbox{on}~S_{1, k}} & \gate{B_{2, k}~\mbox{on}~S_{2, k}} & \gate{\cdots} & \gate{B_{n, k}~\mbox{on}~S_{n, k}} & \qw
} \end{array}\right)\right) =  \\ \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{A_{p\left(1\right), 1}~\mbox{on}~p^{\leftarrow}\left(R_{p\left(1\right), 1}\right)} \qwx[1] & \gate{A_{p\left(2\right), 1}~\mbox{on}~p^{\leftarrow}\left(R_{p\left(2\right), 1}\right)} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{p\left(m\right), 1}~\mbox{on}~p^{\leftarrow}\left(R_{p\left(m\right), 1}\right)} \qwx[1] & \qw \\
& \gate{A_{p\left(1\right), 2}~\mbox{on}~p^{\leftarrow}\left(R_{p\left(1\right), 2}\right)} \qwx[1] & \gate{A_{p\left(2\right), 2}~\mbox{on}~p^{\leftarrow}\left(R_{p\left(2\right), 2}\right)} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{p\left(m\right), 2}~\mbox{on}~p^{\leftarrow}\left(R_{p\left(m\right), 2}\right)} \qwx[1] & \qw \\
& \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \qw \\
& \gate{A_{p\left(1\right), k}~\mbox{on}~p^{\leftarrow}\left(R_{p\left(1\right), k}\right)} & \gate{A_{p\left(2\right), k}~\mbox{on}~p^{\leftarrow}\left(R_{p\left(2\right), k}\right)} & \gate{\cdots} & \gate{A_{p\left(m\right), k}~\mbox{on}~p^{\leftarrow}\left(R_{p\left(m\right), k}\right)} & \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{B_{p\left(1\right), 1}~\mbox{on}~p^{\leftarrow}\left(S_{p\left(1\right), 1}\right)} \qwx[1] & \gate{B_{p\left(2\right), 1}~\mbox{on}~p^{\leftarrow}\left(S_{p\left(2\right), 1}\right)} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{B_{p\left(n\right), 1}~\mbox{on}~p^{\leftarrow}\left(S_{p\left(n\right), 1}\right)} \qwx[1] & \qw \\
& \gate{B_{p\left(1\right), 2}~\mbox{on}~p^{\leftarrow}\left(S_{p\left(1\right), 2}\right)} \qwx[1] & \gate{B_{p\left(2\right), 2}~\mbox{on}~p^{\leftarrow}\left(S_{p\left(2\right), 2}\right)} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{B_{p\left(n\right), 2}~\mbox{on}~p^{\leftarrow}\left(S_{p\left(n\right), 2}\right)} \qwx[1] & \qw \\
& \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \qw \\
& \gate{B_{p\left(1\right), k}~\mbox{on}~p^{\leftarrow}\left(S_{p\left(1\right), k}\right)} & \gate{B_{p\left(2\right), k}~\mbox{on}~p^{\leftarrow}\left(S_{p\left(2\right), k}\right)} & \gate{\cdots} & \gate{B_{p\left(n\right), k}~\mbox{on}~p^{\leftarrow}\left(S_{p\left(n\right), k}\right)} & \qw
} \end{array}\right)\right) \end{array} \end{array}\right)\end{array} \textrm{ if } p \in \textrm{Perm}\left(\{1~\ldotp \ldotp~k\}\right)\right..
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
condition_delimiter'comma' or 'and'commacomma('with_comma_delimiter', 'with_conjunction_delimiter')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Conditionalvalue: 1
condition: 2
1Operationoperator: 3
operand: 7
2Operationoperator: 5
operands: 6
3Literal
4ExprTuple7
5Literal
6ExprTuple105, 8
7Lambdaparameters: 9
body: 10
8Operationoperator: 11
operand: 19
9ExprTuple13, 14, 15, 16
10Operationoperator: 17
operands: 18
11Literal
12ExprTuple19
13ExprRangelambda_map: 20
start_index: 70
end_index: 71
14ExprRangelambda_map: 21
start_index: 70
end_index: 71
15ExprRangelambda_map: 22
start_index: 70
end_index: 71
16ExprRangelambda_map: 23
start_index: 70
end_index: 71
17Literal
18ExprTuple24, 25
19Operationoperator: 26
operands: 27
20Lambdaparameter: 107
body: 28
21Lambdaparameter: 107
body: 29
22Lambdaparameter: 107
body: 30
23Lambdaparameter: 107
body: 31
24Operationoperator: 33
operands: 32
25Operationoperator: 33
operands: 34
26Literal
27ExprTuple70, 71
28ExprRangelambda_map: 35
start_index: 70
end_index: 55
29ExprRangelambda_map: 36
start_index: 70
end_index: 55
30ExprRangelambda_map: 37
start_index: 70
end_index: 57
31ExprRangelambda_map: 38
start_index: 70
end_index: 57
32ExprTuple39, 40
33Literal
34ExprTuple41, 42
35Lambdaparameter: 104
body: 82
36Lambdaparameter: 104
body: 83
37Lambdaparameter: 104
body: 84
38Lambdaparameter: 104
body: 85
39Operationoperator: 46
operands: 43
40Operationoperator: 46
operands: 44
41Operationoperator: 46
operands: 45
42Operationoperator: 46
operands: 47
43ExprTuple48
44ExprTuple49
45ExprTuple50
46Literal
47ExprTuple51
48ExprRangelambda_map: 52
start_index: 70
end_index: 55
49ExprRangelambda_map: 53
start_index: 70
end_index: 57
50ExprRangelambda_map: 54
start_index: 70
end_index: 55
51ExprRangelambda_map: 56
start_index: 70
end_index: 57
52Lambdaparameter: 107
body: 58
53Lambdaparameter: 107
body: 59
54Lambdaparameter: 107
body: 60
55Variable
56Lambdaparameter: 107
body: 61
57Variable
58ExprTuple62
59ExprTuple63
60ExprTuple64
61ExprTuple65
62ExprRangelambda_map: 66
start_index: 70
end_index: 71
63ExprRangelambda_map: 67
start_index: 70
end_index: 71
64ExprRangelambda_map: 68
start_index: 70
end_index: 71
65ExprRangelambda_map: 69
start_index: 70
end_index: 71
66Lambdaparameter: 104
body: 72
67Lambdaparameter: 104
body: 73
68Lambdaparameter: 104
body: 74
69Lambdaparameter: 104
body: 76
70Literal
71Variable
72Operationoperator: 80
operands: 77
73Operationoperator: 80
operands: 78
74Operationoperator: 80
operands: 79
75ExprTuple104
76Operationoperator: 80
operands: 81
77NamedExprselement: 82
targets: 83
78NamedExprselement: 84
targets: 85
79NamedExprselement: 86
targets: 87
80Literal
81NamedExprselement: 88
targets: 89
82IndexedVarvariable: 91
indices: 90
83IndexedVarvariable: 100
indices: 90
84IndexedVarvariable: 93
indices: 90
85IndexedVarvariable: 101
indices: 90
86IndexedVarvariable: 91
indices: 102
87Operationoperator: 94
operand: 96
88IndexedVarvariable: 93
indices: 102
89Operationoperator: 94
operand: 99
90ExprTuple107, 104
91Variable
92ExprTuple96
93Variable
94Operationoperator: 97
operand: 105
95ExprTuple99
96IndexedVarvariable: 100
indices: 102
97Literal
98ExprTuple105
99IndexedVarvariable: 101
indices: 102
100Variable
101Variable
102ExprTuple103, 104
103Operationoperator: 105
operand: 107
104Variable
105Variable
106ExprTuple107
107Variable