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 i, j, m
from proveit.core_expr_types import U_1_to_i, V_1_to_j
from proveit.logic import Equals, Forall, Implies
from proveit.numbers import NaturalPos, one
from proveit.physics.quantum import m_ket_domain, var_ket_psi
from proveit.physics.quantum.circuits import circuit_Ui_Vj, circuit_Ui_psi_m, circuit_psi_m_Vj
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j, m], instance_expr = Forall(instance_param_or_params = [var_ket_psi], instance_expr = Forall(instance_param_or_params = [U_1_to_i, V_1_to_j], instance_expr = Implies(Equals(Prob(circuit_Ui_psi_m), one), Equals(Prob(circuit_Ui_Vj), Prob(circuit_psi_m_Vj)).with_wrapping_at(2)).with_wrapping_at(2)).with_wrapping(), domain = m_ket_domain).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_{i, j, m \in \mathbb{N}^+}~\\
\left[\begin{array}{l}\forall_{\lvert \psi \rangle \in \mathbb{C}^{2^{m}}}~\\
\left[\begin{array}{l}\forall_{U_{1}, U_{2}, \ldots, U_{i}, V_{1}, V_{2}, \ldots, V_{j}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\U_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{i} \\ \downarrow\end{array}} & \qout{\lvert \psi \rangle}
} \end{array}\right) = 1\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\U_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{i} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{j} \\ \downarrow\end{array}} & \qw
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert \psi \rangle} & \gate{\begin{array}{c} \uparrow \\V_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{j} \\ \downarrow\end{array}} & { /^{m} } \qw
} \end{array}\right) \end{array} \end{array}\right) \end{array} \end{array}\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: 22 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	64, 67, 91
4	Conditional	value: 5 condition: 6
5	Operation	operator: 22 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	11, 12, 13
10	Lambda	parameter: 88 body: 15
11	Operation	operator: 24 operands: 16
12	Operation	operator: 24 operands: 17
13	Operation	operator: 24 operands: 18
14	ExprTuple	88
15	Conditional	value: 19 condition: 20
16	ExprTuple	64, 21
17	ExprTuple	67, 21
18	ExprTuple	91, 21
19	Operation	operator: 22 operand: 26
20	Operation	operator: 24 operands: 25
21	Literal
22	Literal
23	ExprTuple	26
24	Literal
25	ExprTuple	88, 27
26	Lambda	parameters: 28 body: 29
27	Operation	operator: 30 operands: 31
28	ExprTuple	59, 61
29	Operation	operator: 32 operands: 33
30	Literal
31	ExprTuple	34, 35
32	Literal
33	ExprTuple	36, 37
34	Literal
35	Operation	operator: 38 operands: 39
36	Operation	operator: 41 operands: 40
37	Operation	operator: 41 operands: 42
38	Literal
39	ExprTuple	43, 91
40	ExprTuple	44, 90
41	Literal
42	ExprTuple	45, 46
43	Literal
44	Operation	operator: 49 operand: 51
45	Operation	operator: 49 operand: 52
46	Operation	operator: 49 operand: 53
47	ExprTuple	51
48	ExprTuple	52
49	Literal
50	ExprTuple	53
51	Operation	operator: 56 operands: 54
52	Operation	operator: 56 operands: 55
53	Operation	operator: 56 operands: 57
54	ExprTuple	59, 58
55	ExprTuple	59, 61
56	Literal
57	ExprTuple	60, 61
58	ExprTuple	62
59	ExprRange	lambda_map: 63 start_index: 90 end_index: 64
60	ExprTuple	65
61	ExprRange	lambda_map: 66 start_index: 90 end_index: 67
62	ExprRange	lambda_map: 68 start_index: 90 end_index: 91
63	Lambda	parameter: 89 body: 69
64	Variable
65	ExprRange	lambda_map: 70 start_index: 90 end_index: 91
66	Lambda	parameter: 89 body: 71
67	Variable
68	Lambda	parameter: 89 body: 72
69	IndexedVar	variable: 73 index: 89
70	Lambda	parameter: 89 body: 74
71	IndexedVar	variable: 75 index: 89
72	Operation	operator: 78 operands: 77
73	Variable
74	Operation	operator: 78 operands: 79
75	Variable
76	ExprTuple	89
77	NamedExprs	element: 80 targets: 82
78	Literal
79	NamedExprs	element: 81 targets: 82
80	Operation	operator: 83 operands: 85
81	Operation	operator: 84 operands: 85
82	Operation	operator: 86 operands: 87
83	Literal
84	Literal
85	NamedExprs	state: 88 part: 89
86	Literal
87	ExprTuple	90, 91
88	Variable
89	Variable
90	Literal
91	Variable

Expression of type Forall¶

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

Expression of type Forall ¶

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