Expression of type Forall ¶

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, ExprRange, IndexedVar, 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
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 = 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()

# 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())

\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}

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

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

Expression of type Forall¶

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

Expression of type Forall ¶

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