1. Introduction
In the realm of scientific computing, the ubiquitous task of solving linear equations assumes paramount significance, emerging as a fundamental and indispensable component across a myriad of academic disciplines. Its essentiality lies in its pivotal role, and its widespread applicability makes it an intricate and indispensable cornerstone of computational research and analysis. Hence, the quest for an algorithm capable of efficiently finding solutions to linear equations is an immensely critical task. As quantum computing technology continues to evolve, some algorithms have demonstrated superior efficiency when executed on quantum computers compared to classical algorithms. For instance, Grover's search algorithm [1] demonstrates its formidable power by efficiently solving the unstructured search problem using only ${ \mathcal O }(\sqrt{N})$ queries when executed on a quantum computer, leading to significant acceleration compared to classical search algorithms. Harrow, Hassidim, and Lloyd (HHL) combined quantum computing with the problem of solving linear systems of equations and introduced the HHL algorithm [2] in 2009, a tailored approach specifically designed for quantum linear system problems (QLSP). Notably, the algorithm has been demonstrated to exhibit acceleration capabilities compared to classical approaches, as substantiated through rigorous analysis. The advent of the HHL algorithm has catalyzed the acceleration of other quantum algorithms that are reliant on QLSP or that draw insights from its principles, spanning both linear and nonlinear domains [3–8]. Moreover, the utility of quantum linear solvers extends across a diverse array of applications, encompassing tasks such as linear algebra [9], data fitting [10], and machine learning [11–16].
Nevertheless, the research and development progress of fault-tolerant quantum computers has been proceeding at a slow pace. The noise in quantum devices and the limited number of quantum bits have posed significant challenges in the implementation of the quantum algorithm [17]. Given a system of linear equations $A\vec{x}=\vec{b}$ with a coefficient matrix of size N × N, the HHL algorithm can find the solution in ${ \mathcal O }(\mathrm{log}N)$ time. In contrast, classical direct and iterative algorithms require ${ \mathcal O }({N}^{3})$ [18] and ${ \mathcal O }({N}^{2})$ [19] time complexities, respectively. Naturally, the attainment of quantum speed-up requires adherence to specific conditions: (i) a quantum state associated with b can be effectively prepared; (ii) the quantum operator e−iAt needs to be capable of being efficiently implemented; (iii) A is well-conditioned; (iv) the focus lies solely on specific numerical values or statistical aspects of the solution $\vec{x}$ [20]. However, even with these constraints, the remarkable performance of the HHL algorithm continues to capture the interest of certain researchers, inspiring them to further explore and develop on its foundations, leading to the proposal of more advanced algorithms [21–26]. Although the theoretical analysis of such algorithms has successfully sparked significant interest in the field of quantum computing, they are currently limited to running in an ideal environment. Specifically, they rely on fault-tolerant quantum devices, which may take a decade or even longer to become accessible. The current implementations of the HHL algorithm [27–31] on real quantum computers are limited to matrices of size 2 × 2. Even the more advanced quantum linear solvers based on adiabatic quantum computing [32, 33] can only handle an 8 × 8 sized problem. Maximization of the potential of noisy intermediate-scale quantum (NISQ) computers [34] has emerged as a crucial concern. Accordingly, the concept of variational quantum algorithms was proposed and has since gained widespread adoption [9, 35–41].
The well-performing variational quantum algorithm for solving linear equations, known as the variational quantum linear solver (VQLS), is introduced in [42]. Four types of loss functions, namely ${\hat{C}}_{G}$, CG, ${\hat{C}}_{L}$, and CL, are derived in the VQLS, employing the projection operator ${H}_{M}=\ I-| \vec{b}\rangle \langle \vec{b}| $. Interestingly, the same mathematical expression that was comparable to that in CG was also independently introduced in [9] for the resolution of linear algebra problems, shortly before the advent of the VQLS. Furthermore, it is worth noting that VQLS is well-trained, even for large systems of linear equations [42]. A schematic diagram of the VQLS is illustrated by the black and green dashed components shown in figure 1. Although the algorithm is capable of identifying the solution vector $\vec{x}$ with notable precision, the computational complexity remains considerable, given that the execution of the loss function necessitates two quantum subroutines. Motivated by this, an innovative loss function is introduced, necessitating only a single quantum subroutine, which effectively trims computational costs while rigorously maintaining precision standards. To be more specific, the schematic diagram of the algorithm is depicted by the black and blue dashed components shown in figure 1. In this figure, the blue box distinctly encompasses a lesser number of subprograms relative to the green box, which suggests that the loss function introduced in this study is calculated with enhanced efficiency.
Figure 1. A schematic diagram of the variational quantum algorithm for solving linear equations. Here, H is the Hadamard gate, Ux(θ) denotes the parameterized quantum circuit employed to encode the solution vector $\vec{x}$, while $F(A)={U}_{{A}_{l}}$, where $A={\sum }_{i}{U}_{{A}_{i}}$. And Ub, a unitary matrix, is utilized to prepare the quantum state $| \vec{b}\rangle ={U}_{b}| 0\rangle $. The loss function CG is constructed based on the projection operator HM, while the degree of overlap (DOP) loss is a method proposed in this paper. Comparatively, the quantum subroutine within the blue box exhibits significantly fewer operations than that within the green box. Consequently, under equivalent conditions, the algorithm represented by the blue box is expected to be more efficient. |
Additionally, novel methodologies are introduced in this work to tackle the challenge of trainability, which is one of the significant obstacles faced by variational quantum algorithms. These approaches encompass a highly expressive parameterized quantum circuit and an advanced variational parameter optimization technique. The first is aimed at the expressiveness of the quantum circuit, which refers to its ability to represent the solution space. The essence of variational quantum algorithms that utilize classical optimizers to optimize the loss function is to adjust parameterized quantum circuits, tuning the state of the circuit to correspond to the optimal or close to the optimal value of the objective function. Hence, the expressive power of the parameterized quantum circuit plays a decisive role in the overall performance of the algorithm. On the other hand, the circuit for preparing arbitrary superposition quantum states is an essential component in quantum computing and quantum information processing. Considering its ability to initialize arbitrary quantum superposition states, the corresponding quantum circuit undoubtedly exhibits a remarkably strong expressive capability within quantum systems of the same size. Taking inspiration from this, we, for the first time, integrate this circuit architecture with variational quantum algorithms, encoding the solutions to linear equation systems into amplitudes of quantum superposition states of corresponding scales in advance. For small-scale linear tasks, a highly expressive parameterized quantum circuit is proposed to be employed in this work. The second technique is employed to optimize the parameters within the parameterized quantum circuit. The circuit involves a substantial number of parameters when confronted with large-scale linear system tasks, posing significant implications for the training process of the variational quantum algorithms. To tackle this issue, we introduce a quantum parameter-sharing strategy that effectively reduces the number of parameters to be optimized while maintaining precision. Moreover, the strong correlation introduced by the quantum parameter-sharing strategy on the amplitude variation trend at each qubit endows the algorithm using this approach with the potential to outperform those that do not, especially in noisy environments.
The remainder of this paper is organized as follows. Section 2 introduces some technical concepts that are necessary for understanding this work, and the main innovative techniques of this research are illustrated. To validate the feasibility of the proposed approach, section 3 presents experimental simulation data and conducts relevant analyses. Finally, in section 4 , the conclusions and future work are discussed.
2. Research methodology
2.1. Degree of overlap (DOP) loss
For simplicity, a system of linear equations in real numbers is being considered. Specifically, for a given N × N matrix A and a vector $\vec{b}={[{b}_{0},{b}_{1},\ldots ,{b}_{N-1}]}^{{\rm{T}}}$, we need to prepare a variational algorithm with the task of finding a normalized quantum state $| \vec{x}\rangle $ such that $A\left|\vec{x}\right\rangle \propto | \vec{b}\rangle $, where $| \vec{b}\rangle =\tfrac{{\sum }_{i=0}^{N-1}{b}_{i}| i\rangle }{\parallel {\sum }_{i=0}^{N-1}{b}_{i}| i\rangle {\parallel }_{2}}$. The current common approach is to construct the projection operator HM [9, 24, 42] and then derive quantum linear equation solving algorithms based on it. In the VQLS, the Hamiltonian HG is obtained based on the foundation provided by HM,
$\begin{eqnarray}{H}_{G}={A}^{\dagger }(I-| \vec{b}\rangle \langle \vec{b}| )A,\end{eqnarray}$
where the ground state of HG corresponds to the solution of the linear equation system. Classical optimization algorithms can be employed to minimize the loss function ${L}_{{\hat{C}}_{G}}$ and determine the optimal solution to the equation system $\begin{eqnarray}{L}_{{\hat{C}}_{G}}={\hat{C}}_{G}=\left\langle \vec{x}\left|{H}_{G}\right|\vec{x}\right\rangle =\langle \vec{\psi }| \vec{\psi }\rangle -| \langle \vec{b}| \vec{\psi }\rangle {| }^{2}.\end{eqnarray}$
And $| \vec{\psi }\rangle =A| \vec{x}\rangle $, where $\vec{x}={U}_{x}(\theta )| 0\rangle $, θ = (θ1, θ2,…,θk) is a set of variable parameters. To avoid the situation where the norm of $A| \vec{x}\rangle $ is very small, leading to small values of ${L}_{{\hat{C}}_{G}}$, further improvements are made to obtain $\begin{eqnarray}{L}_{{C}_{G}}={C}_{G}=1-\displaystyle \frac{| \langle \vec{b}| \vec{\psi }\rangle {| }^{2}}{\langle \vec{\psi }| \vec{\psi }\rangle }.\end{eqnarray}$
The solution vector $\vec{x}$ is encoded into the amplitudes of a quantum state through a parameterized quantum circuit. Therefore, the target task is modeled as a function of the variable θ, and the QLSP is effectively transformed into a minimization problem $\begin{eqnarray}{\theta }_{\mathrm{opt}}=\arg \mathop{\min }\limits_{\theta }{L}_{{C}_{G}}(\theta ).\end{eqnarray}$
Figure 1 illustrates the schematic diagram of solving linear equation systems using the variational quantum algorithm, with the VQLS algorithm flow indicated by green dashed arrows. Theoretically, all loss functions cited in the context of the VQLS are amenable to effective evaluation. Nevertheless, it is readily apparent that algorithms leveraging these loss functions typically exhibit elevated computational costs. To address this problem, we draw inspiration from the similarity measurement between two quantum states and redesign the loss function, introducing the DOP loss. Compared to ${L}_{{C}_{G}}$ that requires two subroutines, the DOP loss only necessitates one subroutine to accomplish the desired task.There are several ways to measure the similarity between quantum states, including methods such as trace distance [43], fidelity [44], and the Hilbert–Schmidt distance [45]. Here, we notice that the fidelity in these methods can be interpreted geometrically as the inner product value between two vectors on the unit sphere. Moreover, given that in the field of quantum computing, all qubit states are normalized to 1. Hence, this paper considers employing the inner product value to quantify the similarity between the predicted state $| \vec{\psi }\rangle $ and the exact state $| \vec{b}\rangle $, which can also be described as the overlap between the two. With the quantum states $| \vec{b}\rangle ={[{b}_{0},{b}_{1},\ldots {b}_{N-1}]}^{{\rm{T}}}$, $| \vec{\psi }\rangle ={[{\psi }_{0},{\psi }_{1},\ldots {\psi }_{N-1}]}^{{\rm{T}}}$, the DOP can be described as
$\begin{eqnarray}\mathrm{DOP}=\displaystyle \sum _{i=0}^{N-1}\displaystyle \sum _{j=0}^{N-1}{b}_{i}{\psi }_{j}^{* }.\end{eqnarray}$
The value of the DOP is 0 when the exact solution $| \vec{b}\rangle $ and the approximate solution $| \vec{\psi }\rangle $ are mutually orthogonal. If there is a certain degree of similarity between them, then 0 < DOP ≤ 1. The equality DOP = 1 holds only when $| \vec{b}\rangle $ and $| \vec{\psi }\rangle $ are identical. According to its characteristics, we can effectively measure the difference between $| \vec{\psi }\rangle $ and $| \vec{b}\rangle $. Specifically, the DOP value can be used to assess the DOP between the approximate solution of the linear equation $| {\vec{x}}_{0}\rangle $ and the exact solution $| {\vec{x}}_{\mathrm{opt}}\rangle $. And then the DOP loss function can be defined as $\begin{eqnarray}{L}_{\mathrm{DOP}}=1-\mathrm{DOP}.\end{eqnarray}$
Since 0 ≤ DOP ≤ 1, the value range of the loss function is 0 ≤ LDOP ≤ 1. In terms of quantum circuit implementation, the Hadamard test circuit structure can be used as a reference to calculate the DOP value, as shown in figure 2.
Figure 2. The quantum circuit for computing the DOP between two quantum states, $| \vec{\psi }\rangle $ and $| \vec{b}\rangle $. Here, the phase gate S† is a transformer, and it will not be included when calculating the real part of the DOP. |
In more detailed terms, the DOP is derived by calculating the average of a special unitary operator HDOP applied to the quantum state ∣0⟩ 5 ) can be rewritten as
$\begin{eqnarray}{H}_{\mathrm{DOP}}={{U}_{b}}^{\dagger }{U}_{A}{U}_{x},\end{eqnarray}$
where UA is the unitary matrix corresponding to the coefficient matrix A in the linear problem, and Ux is used to prepare the quantum state $| \vec{x}\rangle $ on the left-hand side of the linear equation, such that $| \vec{x}\rangle $ = Ux∣0⟩. Meanwhile, ${U}_{b}^{\dagger }$ represents the conjugate transpose matrix of Ub, satisfying $\langle b| \,=\,\langle 0| {{U}_{b}}^{\dagger }$. Considering a matrix of size 2n × 2n that can be effectively expressed as a linear combination of L unitary matrices ${U}_{{A}_{0}},{U}_{{A}_{1}},...,{U}_{{A}_{L-1}}$, i.e. $\begin{eqnarray}A={{\rm{\Sigma }}}_{l=0}^{L-1}\,{c}_{l}\,{U}_{{A}_{l}},\end{eqnarray}$
where ${U}_{{A}_{l}}$ are Pauli matrices and ${c}_{l}\in {\mathbb{C}}$. Then, equation ( $\begin{eqnarray}\mathrm{DOP}=\mathrm{Tr}(| 0\rangle \langle 0| {H}_{\mathrm{DOP}})=\displaystyle \sum _{l}{c}_{l}\langle \vec{b}| {U}_{{A}_{l}}| \vec{x}\rangle .\end{eqnarray}$
Hence, the loss function proposed in this work is defined as ${L}_{\mathrm{DOP}}=1-{\sum }_{l}{c}_{l}\langle \vec{b}| {U}_{{A}_{l}}| \vec{x}\rangle $. It is evident that the LDOP loss function, in contrast to the ${L}_{{C}_{G}}$ loss function, necessitates merely a single quantum program for its computation, thereby yielding reduced computational complexity.2.2. Encoding circuit
Efficient quantum encoding of the target quantum state is another crucial key technique in variational quantum algorithms, as it directly impacts the degree of the target task. In this work, the encoding of the vector $\vec{x}$ is obtained by applying a series of trainable unitary operators Ux(θ) to the state ∣0⟩ to prepare the state $| \vec{x}\rangle $, and the encoding circuit is called an ansatz. Currently, typical ansatz structures can be categorized into two types: the hardware-efficient ansatz (HEA) [46], and the Hamiltonian variational ansatz (HVA) inspired by the quantum approximate optimization algorithm [47]. However, considering that the number of gates used in the actual implementation of the HVA is larger compared with that of the HEA, the latter is preferred in this study.
For a matrix of size N × N, the solution vector can be denoted as $| \vec{x}\rangle ={[{x}_{0},{x}_{1},\ldots ,{x}_{N-1}]}^{{\rm{T}}}$, where N = 2n and n is the number of qubits. The quantum encoding of the solutions to the linear equations is achieved by mapping them into the amplitudes of the computational basis states,
$\begin{eqnarray}| \vec{x}\rangle ={{\rm{\Sigma }}}_{i=0}^{N-1}{x}_{i}\,| i\rangle .\end{eqnarray}$
As the target state is obtained through the ansatz in the variational quantum algorithm, the above expression can be modified to $| \vec{x}\rangle =U(\theta )| 0{\rangle }^{\otimes n}$. The undetermined parameter θ can be adjusted to bring the state closer to its optimal value. The expressive power of the ansatz is a crucial factor to consider when constructing quantum circuits, as it determines whether a given circuit structure can find the target state by optimizing its parameters. Drawing inspiration from the preparation of arbitrary superposition quantum states [48], we introduce an ansatz circuit with a powerful representation capacity specifically tailored for small-scale linear problems. Figure 3 illustrates an ansatz circuit for a matrix of size N = 4. Assuming the solution vector is known as $| \vec{x}\rangle ={[{x}_{0},{x}_{1},{x}_{2},{x}_{3}]}^{{\rm{T}}}$, the process of encoding the superposition quantum state is as follows, $\begin{eqnarray}\begin{array}{l}| 00\rangle \to \sqrt{| {x}_{0}{| }^{2}+| {x}_{1}{| }^{2}| }00\rangle +\sqrt{| {x}_{2}{| }^{2}+| {x}_{3}{| }^{2}}| 10\rangle \\ \to {x}_{0}| 00\rangle +{x}_{1}| 01\rangle +\sqrt{| {x}_{2}{| }^{2}+| {x}_{3}{| }^{2}}| 10\rangle \\ \to {x}_{0}| 00\rangle +{x}_{1}| 01\rangle +{x}_{2}| 10\rangle +{x}_{3}| 11\rangle .\end{array}\end{eqnarray}$
This procedure involves three rotation angles, which are derived from $\vec{x}$ to facilitate the preparation and initialization of arbitrary quantum superposition states, $\begin{eqnarray}\begin{array}{l}{\theta }_{1}=2\ast \arctan \sqrt{\displaystyle \frac{| {x}_{2}{| }^{2}+| {x}_{3}{| }^{2}}{| \,{x}_{0}{| }^{2}+| \,{x}_{1}{| }^{2}}},\\ {\theta }_{2}=2\ast \arccos \,\displaystyle \frac{{x}_{0}}{\sqrt{| {x}_{0}{| }^{2}+| {x}_{1}{| }^{2}}}\ast \,\mathrm{sign}({x}_{1}),\\ {\theta }_{2}=2\ast \arccos \,\displaystyle \frac{{x}_{2}}{\sqrt{| {x}_{2}{| }^{2}+| {x}_{3}{| }^{2}}}\ast \,\mathrm{sign}({x}_{3}).\end{array}\end{eqnarray}$
Figure 3. The encoding circuit structure is applied for the matrix size of 4 × 4. |
Viewed through the lens of variational algorithms, the rotation angles of the circuit can be controlled to obtain any desired quantum state in the given dimension. However, while the ansatz using this method exhibits high expressive capability, it inevitably brings forth new issues. With a large value of n, the ansatz circuit entails an extensive parameter space, gate overhead, and circuit depth, presenting significant challenges for algorithm training. For instance, compared to the circuit structure for n = 2, the circuit structure for n = 3, as depicted in Circuit4 of figure 4, doubles the number of parameters and exhibits even greater growth in gate count and circuit depth. To tackle these difficulties, we make a trade-off by sacrificing some circuit expressiveness to balance the burden imposed by such ansatz structures on the algorithm. Drawing on the working principles of convolutional neural network (CNN) algorithms [51–54], each feature layer is computed using a series of minimal units, which are convolutional kernels with different receptive field sizes. The structure depicted in figure 4 serves as the fundamental building block (unit), on which a new ansatz is constructed to solve larger-scale linear systems. Specifically, figure 11 illustrates an ansatz circuit built from Units that is employed for a matrix of size N = 256.
Figure 4. For the matrix size N = 8, several circuit structures are used in this study to compare their respective expressive capabilities, with each circuit being assigned a circuit ID. Among them, Circuit1 (a) was inspired by [49] as a parameterized quantum circuit structure. Circuit2 (b), as presented in [42], was used to find the target quantum state. Circuit3 (c) corresponds to an ansatz used in [50], which was commonly employed in other variational quantum tasks. Circuit4 (d) is the ansatz proposed in this work. |
2.3. Quantum parameter sharing
In a large-scale linear system, the demand for increased quantum bit resources gives rise to a substantial proliferation of quantum parameters, which are particularly evident when dealing with ansatz circuits of considerable depth. However, having an excessive number of quantum parameters often hinders algorithmic training, leading to issues such as prolonged training cycles and challenges in achieving convergence. On the other hand, parameter sharing is an important parameter optimization strategy in CNNs that significantly reduces the number of parameters. Motivated by this, the implementation of parameter sharing within a variational quantum algorithm is investigated.
Parameter sharing in CNNs refers to the weight sharing across each small patch in a feature map. To provide more details, the left image in figure 6 shows a straightforward weighted output, where ${y}_{0}={w}_{0}^{0}{x}_{1}+{w}_{1}^{0}{x}_{2}+{w}_{2}^{0}{x}_{3}$ and ${y}_{1}={w}_{0}^{1}{x}_{5}+{w}_{1}^{1}{x}_{6}+{w}_{2}^{1}{x}_{7}$. Parameter sharing, which is also known as weight duplication, implies the equality w0 = w1. In an endeavor to improve the efficiency of the variational quantum algorithm, we attempt to introduce the concept of parameter sharing. Based on this approach, the number of parameters awaiting optimization in the ansatz circuit is effectively reduced. The right-hand side of figure 5 is a frame diagram of quantum parameter sharing, and circles of the same color indicate that the parameter values are kept the same during training. Two quantum parameter-sharing strategies are proposed for the ansatz used in this paper, denoted as S1 and S2. To mitigate the impact of closely positioned parameters, S1 involves sharing the first and last parameters, while S2 adopts parameter sharing based on the circuit depth. Consider Circuit2 in figure 4 as an example, where the encoding circuit involves n = 3 qubits. Based on Circuit2, we can observe that the total number of variational parameters is nine, denoted as θ = [θ0, θ1, θ2, θ3, θ4, θ5, θ6, θ7, θ8]. Then, the parameters can be grouped into three sets ${\theta }^{0}=[{\theta }_{0}^{0},{\theta }_{1}^{0},{\theta }_{2}^{0}]$, ${\theta }^{1}=[{\theta }_{3}^{1},{\theta }_{4}^{1},{\theta }_{5}^{1}]$, and ${\theta }^{2}=[{\theta }_{6}^{2},{\theta }_{7}^{2},{\theta }_{8}^{2}]$, based on the circuit depth. After incorporating quantum parameter sharing into the algorithm, one group of parameters becomes entirely or partially identical, such as ${\theta }_{3}^{1}={\theta }_{4}^{1}={\theta }_{5}^{1}$. As a result, the total number of optimization parameters is reduced from nine to seven. For large values of n, this method can substantially decrease the number of parameters in variational quantum algorithms, thereby greatly improving the efficiency of the algorithm. Furthermore, we visualize the variation of parameters during the optimization loop in the algorithm without parameter sharing, as shown in figure 6(a). An intriguing phenomenon observed is that as the algorithm approaches convergence, some parameter values converge and become nearly equal, such as θ3 and θ5. It is astonishing that despite θ3 being approximately equal to θ5, the algorithm achieves an accuracy of 1.0000, which also implies the feasibility of the quantum parameter-sharing strategy.
Figure 5. (a) Illustration of parameter sharing in CNNs. The input data xi represents a feature point on the feature map, and wi denotes the weight parameters required for CNNs. (b) The framework of the proposed variational quantum parameter sharing is depicted. The input data consists of probability amplitudes on each qubit, and a group of input data with the same color belongs to the same qubit. All quantum rotation gates used in the algorithm are represented as a collection, Uθθi, within a cylinder, where each gate is associated with a rotation angle θi. Gates of the same color indicate parameter sharing, implying that they are rotated by the same angle. |
Figure 6. (a) The graph shows the changes in each parameter during the iteration process, and (b) the graph illustrates the accuracy as the algorithm stabilizes. The experiments are carried out in a noiseless environment, where the parameters are randomly initialized, and circuit2, as depicted in figure 5, is adopted as the encoding circuit. The Powell method [55] is employed to facilitate better observation of the parameter variations. |
3. Experiments and discussion
In this section, the methods proposed in this work are implemented using the quantum computing platform provided by IBM. Related numerical simulations were carried out across two distinct tasks. Initially, we utilize a specially crafted matrix A to guarantee its full rank status. Subsequently, the second QLSP, drawing inspiration from the Ising model, is employed in the generation of matrix A. Within the scope of the experiments conducted, matrix A that is utilized is characterized by sparsity, and the range of its singular values is confined between 1/k and 1, in accordance with the standards delineated in [2]. Here, k is the condition number of matrix A.
3.1. Customized matrix
Numerical simulations are utilized here to solve the QLSP using an advanced variational hybrid algorithm, EVQLSE, with the coefficient matrix being defined as9 ). Subsequently, the performance of algorithms using different loss functions can be effectively evaluated based on accuracy, which can be calculated from predicted and exact values. And the accuracy here is defined as9 )), ensuring favorable convergence while concurrently lowering computational complexity. In terms of quantum gate consumption, the DOP requires fewer gates compared to CG. Without loss of generality, assume that ${U}_{{A}_{l}}$ has only one effective Pauli operator P (P = X/Y/Z). The computational circuit cost for CG totals ${ \mathcal O }({{mL}}^{2})$ gates (more precisely, (16 + 3m)L2 gates), while the DOP requires only ${ \mathcal O }({mL})$ (${ \mathcal O }((6+m)L)$ gates). Here, m denotes the number of gates in the ansatz circuit, and L represents the count ${U}_{{A}_{l}}$ of units.
$\begin{eqnarray}A={aI}+{{bX}}_{1}+{{cX}}_{1}{Z}_{2}.\end{eqnarray}$
Herein, a, b, and c can take arbitrary values within the range of real numbers. After adjusting the values of a, b, and c, different matrices A can be randomly generated, each with a unique condition number. Keep in mind that A is subject to a constraint, requiring its maximum eigenvalue to be 1 and the minimum eigenvalue is 1/k, as stipulated in [2]. Here, Pj(P = X/Y/Z) represents the frequently utilized Pauli operator, with their subscripts denoting the corresponding quantum bit upon which the Pauli operator acts. For instance, X1 indicates the application of the X operator on the first qubit. In the specific scenario with a presumed total of n = 3 qubits, X operates on the first qubit, while the identity operator is applied to the remaining qubits. The quantum state on the right side of the linear equation is prepared as $\begin{eqnarray}| \vec{b}\rangle =H| 0{\rangle }^{\otimes n}.\end{eqnarray}$
The quantum circuit shown in figure 2 is implemented by Qiskit to complete the calculation of equation ( $\begin{eqnarray}\mathrm{Accuracy}={\vec{x}}_{\mathrm{opt}}\cdot {\vec{x}}_{0},\end{eqnarray}$
where ${\vec{x}}_{\mathrm{opt}}$ is the predicted value and ${\vec{x}}_{0}$ is the exact value. In the experiment, Circuit2 in figure 4 is selected as the ansatz circuit, which is used in the VQLS algorithm. First, relevant experiments are conducted in a noiseless environment to verify the convergence and effectiveness of the DOP loss function. Figure 7 offers a graphical depiction derived from the experimental data. Through consecutive iterations of the algorithm, the gradual descent of the loss signifies the capacity of the DOP loss to minimize the discrepancy between predicted and exact values to the maximum extent. The dynamic evolution of the loss function values observed during the training epochs distinctly highlights the convergence characteristics inherent in the proposed methodology presented in this work. The results obtained from the experiments are that both CG loss and DOP loss approaches start to converge around the same iteration position, and their respective loss function values become lower than 10−2 when stabilized. However, for CG, it is necessary to evaluate two components, $\begin{eqnarray}| \langle \vec{b}| \vec{\psi }\rangle {| }^{2}=\displaystyle \sum _{{{ll}}^{{\prime} }}{c}_{l}{c}_{{l}^{{\prime} }}^{* }\left\langle 0\left|{U}_{b}^{\dagger }{U}_{{A}_{l}}{U}_{x}\right|0\right\rangle \left\langle 0\left|{U}_{x}^{\dagger }{U}_{{A}_{{l}^{{\prime} }}}^{\dagger }{U}_{b}\right|0\right\rangle ,\end{eqnarray}$
$\begin{eqnarray}\langle \vec{\psi }| \vec{\psi }\rangle =\displaystyle \sum _{{{ll}}^{{\prime} }}{c}_{l}{c}_{{l}^{{\prime} }}^{* }\langle 0| {U}_{x}^{\dagger }{U}_{{A}_{{l}^{{\prime} }}}^{\dagger }{U}_{{A}_{l}}{U}_{x}| 0\rangle .\end{eqnarray}$
Relative to the CG, the DOP is more efficient as it requires merely a single evaluation (equation (
Figure 7. The training loss curve. The experimental data are obtained by simulating the quantum circuit in a noiseless environment, with a matrix condition of k = 2.3. Classical optimization is performed using the gradient-free COBYLA method [56]. |
In addition, the assessment of the ansatz circuit's ability to represent the solution space, based on the characteristics of this work, involves the use of specialized vectors containing zero elements. Experimental verification is performed using a linear system of size 8 × 8. For simplicity, a, b, and c are set to 1, 0, and 0, respectively. Subsequently, the four ansatz circuits provided in figure 4 are individually trained to predict $\vec{x}$, such that $\vec{x}=\vec{b}$. Given this work, accuracy is being considered to quantify the expressive capacity of the circuit. Four sets of experiments are conducted for each ansatz, with 50 repetitions per set. And the variational parameters are randomly initialized for each repetition. Table 1 presents the success rates at various thresholds for the scenario with b3 = 0, where the success rate indicates the ratio of the number of times the accuracy is greater than or equal to the threshold to the total number of repetitions. The results demonstrate that Circuit1 and Circuit4 performed better, with the latter achieving a 100% average success rate across all thresholds. Lastly, the visualization of 16 sets of experimental results is presented in figure 8. When employing Circuit2 and Circuit3 as ansatz circuits, their expressive capabilities show significant fluctuations compared to the other two circuits. In contrast, the proposed circuit in this study not only demonstrates superior stability but also consistently maintains a leading score compared to Circuit1.
Figure 8. Visualization of the expressive capabilities of the four ansatz circuits. The influence of noise on the experiment is avoided, and the results depicted in the figure are obtained in a noiseless environment with randomly initialized parameters. Across the horizontal axis, we depict scenarios when bk = 0 and ${b}_{\bar{k}}\ne 0$, where k ∈ {0, 1, 2, 3}, while the vertical axis corresponds to the expressiveness of the quantum circuit. Here, ${b}_{\bar{k}}\ne 0$ signifies values apart from bk = 0. |
Table 1. The success rates of different ansatz at various thresholds, with the highest accuracy indicated in bold. |
| Circuit ID | 0.975 | 0.980 | 0.985 | 0.990 | 0.995 | 0.999 |
|---|---|---|---|---|---|---|
| Threshold | ||||||
| 1 | 1.00 | 1.00 | 1.00 | 1.00 | 0.00 | 0.00 |
| 2 | 0.84 | 0.76 | 0.64 | 0.46 | 0.34 | 0.16 |
| 3 | 0.64 | 0.52 | 0.00 | 0.00 | 0.00 | 0.00 |
| 4(Ours) | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
On the other hand, we also investigate the impact of applying quantum parameter-sharing strategies on the algorithm and provide further experimental validation of the effectiveness of the proposed strategies, S1 and S2. To ensure the generalizability of the experimental results, the values of a, b, and c are set to a = 1, b = c = 0.2 for data generation. The results, as shown in figure 9, illustrate the variation of the loss function with respect to the number of iterations for both S1 and S2 conditions. It is evident that, apart from Circuit1 adhering to the S1 criterion, the algorithm with parameter sharing does not exhibit a significant decrease in accuracy. Contrasting S1 and S2, in the ansatz circuits exhibiting symmetric structures like Circuit1, 2, and 3, we lean towards employing the latter for parameter sharing. S1 achieves a reduction in a large number of variational parameters while also maintaining the required training accuracy. To delve deeper into its performance, quantum parameter sharing is introduced for the first time in the VQLS algorithm. The experiments are conducted with varying numbers of quantum bits. Based on the experimental results, the conclusion can be drawn that the inclusion of quantum parameter sharing does not adversely affect the final precision or only exerts a minimal influence, as depicted in figure 10. Nevertheless, as evidenced in table 2, the adoption of parameter sharing within an identical circuit framework markedly diminishes the parameter count.
Figure 9. Comparison of the impact of variational parameter sharing under the criteria of S1 and S2 on algorithm performance. The left figure (a) presents the experimental results under the criterion of S1, while the right figure (b) displays the data obtained following the criterion of S2. Both sets of experiments utilized the encoding circuits mentioned in figure 4 and are conducted in a noise-free environment. Note that under S2 criteria, Circuit4 has only one variational parameter for the same depth; hence, there are no experimental data for this circuit in (b). |
Figure 10. Performance evaluation after incorporating quantum parameter sharing (QPS) in VQLS. |
Table 2. Comparison of required parameter quantities for different n. |
| Size n | 4 | 6 | 8 | 10 |
|---|---|---|---|---|
| VQLS | 28 | 46 | 64 | 82 |
| VQLS+QPS | 16 | 26 | 24 | 26 |
| Relative reduct. % ↓ | 42.86% | 43.48% | 62.5% | 68.29% |
Finally, we investigated the performance of the algorithm as the values of n and the matrix condition number k increase, i.e. the scalability. These two factors play a crucial role in evaluating the scalability of quantum linear solving algorithms [2]. The sparse matrix A, generated through formula (13), strictly adheres to the specifications outlined in [2]. In the experimental setup, the ansatz structure shown in figure 11 is utilized. And to achieve better results in a noisy environment, the analytic quantum gradient descent, an optimization algorithm from Qiskit's library [57], is employed. Firstly, the value of n is held constant, and the accuracy is computed for different values of k, as shown in figure 12(a). From the plot, it can be observed that the accuracy of the algorithm decreases with the increasing matrix condition number k and the best accuracy is achieved when k = 20. However, we can infer from [42] that this phenomenon is attributed to the fact that larger condition numbers require more time to complete the task. It should be noted that this observation does not impact the verification of the robustness of our algorithm across different k. Then, the loss for varying n values is calculated while maintaining a constant k. The results indicate that with increasing n, the loss function continues to be effectively optimized, reducing to within 3%, as illustrated in the inset of the graph. The experimental results provide evidence of the algorithm's strong scalability concerning n. Even with n = 20, corresponding to a matrix size of 220 × 220, the optimization of the loss function remains effective, with errors kept within 5%. The important thing to note here is that the loss values less than 0 in the graph are a result of noise interference.
Figure 11. The ansatz circuit is constructed using the unit proposed in this paper. This shows the ansatz circuit for n = 8, where entanglement is introduced between each unit through Z and X operations. |
Figure 12. (a) The accuracy variation with scaling based on k for n = 10, where the blue dashed line represents a threshold boundary at 0.95. With the increase in k, the algorithm's accuracy remains relatively stable, with precision above 95% for each case. Holding k constant at 20, the plot in (b) illustrates the changes in the loss function as n is expanded, with the red dashed line indicating the threshold at a loss function value of 0.05. As the number of qubits increases, the results show that DOP loss can still be effectively trained in the presence of noise. The number of shots for all experiments is set to 1000. |
3.2. Ising-inspired matrix
The second QLSP is inspired by the Ising model, and the sparse matrix A is defined as follows:
$\begin{eqnarray}A=\displaystyle \frac{1}{\zeta }\left(\displaystyle \sum _{j=1}^{n}{X}_{j}+J\displaystyle \sum _{j=1}^{n-1}{Z}_{j}{Z}_{j+1}+\eta {\mathbb{1}}\right),\end{eqnarray}$
where the subscript clearly indicates the qubits upon which the Pauli operators act. The parameter J is determined to be 0.1, a selection grounded on the insights from [42]. The responsible selection of parameters ζ and η is crucial for the matrix A, as it involves ensuring that the largest eigenvalue of the matrix is 1, while the smallest eigenvalue is constrained to 1/k.The performance of the EVQLSE algorithm is further explored by increasing the number of effective quantum bits. Figure 13 demonstrates that the training loss curves exhibit satisfactory convergence across various n values. It is readily apparent that with the escalation in system size, there is a corresponding incremental challenge in algorithm training. Compared to the case with n = 20, the loss curve exhibits a more rapid convergence when n is set to 4. Nonetheless, at n = 20, the EVQLSE algorithm can still find the solution to the linear equations with considerable accuracy.
Figure 13. Exploring the performance of finding the target state with the increase in quantum bit numbers for the Ising-inspired QLSP. |
4. Conclusion and future works
In the field of scientific computing, employing variational quantum algorithms to perform certain computational tasks is an advanced strategy in the NISQ era. This study presents a hybrid quantum–classical algorithm that is designed to efficiently solve systems of linear equations. The main contributions in this work include introducing a more efficient loss function called DOP loss, designing a high expressive ansatz, and utilizing advanced methods for variational parameter optimization. Firstly, the DOP loss function introduced in this work demonstrates reduced computational complexity and enhanced efficiency in comparison to the CG loss function. Additionally, by considering the expressiveness of the ansatz circuit, a parameterized quantum circuit with strong expressive capability is introduced to prepare $\vec{x}$ in the context of small-scale linear systems. Compared to other ansatz circuits, it exhibits both strong and stable expressiveness. Based on this, its topological structure is further developed, which is applied to tackle large-scale QLSPs. The last major innovation lies in the introduction of a novel technique for optimizing variational quantum parameters, known as quantum parameter sharing. This technique offers a new perspective on addressing the challenge of suboptimal training performance caused by the large number of parameters inherent in the algorithm. Note that the quantum parameter-sharing strategy is heuristic in nature. The specific implementation details vary depending on the different ansatz circuits. Therefore, utilizing inappropriate parameter-sharing methods may result in the algorithm failing to converge. Ultimately, the proposed variational quantum algorithm for solving systems of linear equations is implemented in a noisy environment, characterized by prominent readout errors. The obtained accuracy of over 99% further demonstrates the feasibility and advancement of our algorithm in the NISQ era. Notably, the techniques introduced in this paper are not confined to this particular study and can be readily extended to other tasks involving variational quantum algorithms.
Further research endeavors can involve exploring the extensions of this study. An efficient ansatz circuit can improve the training of the algorithm. Hence, it is essential to explore the topological structures that exhibit higher expressiveness and fewer parameters based on the parameterized quantum circuit proposed in this study. Moreover, investigation of more efficient quantum parameter-sharing methods is essential. In this work, we have primarily introduced two parameter-sharing strategies, S1 and S2, for the employed ansatz. Exploring sharing strategies on HVA or other structured HAE requires further investigation. The aim is to transform parameter sharing into a fundamental component, across all variational quantum algorithm tasks, similar to weight sharing in CNNs. Lastly, exploration of the combination of variational quantum parameter initialization with parameter sharing and its noise resilience in the NISQ era constitutes a highly promising and valuable research direction.