1 Introduction
Quantum computers can make possible quantum computations for efficient problem solving [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Gatebased quantum computations represent a way to construct gatemodel quantum computers. In a gatemodel quantum computer architecture, computations are implemented via sequences of unitary operations [12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Gatemodel quantum computers allow establishing experimental quantum computations in nearterm architectures [1, 2, 3, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. Practical demonstrations of gatemodel quantum computers have been already proposed [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and several physicallayer developments are currently in progress.
Finding a stable quantum state of a quantum computer is a challenge, since it requires precise unitaries that yield stable quantum evolutions in the quantum computer. The problem is further increased if the stable system state must be available for a predetermined time or for a predetermined number of running sequences. Particularly, the quantum state of a quantum computer subject to stabilization also coincides with the optimal quantum state. The optimal quantum state of a quantum computer maximizes a particular objective function of an arbitrary computational problem fed into the quantum computer. The problem therefore is to fix the quantum state of the quantum computer in the optimal state for an arbitrary number of running sequences that is determined by the actual environment or by the current problem. Another challenge connected to the problem of stabilization of the system state of a quantum computer is the classification of the sequences of the stabilized quantum states into stabilityclasses. Practically, a solution to these problems can be covered by an unsupervised learning method.
Here, we propose a method for the stabilization of an optimal quantum state of a quantum computer through an arbitrary number of running sequences. We define a solution that utilizes unsupervised learning algorithms to determine the stable quantum states of the quantum computer and to classify the stable quantum states into stability classes. The proposed results are useful for experimental gatebased quantum computations and nearterm quantum computer architectures.
The novel contributions of our manuscript are as follows:

We propose a method for the stabilization of an optimal quantum state of a quantum computer through an arbitrary number of running sequences.

We define a solution that utilizes unsupervised learning algorithms to determine the stable quantum states of the quantum computer.

We evaluate a solution to classify the stable system states into stability classes.
This paper is organized as follows. Section 2 provides the problem statement. Section 3 discusses the stabilization procedure of an optimal quantum state of a quantum computer. Section 4 defines an unsupervised learning method to find the stable quantum states and the stability classes of the stabilized quantum states. In Section 5, a numerical evaluation is proposed. Finally, Section 6 concludes with the results. Supplemental information is included in the Appendix.
2 Problem Statement
Let be the quantum gate structure of a gatemodel quantum computer with a sequence of unitaries [12, 13, 14, 15] with an length input system ,
(1) 
where is the dimension (
=2 for a qubit system),
, and let(2) 
be the optimal system state of the quantum computer that maximizes a particular objective function ,
(3) 
of an arbitrary problem fed into the quantum computer, where is the classical value of the objective function, while
is the gate parameter vector,
(4) 
that identifies the unitaries, , of the quantum circuit of the quantum computer in the optimal state , such that an th unitary, is as [13]
(5) 
where is the gate parameter (real continuous variable) of unitary ,
is a generalized Pauli operator formulated by the tensor product of Pauli operators
[13, 14].The aim is to stabilize the optimal state of the quantum computer through running sequences via unsupervised learning of the evolution of the unitaries in the quantum computer.
The running sequences refers to input systems fed into the input of the quantum computer, such that in an th running sequence, , an th input system, (defined as in (1)), is evolved via the sequence of the uniaries of the quatum computer. The running sequences identify an input system, , formulated via , length quantum systems, as
(6) 
where it is considered that the input systems are unentangled.
Let be the gate parameter vector associated with the stable system state ,
(7) 
as
(8) 
where is the gate parameter of unitary in the stabilized system state , such that the objective function value is stabilized into
(9) 
For the sequences of the quantum computer, we define matrices and as
(10) 
where identifies the quantum state of an th running sequence of the quantum computer, while
(11) 
where , identifies the stabilized quantum state of an th sequence of the quantum computer.
The problem therefore is to find from that stabilizes the optimal state of the quantum computer through sequences as
(12) 
where is a stabilizer matrix,
(13) 
and
is the identity matrix.
The problems to be solved are therefore summarized as follows.
Problem 1
Problem 2
Describe the stability of via unsupervised learning of the stability levels of the quantum states of .
The resolutions of Problems 1 and 2 are proposed in Theorems 1 and 2. The solution framework is defined via a stabilization procedure with an embedded stabilization algorithm (see Theorem 1), and via an classification algorithm that characterizes the stability class of the results of (see Theorem 2). Fig. 1 depicts the system model.
3 Stabilization of the Optimal State of the Quantum Computer
Theorem 1
The matrix for the stabilization of the optimal state of the quantum computer via can be determined via the minimization of an objective function .
Proof. For an th sequence of the quantum computer, define and as
(14) 
and
(15) 
respectively. These vectors formulate and as
(16) 
and
(17) 
respectively. Then, using equations (16) and (17) for the sequences, let be a sum defined as
(18) 
where is the squared norm, is the trace operator, is as given in equation (16), and is as in equation (17).
For the th and th sequences, , with and , let be defined as
(19) 
where is a weight coefficient defined as
(20) 
where and are nonzero parameters.
A sum is defined for the sequences of the quantum computer as
(21) 
At a particular in equations (18) and (21), the stabilization of the optimal state of the quantum computer through the sequences can be reformulated via an objective function , subject to a minimization as
(22) 
where is a regularization constant [31, 32]. The objective function therefore stabilizes the optimal state via the minimization of , while the term achieves stabilization between the sequences.
Then, let be the weight matrix formulated via the coefficients (20) with , and let be a diagonal matrix of the weight coefficients (20) with
(23) 
such that
(24) 
Using and , the objective function in equation (22) can be rewritten as
(25) 
where is as
(26) 
and
(27) 
At a particular (16) and (26), the stabilizer matrix in equation (25) is evaluated via
(28) 
where
is a diagonal matrix of eigenvalues
[31, 32].Algorithm A.1 () gives the method for stabilizing the optimal state of the quantum computer.
4 Learning the Stable Quantum State and Stability Class
Lemma 1
The stabilized sequences of the quantum computer can be determined via unsupervised learning.
Proof. Algorithm 1 with the objective function (25) can be used to formulate an unsupervised learning framework to find the stabilized unitaries. The steps are detailed in Procedure 1 ().
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
(35) 
(36) 
4.1 Learning the Sequence Stability of Stabilized Quantum States
Proposition 1
The stability of a given sequence can be characterized via stability levels. The sequence can be classified into stability classes from set ,
(37) 
where , , is the th stability class.
Theorem 2
The , stability class of a stabilized sequence, , of the quantum computer can be learned via quantities in the highdimensional Hilbert space , where and
is a probability.
Proof. Since the gate parameters are stabilized, the gate parameters and of the th and th sequences must be correlated in the stable system state (7) of the quantum computer.
Let from equation (11) be the stabilized sequences, where is the stabilized gate parameter vector of an th sequence of the quantum computer, and let be the set of all sequences of gate parameters as
(38) 
For an th stabilization class , a probabilistic classifier function [31, 37] can be defined as
(39) 
The goal is to learn a function that maps any sequence to the correct stability class. Applying equation (39) on a given sequence , i.e., therefore maps to a given stability class via the classification of each gate parameter of the sequence.
Thus, an th stabilized gate parameter of an th sequence can be also classified into a particular stabilization class from (37). The , , classifier (39) is trained to classify [37] each of the gate parameters of , via outputting a corresponding probability that belongs to a given class. For a particular , the sum of the probabilities yields
(40) 
for all .
Then, let be a weight parameter associated with a particular and th class , defined as
(41) 
which normalizes into the range of , .
For an th sequence , a collection can be defined as
(42) 
where .
From equations (39) and (42), the evolution of a particular sequence with respect to a th class is defined as
(43) 
Since the term (43) is a nonlinear map, the problem of correlation analysis [31, 37] between the inner products of nonlinear functions and can be reformulated via a kernel machine [34, 35, 36] as , which yields a distance in a highdimensional Hilbert space . This distance in can therefore be used as a metric to describe the correlation between and .
Let be the input space and let be an arbitrary kernel machine, defined for a given via the kernel function
(44) 
where
(45) 
is a nonlinear map from to the highdimensional reproducing kernel Hilbert space (RKHS) associated with . Without a loss of generality, , and we assume that the map in equation (45) has no inverse.
Then, for a and , let be the correlation identifier, as
(46) 
Assuming that is a Gaussian kernel [34, 35, 36] in equation (46), for an th gate parameter the kernel function is
(47) 
where , while yields the distance in ,
(48) 
For a given and , an average is yielded as
(49) 
where refers to the space of symmetric positive semidefinite matrices [35, 36, 37], while the inner products of and are represented in via as
(50) 
The sequence is classified into a given class from set , as given in Algorithm 2 ().
5 Numerical Evaluation
5.1 System Stability
Let be the stabilized system state of the quantum computer formulated by output systems, , , as
(51) 
with gate parameters , as given in (11).
Then, let be a target stabilized system of the quantum computer, as
(52) 
with target gate parameters , as
(53) 
where .
Then, let refer to the gate parameter of an th unitary of an th running sequence of the quantum computer, , , in the state , and let identify the target gate parameter in state .
Then, let be the relative entropy between and , as
(54) 
where , and let be a function that returns the value of the relative entropy function for an th running sequence as
(55) 
Let be a target value for function (55), and let be the difference [33] between and (55), as
(56) 
where is the derivative of .
Using (56), we define a stability parameter to quantify the variation of the stabilized system state of the th running sequence of the quantum computer, as
(57) 
For analytical purposes, let us assume that oscillates between a minimal value , and a maximal value , defined as
(58) 
and
(59) 
therefore can be rewritten as
(60) 
where is a constant, set as
(61) 
while is an expected value of (54), set as
(62) 
while is the number of oscillations.
Then, by using (63), the quantity in (57) is as
(64) 
that identifies the inverse of the number of oscillations.
Therefore, (64) identifies the stability of the system state of the quantum computer in the th running sequence if has the form of (60). For an arbitrary , the stability parameter is evaluated via (57). The high value of indicates that the stabilized system in (51) changes slowly. Particularly, if , where is a target value for , then the system state of the quantum computer is considered as stable.
5.2 Gate Parameter Correlations
Let be the stabilized state of the quantum computer in the th running sequence, with , and let be the target stabilized system state in the th running sequence, with .
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