Method and system for mapping a dataset from a Hilbert space of a given dimension to a Hilbert space of a different dimension
A computer-implemented method is disclosed for mapping a dataset from a Hilbert space of a given dimension to a Hilbert space of a different dimension, the method comprising obtaining a dataset, for each data sample of the dataset, for a plurality of episodes, generating an encoded sample; configuring an adiabatic quantum device by embedding each encoded sample into a q-body Hamiltonian H representative of an adiabatic quantum device, causing the adiabatic quantum device to evolve from an initial state to a final state; and performing a projective measurement along z axis at the final state to determine the value of each qubit; generating a corresponding binary vector representative of the given data sample in a transformed Hilbert space using the determined value of each qubit at each episode and providing a mapped dataset comprising each of the generated corresponding binary vectors.
1. A computer-implemented method for mapping a dataset from a Hilbert space of a given dimension to a Hilbert space of a different dimension, the method comprising:
(a) obtaining a dataset D comprising n data samples x i , x i ∈ p for i∈{1,2, . . . , n}, wherein p is a dimension of each data sample;
(b) for each of n data samples x i of the dataset D,
(i) for a plurality of episodes e,
(A) generating an encoded sample J i =Ax i +b, wherein A is a q×p matrix comprising elements drawn from a first distribution, wherein q is indicative of a number of qubits available in an adiabatic quantum device, and wherein b is a q-dimensional vector comprising elements drawn from a second distribution;
(B) configuring the adiabatic quantum device by embedding the encoded sample into a q-body Ising Hamiltonian H representative of the adiabatic quantum device and defined by:
H
(
t
)
x
i
e
=
a
(
t
)
H
i
+
b
(
t
)
H
f
wherein a(t) and b(t) are classical external fields driving a Hamiltonian H(t) over the time span [0,T], wherein H i is an initial Hamiltonian and H f is a final or an encoding Hamiltonian defined, respectively, by:
H
i
=
∑
v
q
σ
v
x
,
H
f
=
∑
u
q
j
u
e
σ
u
z
+
∑
l
,
m
h
l
,
m
σ
m
z
σ
l
z
wherein σ x , σ z are Pauli-X and Pauli-Z operators, respectively, and wherein h l,m is a parameter defined as a function that depends on values j u e of the encoded sample;
(C) causing the adiabatic quantum device to evolve from an initial state at t i =0 to a final state at t f =t wherein t≤T; and
(D) performing a projective measurement along a z axis at the final state to determine a value of each qubit of the number of qubits of the adiabatic quantum device;
(ii) generating a corresponding binary vector representative of each data sample x i in a transformed Hilbert space using the value of each qubit at each episode e determined in (D), wherein the corresponding binary vector corresponds to a mapped data sample; and
(c) providing a mapped dataset comprising each of the corresponding binary vectors generated in (ii).
2. The computer-implemented method as claimed in claim 1 , wherein the first distribution comprises a parametrized probability distribution; further wherein the elements of the matrix A are drawn from the first distribution using one of a digital computer and a quantum computer.
3. The computer-implemented method as claimed in claim 2 , wherein the elements of the matrix A are drawn from the first distribution using the adiabatic quantum device, further wherein the parametrized probability distribution comprises parameters which parameters are parameters of the q-body Ising Hamiltonian representative of the adiabatic quantum device.
4. The computer-implemented method as claimed in claim 2 , wherein the elements of the matrix A are drawn from the first distribution using a gate-model quantum computer, further wherein the parameterized probability distribution comprises parameters which parameters are parameters of quantum logic gates.
5. The computer-implemented method as claimed in claim 2 , wherein the parametrized probability distribution comprises parameters which parameters are adaptive variables.
6. The computer-implemented method as claimed in claim 5 , wherein the adaptive variables are drawn using a dropout technique.
7. The computer-implemented method as claimed in claim 1 , wherein (a) comprises at least one operation selected from the group consisting of:
receiving the dataset from a user interacting with a digital computer,
obtaining the dataset from a memory unit located in a digital computer, and
obtaining the dataset from a remote processing device operatively connected with a digital computer.
8. The computer-implemented method as claimed in claim 1 , wherein (B) further comprises:
computing a q 2 -body Ising Hamiltonian for q 2 qubits of the adiabatic quantum device, the q 2 -body Ising Hamiltonian comprising a randomness factor; and
generating a global Hamiltonian comprising the q-body Ising Hamiltonian, the q 2 -body Ising Hamiltonian, and interacting terms between the q 2 -body Ising Hamiltonian and the q-body Ising Hamiltonian H;
wherein (B) is performed using the global Hamiltonian.
9. The computer-implemented method as claimed in claim 8 , wherein (B) further comprises:
computing a q 3 -body Ising Hamiltonian for q 3 qubits of the adiabatic quantum device, the q 3 -body Ising Hamiltonian comprising at least two adaptive variables, and
adding the q 3 -body Ising Hamiltonian and interaction terms between the q 3 -body Ising Hamiltonian, the q 2 -body Ising Hamiltonian, and the q-body Ising Hamiltonian H to the global Hamiltonian;
wherein the at least two adaptive variables are updated based on a performance obtained using a machine learning algorithm applied on the mapped dataset generated in (c).
10. The computer-implemented method as claimed in claim 9 , wherein the q 3 -body Ising Hamiltonian is defined by H a e =Σ i α i σ i z +Σ <j,k> β j,k σ j z σ k z , wherein the at least two adaptive variable comprise α and β, and wherein <j, k > goes over pair-wise interacting qubits.
11. The computer-implemented method as claimed in claim 10 , wherein the adaptive variables are drawn using a dropout technique.
12. The computer-implemented method as claimed in claim 9 , wherein the adaptive variables are drawn using a dropout technique.
13. The computer-implemented method as claimed in claim 8 , wherein the q 2 -body Ising Hamiltonian is defined by H r e =Σ g q 2 J g σ g z +Σ <k,w> h k,w σ k z σ w z , wherein r is a randomness factor, J g and h k,w are real numbers, and <k, w> goes over pair-wise interacting qubits.
14. The computer-implemented method as claimed in claim 13 , wherein the J g and h k,w are drawn randomly from a classical probability distribution.
15. The computer-implemented method as claimed in claim 14 , wherein J g and h k,w are drawn randomly from {0,1,−1}.
16. The method of claim 8 , wherein the dataset D is used for training a machine learning model, and wherein the method further comprises:
obtaining the machine learning model; and
training the machine learning model using the mapped dataset.
17. The computer-implemented method as claimed in claim 1 , wherein is equal to an absolute mean of the values j u e of the encoded sample.
18. The method of claim 1 , wherein the dataset D is used for training a machine learning model, and wherein the method further comprises:
obtaining the machine learning model; and
training the machine learning model using the mapped dataset, thereby generating a trained machine learning model.
19. The method of claim 18 , further comprising: performing a machine learning task using the trained machine learning model.
20. A digital computer operatively connected to an adiabatic quantum device via a communication port, the digital computer comprising:
a central processing unit; and
a memory unit comprising instructions which when executed by the central processing unit are configured to implement a computer-implemented method for mapping a dataset from a Hilbert space of a given dimension to a Hilbert space of a different dimension, the computer-implemented method comprising:
(a) obtaining a dataset D comprising n data samples x i , x i ∈ p for i∈{1,2, . . . , n}, wherein p is a dimension of each data sample;
(b) for each of n data samples x i of the dataset D,
(i) for a plurality of episodes e,
(A) generating an encoded sample J i =Ax i +b, wherein A is a q×p matrix comprising elements drawn from a first distribution, wherein q is indicative of a number of qubits available in an adiabatic quantum device, and wherein b is a q-dimensional vector comprising elements drawn from a second distribution;
(B) configuring the adiabatic quantum device by embedding the encoded sample into a q-body Hamiltonian H(t) representative of an adiabatic quantum device and defined by:
H
(
t
)
x
i
e
=
a
(
t
)
H
i
+
b
(
t
)
H
f
wherein a(t) and b(t) are classical external fields driving a Hamiltonian H(t) over the time span [0,T], wherein H i is an initial Hamiltonian and H f is a final or an encoding Hamiltonian defined, respectively, by
H
i
=
∑
v
q
σ
v
x
,
H
f
=
∑
u
q
j
u
e
σ
u
z
+
∑
l
,
m
h
l
,
m
σ
m
z
σ
l
z
wherein σ x , σ z are Pauli-X and Pauli-Z operators, respectively, and wherein h l,m is a parameter defined as a function that depends on values j u e of the encoded sample;
(C) causing the adiabatic quantum device to evolve from an initial state at t i =0 to a final state at t f =t wherein t≤T; and
(D) performing a projective measurement along a z axis at the final state to determine a value of each qubit of the number of qubits of the adiabatic quantum device;
(ii) generating a corresponding binary vector representative of each data sample x i in a transformed Hilbert space using the value of each qubit at each episode e determined in (D), wherein the corresponding binary vector corresponds to a mapped data sample; and
(c) providing a mapped dataset comprising each of the corresponding binary vectors generated in (ii).