Expression of type ExprTuple ¶

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 A, Conditional, ExprRange, ExprTuple, IndexedVar, Lambda, N, Variable, VertExprArray, m, n
from proveit.core_expr_types import A_1_to_m, n_1_to_m
from proveit.linear_algebra import TensorProd
from proveit.logic import And, CartExp, Forall, InSet
from proveit.numbers import Add, Complex, Exp, Interval, Natural, NaturalPos, one, subtract, two
from proveit.physics.quantum.circuits import MultiQubitElem, N_0_to_m, N_m, Output, Qcircuit, QcircuitEquiv, each_Nk_is_partial_sum

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr3 = Variable("_c", latex_format = r"{_{-}c}")
sub_expr4 = IndexedVar(n, sub_expr2)
expr = ExprTuple(Lambda([n_1_to_m], Conditional(Forall(instance_param_or_params = [A_1_to_m], instance_expr = Forall(instance_param_or_params = [N_0_to_m], instance_expr = QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, ExprRange(sub_expr2, MultiQubitElem(element = Output(state = IndexedVar(A, sub_expr1), part = sub_expr2), targets = Interval(Add(IndexedVar(N, subtract(sub_expr1, one)), one), IndexedVar(N, sub_expr1))), one, IndexedVar(n, sub_expr1)).with_wrapping_at(2,6), one, m)])), Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr3, ExprRange(sub_expr1, MultiQubitElem(element = Output(state = TensorProd(A_1_to_m), part = sub_expr1), targets = Interval(one, N_m)), Add(IndexedVar(N, subtract(sub_expr3, one)), one), IndexedVar(N, sub_expr3)).with_wrapping_at(2,6), one, m)]))), domain = Natural, condition = each_Nk_is_partial_sum).with_wrapping(), domains = [ExprRange(sub_expr2, CartExp(Complex, Exp(two, sub_expr4)), one, m)]).with_wrapping(), And(ExprRange(sub_expr2, InSet(sub_expr4, NaturalPos), one, m)))))

# 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(\left(n_{1}, n_{2}, \ldots, n_{m}\right) \mapsto \left\{\begin{array}{l}\forall_{\left(A_{1} \in \mathbb{C}^{2^{n_{1}}}\right), \left(A_{2} \in \mathbb{C}^{2^{n_{2}}}\right), \ldots, \left(A_{m} \in \mathbb{C}^{2^{n_{m}}}\right)}~\\
\left[\begin{array}{l}\forall_{N_{0}, N_{1}, \ldots, N_{m} \in \mathbb{N}~|~\left(N_{0} = 0\right)\land \left(N_{1} = \left(N_{1 - 1} + n_{1}\right)\right) \land  \left(N_{2} = \left(N_{2 - 1} + n_{2}\right)\right) \land  \ldots \land  \left(N_{m} = \left(N_{m - 1} + n_{m}\right)\right)}~\\
\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qout{A_{1}~\mbox{part}~1~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} \\
& \qout{A_{1}~\mbox{part}~2~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} \\
& \qout{\vdots} \\
& \qout{A_{1}~\mbox{part}~n_{1}~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} \\
& \qout{A_{2}~\mbox{part}~1~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} \\
& \qout{A_{2}~\mbox{part}~2~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} \\
& \qout{\vdots} \\
& \qout{A_{2}~\mbox{part}~n_{2}~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} \\
& \qout{\begin{array}{c}\vdots\\ \vdots\end{array}} \\
& \qout{A_{m}~\mbox{part}~1~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} \\
& \qout{A_{m}~\mbox{part}~2~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} \\
& \qout{\vdots} \\
& \qout{A_{m}~\mbox{part}~n_{m}~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}}
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1 - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1 - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{\vdots} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2 - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2 - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{\vdots} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{\begin{array}{c}\vdots\\ \vdots\end{array}} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{\vdots} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}}
} \end{array}\right)\right)\end{array}\right]\end{array} \textrm{ if } \left(n_{1} \in \mathbb{N}^+\right) \land  \left(n_{2} \in \mathbb{N}^+\right) \land  \ldots \land  \left(n_{m} \in \mathbb{N}^+\right)\right..\right)

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	ExprTuple	1
1	Lambda	parameters: 2 body: 3
2	ExprTuple	4
3	Conditional	value: 5 condition: 6
4	ExprRange	lambda_map: 7 start_index: 139 end_index: 127
5	Operation	operator: 17 operand: 10
6	Operation	operator: 45 operands: 9
7	Lambda	parameter: 136 body: 93
8	ExprTuple	10
9	ExprTuple	11
10	Lambda	parameters: 119 body: 12
11	ExprRange	lambda_map: 13 start_index: 139 end_index: 127
12	Conditional	value: 14 condition: 15
13	Lambda	parameter: 136 body: 16
14	Operation	operator: 17 operand: 21
15	Operation	operator: 45 operands: 19
16	Operation	operator: 58 operands: 20
17	Literal
18	ExprTuple	21
19	ExprTuple	22
20	ExprTuple	93, 23
21	Lambda	parameters: 24 body: 25
22	ExprRange	lambda_map: 26 start_index: 139 end_index: 127
23	Literal
24	ExprTuple	27
25	Conditional	value: 28 condition: 29
26	Lambda	parameter: 136 body: 30
27	ExprRange	lambda_map: 31 start_index: 80 end_index: 127
28	Operation	operator: 32 operands: 33
29	Operation	operator: 45 operands: 34
30	Operation	operator: 58 operands: 35
31	Lambda	parameter: 136 body: 81
32	Literal
33	ExprTuple	36, 37
34	ExprTuple	38, 39
35	ExprTuple	129, 40
36	Operation	operator: 42 operand: 49
37	Operation	operator: 42 operand: 50
38	ExprRange	lambda_map: 44 start_index: 80 end_index: 127
39	Operation	operator: 45 operands: 46
40	Operation	operator: 47 operands: 48
41	ExprTuple	49
42	Literal
43	ExprTuple	50
44	Lambda	parameter: 136 body: 51
45	Literal
46	ExprTuple	52, 53
47	Literal
48	ExprTuple	54, 55
49	ExprTuple	56
50	ExprTuple	57
51	Operation	operator: 58 operands: 59
52	Operation	operator: 73 operands: 60
53	ExprRange	lambda_map: 61 start_index: 139 end_index: 127
54	Literal
55	Operation	operator: 62 operands: 63
56	ExprRange	lambda_map: 64 start_index: 139 end_index: 127
57	ExprRange	lambda_map: 65 start_index: 139 end_index: 127
58	Literal
59	ExprTuple	81, 66
60	ExprTuple	67, 80
61	Lambda	parameter: 136 body: 68
62	Literal
63	ExprTuple	69, 93
64	Lambda	parameter: 134 body: 70
65	Lambda	parameter: 121 body: 71
66	Literal
67	IndexedVar	variable: 124 index: 80
68	Operation	operator: 73 operands: 74
69	Literal
70	ExprRange	lambda_map: 75 start_index: 139 end_index: 76
71	ExprRange	lambda_map: 77 start_index: 78 end_index: 79
72	ExprTuple	80
73	Literal
74	ExprTuple	81, 82
75	Lambda	parameter: 136 body: 83
76	IndexedVar	variable: 100 index: 134
77	Lambda	parameter: 134 body: 84
78	Operation	operator: 130 operands: 85
79	IndexedVar	variable: 124 index: 121
80	Literal
81	IndexedVar	variable: 124 index: 136
82	Operation	operator: 130 operands: 87
83	Operation	operator: 89 operands: 88
84	Operation	operator: 89 operands: 90
85	ExprTuple	91, 139
86	ExprTuple	121
87	ExprTuple	92, 93
88	NamedExprs	element: 94 targets: 95
89	Literal
90	NamedExprs	element: 96 targets: 97
91	IndexedVar	variable: 124 index: 107
92	IndexedVar	variable: 124 index: 108
93	IndexedVar	variable: 100 index: 136
94	Operation	operator: 103 operands: 101
95	Operation	operator: 105 operands: 102
96	Operation	operator: 103 operands: 104
97	Operation	operator: 105 operands: 106
98	ExprTuple	107
99	ExprTuple	108
100	Variable
101	NamedExprs	state: 109 part: 136
102	ExprTuple	110, 111
103	Literal
104	NamedExprs	state: 112 part: 134
105	Literal
106	ExprTuple	139, 113
107	Operation	operator: 130 operands: 114
108	Operation	operator: 130 operands: 115
109	IndexedVar	variable: 132 index: 134
110	Operation	operator: 130 operands: 116
111	IndexedVar	variable: 124 index: 134
112	Operation	operator: 118 operands: 119
113	IndexedVar	variable: 124 index: 127
114	ExprTuple	121, 135
115	ExprTuple	136, 135
116	ExprTuple	122, 139
117	ExprTuple	134
118	Literal
119	ExprTuple	123
120	ExprTuple	127
121	Variable
122	IndexedVar	variable: 124 index: 128
123	ExprRange	lambda_map: 126 start_index: 139 end_index: 127
124	Variable
125	ExprTuple	128
126	Lambda	parameter: 136 body: 129
127	Variable
128	Operation	operator: 130 operands: 131
129	IndexedVar	variable: 132 index: 136
130	Literal
131	ExprTuple	134, 135
132	Variable
133	ExprTuple	136
134	Variable
135	Operation	operator: 137 operand: 139
136	Variable
137	Literal
138	ExprTuple	139
139	Literal

Expression of type ExprTuple¶

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

Expression of type ExprTuple ¶

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