Computing using unknown values
A method of computing includes defining a first atomic random variable (ARV) and first random variable (RV) in a programming language system. The first ARV having a non-deterministic value of either zero according to a second probability or one according to a first probability. A sum of the first probability and the second probability is one. A covariance of the first ARV and a second ARV is zero. The first RV has a first indefinite value at a first definite probability and includes a polynomial of one or more atomic random variables (ARVS) that includes the first ARV. The method includes executing a computer instruction that includes a mathematical operation involving the first RV as a basic data type and produces a second RV having a second indefinite value at a second definite probability, represents a result distribution, and tracks a response to the one or more ARVS.
1. A method of quantum-inspired computing, the method comprising:
defining a first atomic random variable (ARV) in a programming language system, the first ARV having a non-deterministic value of either zero or one, a first probability of having a value of one, and a second probability of having a value of zero; a sum of the first probability and the second probability is one; and a covariance of the first ARV and a second ARV is zero;
defining a first random variable (RV) in the programming language system, the first RV having a first indefinite value at a first definite probability and includes a polynomial of one or more atomic random variables (ARVS) that includes the first ARV; and
executing a computer instruction that includes a mathematical operation involving the first RV as a basic data type, the executing producing a second RV that has a second indefinite value at a second definite probability, represents a result distribution, and tracks a response to the one or more ARVS.
2. The method of claim 1 , wherein:
a mean of the first ARV is equal to the first probability;
a mean square of the first ARV is equal to the first probability;
an n-th moment of the first ARV is equal to the first probability for all values of n;
a variance of the first ARV is equal to a product of the first probability and the second probability; and
an m-th power of the first ARV is equal to the first ARV for all values of m.
3. The method of claim 1 , wherein the first ARV is defined according to ARV expressions:
X i ∈{0,1};
0 ≤p i ≤1;
Pr[ X i =1]= p i ;
Pr[ X i =0]=1 −p i ; and
Cov( X i ,X k )=0 for i≠k ; in which:
i and k represent indexing variables;
p i represents a first probability indexed according to the indexing variable i;
X i represents an ARV indexed according to the indexing variable i;
X k represents an ARV indexed according to the indexing variable k;
Pr[ ] represents a probability function; and
Cov( ) represents a covariance function.
4. The method of claim 3 , wherein entropy of the ARV is approximated according to an approximate entropy expression:
S [ X i ]≃ log 2−2 q i 2 ; in which:
S[X i ] represents the entropy,
q i represents p i −½, and
log 2 represents a natural log of 2 base Napier's constant (e).
5. The method of claim 3 , wherein:
the first random variable is a quadratic random variable (QRV); and
the QRV is defined according to the QRV expressions:
Y
A
≡
X
→
T
A
X
→
;
X
→
≡
[
X
0
X
1
X
2
⋯
X
m
]
T
;
A
≡
[
a
0
,
0
a
0
,
1
a
0
,
2
…
a
0
,
m
a
1
,
0
a
1
,
1
a
1
,
2
…
a
1
,
m
a
2
,
0
a
2
,
1
a
2
,
2
…
a
2
,
m
⋮
⋮
⋮
⋱
⋮
a
m
,
0
a
m
,
1
a
m
,
2
…
a
m
,
m
]
;
a
k
,
l
∈
ℝ
(
or
ℂ
)
;
m
∈
ℕ
;
and
p
0
=
1
;
in which:
l represents an indexing variable;
Y A represents the QRV;
X k represents an ARV indexed according to the indexing variable k;
a k,l represents a deterministic real or complex value that represents a response to the ARVS indexed according to the indexing variables k and l;
represents the set of real numbers;
represents the set of complex numbers;
represents the set of natural numbers; and
X 0 a 0,0 X 0 is reserved for a constant term with p 0 =1 such that X 0 is equal to 1.
6. The method of claim 3 , wherein:
the first random variable is a high-order random variable (HRV); and
the HRV is defined according to HRV expressions:
Y
A
≡
∑
k
1
=
0
m
∑
k
2
=
0
m
…
∑
k
d
=
0
m
a
k
1
,
k
2
,
⋯
,
k
d
X
k
1
X
k
2
…
X
k
d
;
a
k
1
,
k
2
,
⋯
,
k
d
∈
ℝ
(
or
ℂ
)
;
m
∈
ℕ
;
d
≤
m
;
and
p
0
=
1
;
in which:
Y A represents the HRV;
k and d represent indexing variables;
X k i represents an ARV indexed according to the indexing variable k i ;
a k 1 , k 2 , . . . , k d represents a deterministic real or complex value that represents a response to the ARVS indexed according to the indexing variables k 1 , k 2 , . . . , k d ;
represents the set of real numbers;
represents the set of complex numbers;
represents the set of natural numbers; and
a 0, 0, . . . , 0 X 0 X 0 . . . X 0 is reserved for the constant term with p 0 =1 such that X 0 is equal to 1.
7. The method of claim 1 , wherein:
the mathematical operation includes an RV addition operation of the first RV with a third RV; and
the RV addition operation is defined according to RV addition operation expressions:
+: Y A +Y B →Y C ; such that:
Y
A
=
a
0
+
∑
k
1
=
1
m
a
k
1
X
k
1
+
∑
k
1
=
1
m
-
1
∑
k
2
=
k
1
+
1
m
a
k
1
,
k
2
X
k
1
X
k
2
+
…
+
∑
k
1
=
1
m
-
d
A
+
1
∑
k
2
=
k
1
+
1
m
-
d
A
+
2
∑
k
3
=
k
2
+
1
m
-
d
A
+
3
⋯
∑
k
d
A
=
k
d
A
-
1
+
1
m
a
k
1
,
k
2
⋯
,
k
d
A
X
k
1
X
k
2
…
X
k
d
A
;
Y
B
=
b
0
+
∑
k
1
=
1
m
b
k
1
X
k
1
+
∑
k
1
=
1
m
-
1
∑
k
2
=
k
1
+
1
m
b
k
1
,
k
2
X
k
1
X
k
2
+
…
+
∑
k
1
=
1
m
-
d
B
+
1
∑
k
2
=
k
1
+
1
m
-
d
B
+
2
∑
k
3
=
k
2
+
1
m
-
d
B
+
3
…
∑
k
d
B
=
k
d
B
-
1
+
1
m
b
k
1
,
k
2
⋯
,
k
d
B
X
k
1
X
k
2
…
X
k
d
B
;
Y
C
≡
c
0
+
∑
k
1
=
1
m
c
k
1
X
k
1
+
∑
k
1
=
1
m
-
1
∑
k
2
=
k
1
+
1
m
c
k
1
,
k
2
X
k
1
X
k
2
+
…
+
∑
k
1
=
1
m
-
d
C
+
1
∑
k
2
=
k
1
+
1
m
-
d
C
+
2
∑
k
3
=
k
2
+
1
m
-
d
C
+
3
…
∑
k
d
C
=
k
d
C
-
1
+
1
m
C
k
1
,
k
2
⋯
,
k
d
C
X
k
1
X
k
2
…
X
k
d
C
;
d
A
,
d
B
≤
m
;
d
C
≡
max
(
d
A
,
d
B
)
;
a
k
1
,
k
2
,
⋯
k
d
A
∈
ℝ
(
or
ℂ
)
;
b
k
1
,
k
2
,
⋯
k
d
B
∈
ℝ
(
or
ℂ
)
;
m
∈
ℕ
;
c
o
≡
a
0
+
b
0
;
and
c
k
1
,
k
2
⋯
,
k
l
≡
{
a
k
1
,
k
2
⋯
,
k
l
+
b
k
1
,
k
2
⋯
,
k
l
for
(
1
≤
l
≤
d
A
)
⩓
(
1
≤
l
≤
d
B
)
a
k
1
,
k
2
⋯
,
k
l
for
(
d
B
<
l
≤
d
A
)
b
k
1
,
k
2
⋯
,
k
l
for
(
d
A
<
l
≤
d
B
)
;
in which:
Y A , Y B , and Y C represents the first RV, the third RV, and the second RV, respectively;
the Y A , Y B , and Y C are high-order random variable (HRV);
k i represents an indexing variable;
X k i represents an ARV indexed according to the indexing variable k i ;
a k 1 , k 2 , . . . , k l , b k 1 , k 2 , . . . , k l , and c k 1 , k 2 , . . . , k l represent deterministic real or complex values that represents a response to the ARVS indexed according to the indexing variables k 1 . . . k l ;
represents the set of real numbers;
represents the set of complex numbers; and
represents the set of natural numbers.
8. The method of claim 1 , wherein:
the mathematical operation includes an RV multiplication operation of the first RV with a third RV; and
the RV multiplication operation is defined according to RV multiplication operation expressions:
×: Y A +Y B →Y C ; such that:
Y
A
=
∑
k
1
=
0
m
…
∑
k
α
=
0
m
a
k
1
,
⋯
k
α
X
k
1
…
X
k
α
;
Y
B
=
∑
l
1
=
0
m
…
∑
l
β
=
0
m
b
l
1
,
⋯
l
β
X
l
1
…
X
l
β
;
Y
C
≡
∑
k
1
=
0
m
…
∑
k
α
=
0
m
∑
l
1
=
0
m
…
∑
l
β
=
0
m
c
k
1
,
⋯
k
α
,
l
1
,
⋯
,
l
α
X
k
1
…
X
k
α
X
l
1
…
X
l
β
;
c
k
1
,
⋯
,
k
α
,
l
1
,
⋯
l
α
≡
a
k
1
,
⋯
k
α
b
k
1
,
⋯
l
β
;
and
m
∈
ℕ
;
in which:
Y A , Y B , and Y C represents the first RV, the third RV, and the second RV, respectively;
the Y A , Y B , and Y C are high-order random variables (HRV);
the Y A is an αth-order HRV;
the Y B is an βth-order HRV;
the Y C is an (α-β)th-order HRV;
k 1 . . . k α and l 1 . . . l β represent indexing variables;
X k i represents an ARV indexed according to the indexing variables k i ;
X l i represents an ARV indexed according to the indexing variables l i ;
a k 1 , . . . , k α , b l 1 , . . . , l β , and c k 1 , . . . , k α , l 1 , . . . , l β represent deterministic real or complex values that represent a response to the ARV indexed according to the indexing variables k 1 . . . k α and l 1 . . . l β ;
represents the set of real numbers;
represents the set of complex numbers; and
represents the set of natural numbers.
9. The method of claim 1 , wherein:
the mathematical operation includes an RV derivative operation of the first RV; and
the RV derivative operation is defined according to RV derivative operation expressions:
d
dS
x
:
d
dS
x
Y
A
→
Y
B
;
such that:
Y B ≡( Y A |X i =1 for ∀ i∈S X )−( Y A |X i =0 for ∀ i∈S X ); in which:
Y A and Y B represents the first RV and the second RV, respectively;
∀ represents a “for all” operator; and
| represents a “such that” operator.
10. The method of claim 1 , wherein the computer instruction is included in an application implemented as:
a quantum simulation for a drug design application;
a quantum simulation for a material design application;
a combinatory optimization for a place and a route in computer aided design (CAD);
a statistical eye simulation of signal integrity analysis for high-speed signal transmission, or
a statistical static timing analysis for CAD which explores the solution space.
11. A non-transitory computer-readable medium having encoded therein programming code executable by one or more processors to perform operations comprising:
defining a first atomic random variable (ARV) in a programming language system, the first ARV having a non-deterministic value of either zero or one, a first probability of having a value of one, and a second probability of having a value of zero; a sum of the first probability and the second probability is one; and a covariance of the first ARV and a second ARV is zero;
defining a first random variable (RV) in the programming language system, the first RV having a first indefinite value at a first definite probability and includes a polynomial of one or more atomic random variables (ARVS) that includes the first ARV; and
executing a computer instruction that includes a mathematical operation involving the first RV as a basic data type, the executing producing a second RV that has a second indefinite value at a second definite probability, represents a result distribution, and tracks a response to the one or more ARVS.
12. The non-transitory computer-readable medium of claim 11 , wherein:
a mean of the first ARV is equal to the first probability;
a mean square of the first ARV is equal to the first probability;
an n-th moment of the first ARV is equal to the first probability for all values of n;
a variance of the first ARV is equal to a product of the first probability and the second probability; and
an m-th power of the first ARV is equal to the first ARV for all values of m.
13. The non-transitory computer-readable medium of claim 11 , wherein the first ARV is defined according to ARV expressions:
X i ∈{0,1};
0 ≤p i ≤1;
Pr[ X i =1]= p i ;
Pr[ X i =0]=1 −p i ; and
Cov( X i ,X k )=0 for i≠k ; in which:
i and k represent indexing variables;
p i represents a first probability indexed according to the indexing variable i;
X i represents an ARV indexed according to the indexing variable i;
X k represents an ARV indexed according to the indexing variable k;
Pr[ ] represents a probability function; and
Cov( ) represents a covariance function.
14. The non-transitory computer-readable medium of claim 13 , wherein entropy of the ARV is approximated according to an approximate entropy expression:
S [ X i ]≃ log 2−2 q i 2 ; in which:
S[X i ] represents the entropy,
q i represents p i −½, and
log 2 represents a natural log of 2 base Napier's constant (e).
15. The non-transitory computer-readable medium of claim 13 , wherein:
the first random variable is a quadratic random variable (QRV); and
the QRV is defined according to the QRV expressions:
Y
A
≡
X
→
T
A
X
→
;
X
→
≡
[
X
0
X
1
X
2
⋯
X
m
]
T
;
A
≡
[
a
0
,
0
a
0
,
1
a
0
,
2
…
a
0
,
m
a
1
,
0
a
1
,
1
a
1
,
2
…
a
1
,
m
a
2
,
0
a
2
,
1
a
2
,
2
…
a
2
,
m
⋮
⋮
⋮
⋱
⋮
a
m
,
0
a
m
,
1
a
m
,
2
…
a
m
,
m
]
;
a
k
,
l
∈
ℝ
(
or
ℂ
)
;
m
∈
ℕ
;
and
p
0
=
1
;
in which:
l represents an indexing variable;
Y A represents the QRV;
X k represents an ARV indexed according to the indexing variable k;
a k,l represents a deterministic real or complex value that represents a response to the ARVS indexed according to the indexing variables k;
represents the set of real numbers;
represents the set of complex numbers;
represents the set of natural numbers; and
X 0 a 0,0 X 0 is reserved for a constant term with p 0 =1 such that X 0 is equal to 1.
16. The non-transitory computer-readable medium of claim 13 , wherein:
the first random variable is a high-order random variable (HRV); and
the HRV is defined according to HRV expressions:
Y
A
≡
∑
k
1
=
0
m
∑
k
2
=
0
m
…
∑
k
d
=
0
m
a
k
1
,
k
2
,
⋯
,
k
d
X
k
1
X
k
2
…
X
k
d
;
a
k
1
,
k
2
,
⋯
k
d
∈
ℝ
(
or
ℂ
)
;
m
∈
ℕ
;
d
≤
m
;
and
p
0
=
1
;
in which:
Y A represents the HRV;
k and d represent indexing variables;
X k i represents an ARV indexed according to the indexing variable k i ;
a k 1 , k 2 , . . . , k d represents a deterministic real or complex value that represents a response to the ARVS indexed according to the indexing variables k 1 , k 2 , . . . , k d ;
represents the set of real numbers;
represents the set of complex numbers;
represents the set of natural numbers; and
a 0, 0, . . . , 0 X 0 X 0 . . . X 0 is reserved for the constant term with p 0 =1 such that X 0 is equal to 1.
17. The non-transitory computer-readable medium of claim 11 , wherein:
the mathematical operation includes an RV addition operation of the first RV with a third RV; and
the RV addition operation is defined according to RV addition operation expressions:
+: Y A +Y B →Y C ; such that:
Y
A
=
a
0
+
∑
k
1
=
1
m
a
k
1
X
k
1
+
∑
k
1
=
1
m
-
1
∑
k
2
=
k
1
+
1
m
a
k
1
,
k
2
X
k
1
X
k
2
+
…
+
∑
k
1
=
1
m
-
d
A
+
1
∑
k
2
=
k
1
+
1
m
-
d
A
+
2
∑
k
3
=
k
2
+
1
m
-
d
A
+
3
…
∑
k
d
A
=
k
d
A
-
1
+
1
m
a
k
1
,
k
2
⋯
,
k
d
A
X
k
1
X
k
2
…
X
k
d
A
;
Y
B
=
b
0
+
∑
k
1
=
1
m
b
k
1
X
k
1
+
∑
k
1
=
1
m
-
1
∑
k
2
=
k
1
+
1
m
b
k
1
,
k
2
X
k
1
X
k
2
+
…
+
∑
k
1
=
1
m
-
d
B
+
1
∑
k
2
=
k
1
+
1
m
-
d
B
+
2
∑
k
3
=
k
2
+
1
m
-
d
B
+
3
…
∑
k
d
B
=
k
d
B
-
1
+
1
m
b
k
1
,
k
2
⋯
,
k
d
B
X
k
1
X
k
2
…
X
k
d
B
;
Y
C
≡
c
0
+
∑
k
1
=
1
m
c
k
1
X
k
1
+
∑
k
1
=
1
m
-
1
∑
k
2
=
k
1
+
1
m
c
k
1
,
k
2
X
k
1
X
k
2
+
…
+
∑
k
1
=
1
m
-
d
C
+
1
∑
k
2
=
k
1
+
1
m
-
d
C
+
2
∑
k
3
=
k
2
+
1
m
-
d
C
+
3
…
∑
k
d
C
=
k
d
C
-
1
+
1
m
C
k
1
,
k
2
⋯
,
k
d
C
X
k
1
X
k
2
…
X
k
d
C
;
d
A
,
d
B
≤
m
;
d
C
≡
max
(
d
A
,
d
B
)
;
a
k
1
,
k
2
,
⋯
k
d
A
∈
ℝ
(
or
ℂ
)
;
b
k
1
,
k
2
,
⋯
k
d
B
∈
ℝ
(
or
ℂ
)
;
m
∈
ℕ
;
c
o
≡
a
0
+
b
0
;
and
c
k
1
,
k
2
⋯
,
k
l
≡
{
a
k
1
,
k
2
⋯
,
k
l
+
b
k
1
,
k
2
⋯
,
k
l
for
(
1
≤
l
≤
d
A
)
⩓
(
1
≤
l
≤
d
B
)
a
k
1
,
k
2
⋯
,
k
l
for
(
d
B
<
l
≤
d
A
)
b
k
1
,
k
2
⋯
,
k
l
for
(
d
A
<
l
≤
d
B
)
;
in which:
Y A , Y B , and Y C represents the first RV, the third RV, and the second RV, respectively;
the Y A , Y B , and Y C are high-order random variable (HRV);
k i represents an indexing variable;
X k i represents an ARV indexed according to the indexing variable k i ;
a k 1 , k 2 , . . . , k l , b k 1 , k 2 , . . . , k l , and c k 1 , k 2 , . . . , k l represent deterministic real or complex values that represents a response to the ARVS indexed according to the indexing variables k 1 . . . k l ;
represents the set of real numbers;
represents the set of complex numbers; and
represents the set of natural numbers.
18. The non-transitory computer-readable medium of claim 11 , wherein:
the mathematical operation includes an RV multiplication operation of the first RV with a third RV; and
the RV addition operation is defined according to RV multiplication operation expressions:
×: Y A +Y B →Y C ; such that:
Y
A
=
∑
k
1
=
0
m
…
∑
k
α
=
0
m
a
k
1
,
⋯
k
α
X
k
1
…
X
k
α
;
Y
B
=
∑
l
1
=
0
m
…
∑
l
β
=
0
m
b
l
1
,
⋯
l
β
X
l
1
…
X
l
β
;
Y
C
≡
∑
k
1
=
0
m
…
∑
k
α
=
0
m
∑
l
1
=
0
m
…
∑
l
β
=
0
m
c
k
1
,
⋯
k
α
,
l
1
,
⋯
,
l
α
X
k
1
…
X
k
α
X
l
1
…X
l
β
;
c
k
1
,
⋯
,
k
α
,
l
1
,
⋯
l
α
≡
a
k
1
,
⋯
k
α
b
l
1
,
⋯
l
β
;
and
m
∈
ℕ
;
in which:
Y A , Y B , and Y C represents the first RV, the third RV, and the second RV, respectively;
the Y A , Y B , and Y C are high-order random variables (HRV);
the Y A is an αth-order HRV;
the Y B is an βth-order HRV;
the Y C is an (α-β)th-order HRV;
k 1 . . . k α and l 1 . . . l β represent indexing variables;
X k i represents an ARV indexed according to the indexing variable k i ;
X l i represents an ARV indexed according to the indexing variable l i ;
a k 1 , . . . , k α , b l 1 , . . . , l β , and c k 1 , . . . , k α , l 1 , . . . , l β represent deterministic real or complex values that represent a response to the ARV indexed according to the indexing variables k 1 . . . k α and l 1 . . . l β ;
represents the set of real numbers;
represents the set of complex numbers; and
represents the set of natural numbers.
19. The non-transitory computer-readable medium of claim 11 , wherein:
the mathematical operation includes an RV derivative operation of the first RV; and
the RV derivative operation is defined according to RV derivative operation expressions:
d
dS
x
:
d
dS
x
Y
A
→
Y
B
;
such that:
Y B ≡( Y A |X i =1 for ∀ i∈S X )−( Y A |X i =0 for ∀ i∈S X ); in which:
Y A and Y B represents the first RV and the second RV, respectively;
∀ represents a “for all” operator; and
| represents a “such that” operator.
20. The non-transitory computer-readable medium of claim 11 , wherein the computer instruction is included in an application implemented as:
a quantum simulation for a drug design application;
a quantum simulation for a material design application;
a combinatory optimization for a place and a route in computer aided design (CAD);
a statistical eye simulation of signal integrity analysis for high-speed signal transmission, or
a statistical static timing analysis for CAD which explores the solution space.