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 A, Conditional, ExprRange, IndexedVar, Lambda, N, Variable, VertExprArray, m, n
from proveit.core_expr_types import A_1_to_m
from proveit.linear_algebra import TensorProd
from proveit.logic import And, InSet
from proveit.numbers import Add, Interval, Natural, one, subtract, zero
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}")
expr = Lambda([N_0_to_m], Conditional(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)]))), And(ExprRange(sub_expr2, InSet(IndexedVar(N, sub_expr2), Natural), zero, m), each_Nk_is_partial_sum)))

# 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(N_{0}, N_{1}, \ldots, N_{m}\right) \mapsto \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) \textrm{ if } \left(N_{0} \in \mathbb{N}\right) ,  \left(N_{1} \in \mathbb{N}\right) ,  \ldots ,  \left(N_{m} \in \mathbb{N}\right) ,  \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)\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	3
2	Conditional	value: 4 condition: 5
3	ExprRange	lambda_map: 6 start_index: 46 end_index: 93
4	Operation	operator: 7 operands: 8
5	Operation	operator: 18 operands: 9
6	Lambda	parameter: 102 body: 47
7	Literal
8	ExprTuple	10, 11
9	ExprTuple	12, 13
10	Operation	operator: 15 operand: 20
11	Operation	operator: 15 operand: 21
12	ExprRange	lambda_map: 17 start_index: 46 end_index: 93
13	Operation	operator: 18 operands: 19
14	ExprTuple	20
15	Literal
16	ExprTuple	21
17	Lambda	parameter: 102 body: 22
18	Literal
19	ExprTuple	23, 24
20	ExprTuple	25
21	ExprTuple	26
22	Operation	operator: 27 operands: 28
23	Operation	operator: 39 operands: 29
24	ExprRange	lambda_map: 30 start_index: 105 end_index: 93
25	ExprRange	lambda_map: 31 start_index: 105 end_index: 93
26	ExprRange	lambda_map: 32 start_index: 105 end_index: 93
27	Literal
28	ExprTuple	47, 33
29	ExprTuple	34, 46
30	Lambda	parameter: 102 body: 35
31	Lambda	parameter: 100 body: 36
32	Lambda	parameter: 87 body: 37
33	Literal
34	IndexedVar	variable: 90 index: 46
35	Operation	operator: 39 operands: 40
36	ExprRange	lambda_map: 41 start_index: 105 end_index: 42
37	ExprRange	lambda_map: 43 start_index: 44 end_index: 45
38	ExprTuple	46
39	Literal
40	ExprTuple	47, 48
41	Lambda	parameter: 102 body: 49
42	IndexedVar	variable: 66 index: 100
43	Lambda	parameter: 100 body: 50
44	Operation	operator: 96 operands: 51
45	IndexedVar	variable: 90 index: 87
46	Literal
47	IndexedVar	variable: 90 index: 102
48	Operation	operator: 96 operands: 53
49	Operation	operator: 55 operands: 54
50	Operation	operator: 55 operands: 56
51	ExprTuple	57, 105
52	ExprTuple	87
53	ExprTuple	58, 59
54	NamedExprs	element: 60 targets: 61
55	Literal
56	NamedExprs	element: 62 targets: 63
57	IndexedVar	variable: 90 index: 73
58	IndexedVar	variable: 90 index: 74
59	IndexedVar	variable: 66 index: 102
60	Operation	operator: 69 operands: 67
61	Operation	operator: 71 operands: 68
62	Operation	operator: 69 operands: 70
63	Operation	operator: 71 operands: 72
64	ExprTuple	73
65	ExprTuple	74
66	Variable
67	NamedExprs	state: 75 part: 102
68	ExprTuple	76, 77
69	Literal
70	NamedExprs	state: 78 part: 100
71	Literal
72	ExprTuple	105, 79
73	Operation	operator: 96 operands: 80
74	Operation	operator: 96 operands: 81
75	IndexedVar	variable: 98 index: 100
76	Operation	operator: 96 operands: 82
77	IndexedVar	variable: 90 index: 100
78	Operation	operator: 84 operands: 85
79	IndexedVar	variable: 90 index: 93
80	ExprTuple	87, 101
81	ExprTuple	102, 101
82	ExprTuple	88, 105
83	ExprTuple	100
84	Literal
85	ExprTuple	89
86	ExprTuple	93
87	Variable
88	IndexedVar	variable: 90 index: 94
89	ExprRange	lambda_map: 92 start_index: 105 end_index: 93
90	Variable
91	ExprTuple	94
92	Lambda	parameter: 102 body: 95
93	Variable
94	Operation	operator: 96 operands: 97
95	IndexedVar	variable: 98 index: 102
96	Literal
97	ExprTuple	100, 101
98	Variable
99	ExprTuple	102
100	Variable
101	Operation	operator: 103 operand: 105
102	Variable
103	Literal
104	ExprTuple	105
105	Literal

Expression of type Lambda¶

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

Expression of type Lambda ¶

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