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
from proveit.linear_algebra import TensorProd
from proveit.logic import CartExp, Forall
from proveit.numbers import Add, Complex, Exp, Interval, Natural, 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 = [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()

# 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_{\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}

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