SYSTEM AND METHOD OF BOSONIC QUBITS SIMULATION
A method for simulating a bosonic quantum bit (qubit) on a classical computer are described. The method determines a phase space representation of the qubit in the form of a linear combination of Gaussian functions, each of which is characterized by a mean, a covariance matrix, and a weight coefficient determined from user defined energy parameter and qubit class of the qubit. The qubit may be simulated on a classical computer by applying transformations of quantum logic gates and measurements to update the weight coefficient, mean, and covariance matrix of each of the Gaussian functions.
1 . A method for simulating a quantum bit (qubit) on a classical computer, the method comprising:
obtaining, by the classical computer, an energy parameter and a qubit class of the qubit to be simulated;
determining, by the classical computer from the energy parameter and the qubit class, a mean, a covariance matrix, and a weight coefficient for each of at least one Gaussian function in phase space, wherein a linear combination of the at least one Gaussian function is a phase space representation of the qubit to be simulated;
simulating the linear combination of the at least one Gaussian function by applying transformations of quantum logic gates and measurements to update the weight coefficient, mean, and covariance matrix of each of the at least one Gaussian function; and
storing the updated weight coefficient, mean, and covariance matrix of each of the at least one Gaussian function on the classical computer.
2 . The method of claim 1 , wherein the qubit is a bosonic qubit.
3 . The method of claim 2 , wherein the qubit class includes Gottesman-Kitaev-Preskill (GKP) state, cat state, and Fock state.
4 . The method of claim 1 , further comprises constructing a Wigner function of the qubit from the updated weight coefficient, mean, and covariance matrix of each of the linear combination of the Gaussian functions.
5 . The method of claim 1 , further comprises sampling outcomes of the linear combination of the at least one Gaussian function.
6 . The method of claim 1 , wherein the linear combination of Gaussian functions is:
W
(
ξ
;
ρ
ˆ
)
=
c
m
G
μ
m
∑
m
(
ξ
)
wherein W is a Wigner function of the of the qubit, is the set of indices enumerating the Gaussian functions, ζ∈ 2n is a phase space variable for an n-mode continuous variable (CV) quantum system, {circumflex over (ρ)} is a density matrix operator in Hilbert space that is representative of the energy parameter and the qubit class of the qubit, c m is the weight coefficient, μ m is the mean, and Σ m is the covariance matrix, and G is a normalized multivariate Gaussian distribution function.
7 . The method of claim 1 , wherein the qubit class is a GKP state, the obtaining further comprises receiving a GKP state representation.
8 . The method of claim 7 , wherein the GKP state representation is a real-valued GKP state, the weight coefficient, mean, and covariance matrix of each of the at least one Gaussian function are determined by:
c
m
(
ϵ
;
θ
,
ϕ
)
=
exp
[
-
1
-
e
-
2
ϵ
ℏ
(
1
+
e
-
2
ϵ
)
μ
m
T
μ
m
]
;
μ
m
(
ϵ
)
=
2
e
-
ϵ
1
+
e
-
2
ϵ
)
μ
m
;
and
Σ
m
(
ϵ
)
=
ℏ
2
1
-
e
-
2
ϵ
(
1
+
e
-
2
ϵ
)
𝟙
;
Where the energy parameter is specified by ∈, θ and ϕ are the polar and azimuthal angles of the qubit in the Bloch sphere representation, is an overall normalization constant defined to satisfy a condition of as Σ m∈M c m =1, n is Planck's constant, and is an identity matrix.
9 . The method of claim 7 , wherein the GKP representation is a complex-valued GKP state, the weight coefficient, mean, and covariant matrix of each of the at least one Gaussian function are determined by:
c
m
=
exp
{
-
α
π
2
[
(
s
+
2
l
)
2
+
(
t
+
2
k
)
2
]
}
×
[
β
2
π
(
t
+
s
+
2
l
+
s
k
)
2
4
α
]
;
μ
m
(
ϵ
)
=
-
β
πℏ
2
(
(
t
+
s
+
2
l
+
2
k
)
α
i
(
s
-
t
+
2
l
-
2
k
)
)
;
and
Σ
m
(
ϵ
)
=
ℏ
2
(
1
α
0
0
α
)
,
(
α
,
β
)
=
(
coth
(
ϵ
)
,
-
csch
(
ϵ
)
)
;
Where the energy parameter is specified by ∈, ={m≡(k, l, s, t)|s,t ∈ {0,1}&k,l ∈ }, denotes set of all integers, α s is and α t are derived from |ψ(α) =α 0 |0 gkp +α 1 |1 gkp , (α,∈) is an overall normalization constant defined to satisfy the condition of c m =1, n is Planck's constant.
10 . The method of claim 7 , wherein the GKP representation is a squeezed comb state, the weight coefficient, mean, and covariant matrix of each of the at least one Gaussian function are determined by:
=
{
m
≡
(
k
,
l
)
❘
k
,
l
∈
1
…
,
d
}
,
μ
m
=
1
2
(
q
_
k
+
q
_
l
,
i
ϵ
2
r
[
q
_
k
-
q
_
l
]
)
,
∑
m
=
ℏ
2
[
e
-
2
r
0
0
e
2
r
]
e
m
=
exp
(
-
1
4
ℏ
e
2
r
(
q
_
k
-
q
_
l
)
2
)
.
Where is a normalization constant, q n are the locations of the peaks in the position quadrature where
q
_
n
=
-
(
N
+
1
)
d
2
+
nd
,
and N is the number of peaks in the comb.
11 . The method of claim 1 , wherein the qubit class is a cat state, the weight coefficient, mean, and covariant matrix of each of the at least one Gaussian function are determined by:
c
±
=
,
c
z
=
(
c
z
_
)
⋆
=
e
-
i
π
k
-
2
❘
"\[LeftBracketingBar]"
α
❘
"\[RightBracketingBar]"
2
;
μ
±
=
±
2
ℏ
(
(
α
)
,
(
α
)
)
,
μ
z
=
(
μ
z
_
)
=
2
ℏ
(
i
(
α
)
,
-
i
(
α
)
,
)
;
and
∑
vac
=
ℏ
2
.
Wherein the energy parameter is specified by α, +, −, , z are indices for the Gaussian functions in phase space corresponding to the four terms from the cat state density matrix such that ={+, −, , z }, is an overall normalization constant defined to satisfy the condition of (∈; θ,ϕ)=1, n is Planck's constant, and is an identity matrix.
12 . The method of claim 1 , wherein the qubit class is a Fock state, the weight coefficient, mean, and covariant matrix of each of the at least one Gaussian function are determined by:
c
m
=
(
n
m
)
[
1
-
nr
2
1
-
(
n
-
m
)
r
2
]
;
μ
m
=
0
;
and
Σ
m
=
ℏ
2
1
+
(
n
-
m
)
r
2
1
-
(
n
-
m
)
r
2
;
Wherein the energy parameter is specified by n, ={0, . . . , n}, r is a parameter quantifying the quality of the approximation, is the index fro reach Gaussian function, is an overall normalization constant defined as
=
n
!
(
n
(
-
r
2
-
1
r
2
)
!
+
(
-
1
r
2
)
!
)
(
n
r
2
-
1
r
2
)
!
,
n is Planck's constant, and is an identity matrix.
13 . A method of simulating a multi-mode quantum state on a classical computer, the method comprising:
obtaining, by the classical computer, an energy parameter and a qubit class of each mode of the multi-mode quantum state to be simulated;
initializing, by the classical computer from the energy parameter and the qubit class, a mean, covariance matrix, and a weight coefficient of at least one first Gaussian function of each mode in phase space, wherein a linear combination of the at least one first Gaussian function is a phase space representation of a mode of the multi-mode quantum state;
combining the weight coefficient, mean, and covariance matrix of the at least one first Gaussian function of each mode into weight coefficients, means, and covariance matrices of at least one second Gaussian function, wherein a linear combination of the at least one second Gaussian function is a phase space representation of the multi-mode quantum system;
simulating the multi-mode quantum system by updating the weight coefficients, means, and covariance matrices of the at least one second Gaussian function by applying transformations of quantum logic gates and measurements expressed in phase space; and
storing the updated weight coefficients, means, and covariance matrices of the at least one second Gaussian function of the multi-mode quantum system on the classical computer.
14 . The method of claim 13 , wherein the combining further comprises recursively combining the weight coefficient, mean, and covariance matrix of the at least one first Gaussian function into the weight coefficients, means, and covariance matrices of the at least one second Gaussian function of the multi-mode quantum system.
15 . The method of claim 13 , when the qubit class is GKP state or cat state, wherein the initializing further comprises initializing one covariance matrix for the at least one first Gaussian function.
16 . The method of claim 13 , wherein the combining further comprises:
combining the weight coefficients as a product of two weight coefficients;
combining the means as a direct sum of two means; and
combining the covariance matrix as a direct sum of two covariance matrices.
17 . The method of claim 13 , where in the linear combination of the at least one first Gaussian function and the linear combination of the at least one second Gaussian function are of the form:
W
(
ξ
;
ρ
ˆ
)
=
c
m
G
μ
m
∑
m
(
ξ
)
wherein W is a Wigner function of the of the qubit, is the set of indices enumerating the Gaussian functions, ζ∈ 2n is a phase space variable for an n-mode continuous variable (CV) quantum system, {circumflex over (ρ)} is a density matrix operator in Hilbert space that is representative of the energy parameter and the qubit class of the qubit, c m is the weight coefficient, μ m is the mean, and Σ m is the covariance matrix, and G is a normalized multivariate Gaussian distribution function.
18 . A system for simulating a quantum bit (qubit) on a classical computer, the system comprises:
a Gaussian function constructor configured to determine, using inputs of an energy parameter and a qubit class, a mean, a covariance matrix, and a weight coefficient for each of at least one Gaussian function in phase space, wherein a linear combination of the one Gaussian function is the phase space representation of the qubit to be simulated; and
a Gaussian function transformer configured to simulate the qubit by applying transformations of quantum logic gates and measurements to the linear combination of the at least one Gaussian function on the classical computer, thereby updating the weight coefficient, mean, and covariance matrix of each of the at least one Gaussian function.
19 . The system of claim 18 , wherein the Gaussian function constructor is further configured to determine the linear combination of the at least one Gaussian function in phase space are of the form:
W
(
ξ
;
ρ
ˆ
)
=
c
m
G
μ
m
∑
m
(
ξ
)
wherein W is a Wigner function of the of the qubit, is the set of indices enumerating the Gaussian functions, ζ∈ 2n is a phase space variable for an n-mode continuous variable (CV) quantum system, {circumflex over (ρ)} is a density matrix operator in Hilbert space that is representative of the energy parameter and the qubit class of the qubit, c m is the weight coefficient, μ m is the mean, and Σ m is the covariance matrix, and G is a normalized multivariate Gaussian distribution function.
20 . A non-transitory machine-readable medium having tangibly stored thereon executable instructions for execution by a processor of a classical computer, wherein the executable instructions, when executed by the processor, cause the classical computer to:
obtain, by the classical computer, an energy parameter and a qubit class of the qubit to be simulated;
determine, by the classical computer from the energy parameter and the qubit class, a mean, a covariance matrix, and a weight coefficient for each of at least one Gaussian function in phase space, wherein a linear combination of the at least one Gaussian function is a phase space representation of the qubit to be simulated;
simulate the linear combination of the at least one Gaussian function by applying transformations of quantum logic gates and measurements to update the weight coefficient, mean, and covariance matrix of each of the at least one Gaussian function; and
store the updated weight coefficient, mean, and covariance matrix of each of the at least one Gaussian function on the classical computer.