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, n_1_to_m
from proveit.linear_algebra import TensorProd
from proveit.logic import 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}")
expr = Lambda(m, Conditional(Forall(instance_param_or_params = [n_1_to_m], instance_expr = 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, IndexedVar(n, sub_expr2))), one, m)]).with_wrapping(), domain = NaturalPos).with_wrapping(), InSet(m, NaturalPos)))

# 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())

m \mapsto \left\{\begin{array}{l}\forall_{n_{1}, n_{2}, \ldots, n_{m} \in \mathbb{N}^+}~\\
\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}\right]\end{array} \textrm{ if } m \in \mathbb{N}^+\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Lambda	parameter: 132 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 22 operand: 6
3	Operation	operator: 63 operands: 5
4	ExprTuple	6
5	ExprTuple	132, 28
6	Lambda	parameters: 7 body: 8
7	ExprTuple	9
8	Conditional	value: 10 condition: 11
9	ExprRange	lambda_map: 12 start_index: 144 end_index: 132
10	Operation	operator: 22 operand: 15
11	Operation	operator: 50 operands: 14
12	Lambda	parameter: 141 body: 98
13	ExprTuple	15
14	ExprTuple	16
15	Lambda	parameters: 124 body: 17
16	ExprRange	lambda_map: 18 start_index: 144 end_index: 132
17	Conditional	value: 19 condition: 20
18	Lambda	parameter: 141 body: 21
19	Operation	operator: 22 operand: 26
20	Operation	operator: 50 operands: 24
21	Operation	operator: 63 operands: 25
22	Literal
23	ExprTuple	26
24	ExprTuple	27
25	ExprTuple	98, 28
26	Lambda	parameters: 29 body: 30
27	ExprRange	lambda_map: 31 start_index: 144 end_index: 132
28	Literal
29	ExprTuple	32
30	Conditional	value: 33 condition: 34
31	Lambda	parameter: 141 body: 35
32	ExprRange	lambda_map: 36 start_index: 85 end_index: 132
33	Operation	operator: 37 operands: 38
34	Operation	operator: 50 operands: 39
35	Operation	operator: 63 operands: 40
36	Lambda	parameter: 141 body: 86
37	Literal
38	ExprTuple	41, 42
39	ExprTuple	43, 44
40	ExprTuple	134, 45
41	Operation	operator: 47 operand: 54
42	Operation	operator: 47 operand: 55
43	ExprRange	lambda_map: 49 start_index: 85 end_index: 132
44	Operation	operator: 50 operands: 51
45	Operation	operator: 52 operands: 53
46	ExprTuple	54
47	Literal
48	ExprTuple	55
49	Lambda	parameter: 141 body: 56
50	Literal
51	ExprTuple	57, 58
52	Literal
53	ExprTuple	59, 60
54	ExprTuple	61
55	ExprTuple	62
56	Operation	operator: 63 operands: 64
57	Operation	operator: 78 operands: 65
58	ExprRange	lambda_map: 66 start_index: 144 end_index: 132
59	Literal
60	Operation	operator: 67 operands: 68
61	ExprRange	lambda_map: 69 start_index: 144 end_index: 132
62	ExprRange	lambda_map: 70 start_index: 144 end_index: 132
63	Literal
64	ExprTuple	86, 71
65	ExprTuple	72, 85
66	Lambda	parameter: 141 body: 73
67	Literal
68	ExprTuple	74, 98
69	Lambda	parameter: 139 body: 75
70	Lambda	parameter: 126 body: 76
71	Literal
72	IndexedVar	variable: 129 index: 85
73	Operation	operator: 78 operands: 79
74	Literal
75	ExprRange	lambda_map: 80 start_index: 144 end_index: 81
76	ExprRange	lambda_map: 82 start_index: 83 end_index: 84
77	ExprTuple	85
78	Literal
79	ExprTuple	86, 87
80	Lambda	parameter: 141 body: 88
81	IndexedVar	variable: 105 index: 139
82	Lambda	parameter: 139 body: 89
83	Operation	operator: 135 operands: 90
84	IndexedVar	variable: 129 index: 126
85	Literal
86	IndexedVar	variable: 129 index: 141
87	Operation	operator: 135 operands: 92
88	Operation	operator: 94 operands: 93
89	Operation	operator: 94 operands: 95
90	ExprTuple	96, 144
91	ExprTuple	126
92	ExprTuple	97, 98
93	NamedExprs	element: 99 targets: 100
94	Literal
95	NamedExprs	element: 101 targets: 102
96	IndexedVar	variable: 129 index: 112
97	IndexedVar	variable: 129 index: 113
98	IndexedVar	variable: 105 index: 141
99	Operation	operator: 108 operands: 106
100	Operation	operator: 110 operands: 107
101	Operation	operator: 108 operands: 109
102	Operation	operator: 110 operands: 111
103	ExprTuple	112
104	ExprTuple	113
105	Variable
106	NamedExprs	state: 114 part: 141
107	ExprTuple	115, 116
108	Literal
109	NamedExprs	state: 117 part: 139
110	Literal
111	ExprTuple	144, 118
112	Operation	operator: 135 operands: 119
113	Operation	operator: 135 operands: 120
114	IndexedVar	variable: 137 index: 139
115	Operation	operator: 135 operands: 121
116	IndexedVar	variable: 129 index: 139
117	Operation	operator: 123 operands: 124
118	IndexedVar	variable: 129 index: 132
119	ExprTuple	126, 140
120	ExprTuple	141, 140
121	ExprTuple	127, 144
122	ExprTuple	139
123	Literal
124	ExprTuple	128
125	ExprTuple	132
126	Variable
127	IndexedVar	variable: 129 index: 133
128	ExprRange	lambda_map: 131 start_index: 144 end_index: 132
129	Variable
130	ExprTuple	133
131	Lambda	parameter: 141 body: 134
132	Variable
133	Operation	operator: 135 operands: 136
134	IndexedVar	variable: 137 index: 141
135	Literal
136	ExprTuple	139, 140
137	Variable
138	ExprTuple	141
139	Variable
140	Operation	operator: 142 operand: 144
141	Variable
142	Literal
143	ExprTuple	144
144	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 ¶