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 And, CartExp, Forall, InSet
from proveit.numbers import Add, Complex, Exp, Interval, Natural, NaturalPos, 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}")
sub_expr4 = IndexedVar(n, sub_expr2)
expr = 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 = 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(), 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(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{
\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}\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..

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