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, CartExp, Forall, InSet
from proveit.numbers import Add, Complex, Exp, Interval, Natural, one, subtract, two
from proveit.physics.quantum.circuits import Input, MultiQubitElem, N_0_to_m, N_m, 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([A_1_to_m], Conditional(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 = Input(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 = Input(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(), And(ExprRange(sub_expr2, InSet(IndexedVar(A, sub_expr2), CartExp(Complex, Exp(two, IndexedVar(n, sub_expr2)))), 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(A_{1}, A_{2}, \ldots, A_{m}\right) \mapsto \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{
\qin{A_{1}~\mbox{part}~1~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} & \qw \\
\qin{A_{1}~\mbox{part}~2~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1}~\mbox{part}~n_{1}~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} & \qw \\
\qin{A_{2}~\mbox{part}~1~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} & \qw \\
\qin{A_{2}~\mbox{part}~2~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{2}~\mbox{part}~n_{2}~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} & \qw \\
\qin{\begin{array}{c}\vdots\\ \vdots\end{array}} & \qw \\
\qin{A_{m}~\mbox{part}~1~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{m}~\mbox{part}~2~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{m}~\mbox{part}~n_{m}~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} & \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1 - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1 - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2 - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2 - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\begin{array}{c}\vdots\\ \vdots\end{array}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw
} \end{array}\right)\right)\end{array} \textrm{ if } \left(A_{1} \in \mathbb{C}^{2^{n_{1}}}\right) \land  \left(A_{2} \in \mathbb{C}^{2^{n_{2}}}\right) \land  \ldots \land  \left(A_{m} \in \mathbb{C}^{2^{n_{m}}}\right)\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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