1. Introduction

Many important problems in science and engineering can be reduced to the problem of solving linear equations. Thus, it is very important to develop a fast algorithm for solving them. The best known classical algorithm to solve linear equations on classical computers with N variables requires a time that scales as $O(N)$ [1]. Recently, Harrow et al. proposed a quantum algorithm (HHL algorithm) that is able to solve linear equations in $O(log(N))$ time, which is an exponential speedup over the classical algorithm [2]. Such an exponential speedup promises widespread applications in large-scale systems, such as data processing [3], numerical calculation [4], and artificial intelligence [5,6].

The HHL algorithm has been experimentally demonstrated with parametric down-converted photon schemes [7,8], liquid nuclear magnetic resonance [9] and superconducting quantum processor [10]. All of these experimental schemes are based on encoding the qubit with the same degree of freedom (DoF) of multiparticle. For example, using the polarization of photon to encode one qubit has been reported for the multi-photon scheme described in [7]. In such a case, four photons are required to solve a linear equation with two variables, and seven photons are required to solve the linear equation with four variables. As for linear equations with$2^n$variables, $3n+1$ photons are required. While to this end, the reported largest number of experimentally preparing multipartite entanglement are 14 trapped ions [11], 10 photons [12,13], and 10 superconducting qubits [14]. That is to say, the computation task with larger number of variables is difficult to be experimentally demonstrated in present technology. To overcome this difficulty, one way is to develop technology to prepare larger number of photons, the other is to reduce the number of photons required to solve linear equations.

Recently, some investigations have shown that the introduction of multiple DoFs of photons can increase the number of entangled qubits [15–17], enhance the violation of local realism [18,19], simplify quantum logic gates [20]. Motivated by these studies, in this work we provide an experimental realization of the HHL algorithm with multiple DoFs of photons. The path and polarization DoFs of single photons are used to encode qubit. As an example for solving a $2\times 2$ linear equations, we describe the detailed experimental process of the HHL algorithm in such a scheme. Compared with the previous studies, our scheme not only presents excellent efficiency, it requires less photons to solve the same problem.

2. Theoretical description of the quantum HHL algorithm

The HHL algorithm aims to solve a system of linear equations $A\overrightarrow{x}=\overrightarrow{b}$ to find $\overrightarrow{x}$ for a given $N\times N$ Hermitian matrix $A$ and a N-dimension unit vector $\overrightarrow{b}$, where N represents the number of variables. The process involves three subsystems as shown in Fig. 1: a single ancilla qubit initialized in $|0〉$, a register of n qubits of working memory initialized in ${|0〉}^{\otimes n}$, and an input state initialized in $|b〉$. The input state $|b〉$ can be expanded in the basis of $|u_j〉$ as $|b〉={\displaystyle \sum _{j-1}^N{\beta }_j|u_j〉}$, where $|u_j〉$ is the eigenstate of $A$, and ${\beta }_j=〈u_j|b〉$ . That is to say, the initial state of this system is prepared at ${|0〉}_a{|0〉}^{\otimes n}{}_r{{\displaystyle \sum _{j-1}^N{\beta }_j|u_j〉}}_b$, the subscript a, r and b represents the ancilla, register and input qubit, respectively. The algorithm can be decomposed into the following three subroutines as shown in Fig. 1: (1) phase estimation [21,22], (2) controlled rotation, and (3) inverse phase estimation.

 

Fig. 1 The overall outline of the HHL algorithm. The phase estimation operation between the Register and input is realized by a $H^{\otimes n}$gate on the Register qubit, a C-U gate between input and register qubit, and a$F{T}^{†}$on the Register qubit. $H^{\otimes n}$ is the n-bit Hadamard gate on the register. U is determined by the matrix A. $FT$and $F{T}^{†}$represent the Fourier transformation and inverse Fourier transformation. The C-Rotation operation is used to obtain the factor of $\frac{1}{{\lambda }_j}$. Finally, the state is post-selected by ${|1〉}_a{|0〉}_r^{\otimes n}$ at the Ancilla and Register to obtain the expected results $x$ at the input qubit.

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In the step (1), the phase estimation operation is applied to the input state, using the working memory as control, to transfer the state into ${|0〉}_a{{\displaystyle \sum _{j=1}^N{\beta }_j{|{\lambda }_j〉}_r|u_j〉}}_b$ where ${|{\lambda }_j〉}_r$ represents the binary representation of the eigenvalue ${\lambda }_j$ of $A$. In the step (2), an appropriate controlled rotation operation $R({\lambda }^{-1})$ is used to extract the eigenvalues ${\lambda }_j^{-1}$ of $A^{-1}$. The control and controlled qubits are the register and the ancilla, respectively. After this step, the state of the system is ${\displaystyle \sum _{j=1}^N{\beta }_j(\sqrt{1-\frac{C^2}{{\lambda }_j^2}}{|0〉}_a+\frac{C}{{\lambda }_j}{|1〉}_a)}{|{\lambda }_j〉}_r{|u_j〉}_b$, where$C$is an appropriate constant. In the final step of the inverse phase estimation, the register qubit is transformed back to ${|0〉}^{\otimes n}$. Then we obtain the state ${\displaystyle \sum _{j=1}^N{\beta }_j(\sqrt{1-\frac{C^2}{{\lambda }_j^2}}{|0〉}_a+\frac{C}{{\lambda }_j}{|1〉}_a)}{|0〉}_r{|u_j〉}_b$. So, making a projection measurement to obtain ${|1〉}_a{|0〉}_r$ at the ancilla and register qubits, we can get the state ${\displaystyle \sum _{j=1}^N{\beta }_j\frac{C}{{\lambda }_j}}|u_j〉$, which corresponds to the expected results of the linear equations $|x〉={\displaystyle \sum _{j=1}^N\frac{{\beta }_j}{{\lambda }_j}}|u_j〉$.

3. Experimental realization of the HHL algorithm for a $2\times 2$ linear equations with multiple DoFs of single photons

To illustrate the above algorithm, we consider an concrete example with a matrix $A=\left(\begin{array}{cc}1.5& 0.5\\ 0.5& 1.5\end{array}\right)$, three input states $|b_1〉=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)$, $|b_2〉=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -1\end{array}\right)$ and $|b_3〉=\left(\begin{array}{c}1\\ 0\end{array}\right)$. Following the quantum multiphoton realization of this algorithm in [7], the optimized quantum circuit for solving the linear equation $\left(\begin{array}{cc}1.5& 0.5\\ 0.5& 1.5\end{array}\right)|x〉=|b〉$ is shown in Fig. 2(a), which corresponds to the process shown in Fig. 1. The compiled circuit realization of the phase estimation step consists of two C-Not gate between input and register qubits, two Hadamard gate on input qubit, a swap gate between the two register qubits and a X gate on register gate. The control rotation step is realized by two $C-H(\theta )$ gate between the register and aniclla qubits, where $H(\theta )=\left(\begin{array}{cc}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array}\right)$. The final step is realized by two Hadamard gates and post-selection measurement on two register qubits, which is a semiclassical version that employs single-qubit rotation conditioned on measurement outcomes [23].

 

Fig. 2 (a). The optimized circuit for solving the linear equation $A|\overrightarrow{x}〉=|b〉$, where $A=\left(\begin{array}{cc}1.5& 0.5\\ 0.5& 1.5\end{array}\right)$ . To describe the detailed realization of this quantum circuit in the experiment, we label all the quantum gates with A-J. (b). The experimental realization for solving linear equation. The experiment contains three parts, the Initial State Preparation part, the HHL Algorithm Evolution part, and the Measurement part. The GL is Glan Laser polarizer. $HWP@\theta$ represents a half wave plate with the optical axis oriented at$\theta$. The BS is the beam splitter. The PBS is polarizing beam splitter. Detector is single photon detection module. The double sided arrow in the measurement module represents the phase adjustment between the two arms of the Mach-Zehnder interferometer.

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In our experimental scheme, the ancilla, register1 and input qubits are encoded by multipath of single photon. And register2 qubit is encoded by the polarization of the same photon. The whole experimental realization of this optimized quantum circuit is shown in Fig. 2(b). It contains three parts, the initial state preparation part to prepare the initial state, the algorithm evolution part including three subroutines to execute the phase estimation, control rotation and inverse phase estimation operation, and the measurement part to make post-selection measurement to obtain the expected state $|x〉$.

In the part of initial state preparation, we use ultraviolet laser pulses with a central wavelength of 390nm passing through a β-barium borate (BBO) crystal to produce two photon pairs. Before the photons are collected by single mode fibers, they are filtered by narrow bandwidth filters (780 nm, $\pm 5$nm), which is not shown in Fig. 2(b). The total coincident counts are about 15000 per second. The collection time is 30 seconds for each measurement. One photon of the photon pairs is used as signal photon, and the other is used as idler photon. The signal photon passes through a HWP (half wave plate) with its optical axis oriented at ${\theta }_1$, a PBS (polarizing beam splitter) and a HWP with its axis oriented at ${45}^{\circ }$on the vertical output path of the PBS to prepare the input state $|b〉$. We can obtain the required input state $|b_1〉=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)$, $|b_2〉=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ -1\end{array}\right)$ and $|b_3〉=\left(\begin{array}{c}1\\ 0\end{array}\right)$ with the ancilla and register qubit on $|0〉$, when the angle ${\theta }_1$ is taken as ${22.5}^{\circ }$, ${67.5}^{\circ }$ and $0^{\circ }$, respectively.

In the Algorithm Evolution part, the whole process corresponds to the quantum circuit shown in Fig. 2(a). The phase estimation subroutine in Fig. 2(a) is executed using the optical devices in the gray dotted box at the top left corner in Fig. 2(b). We find that the required quantum gates are the Hadamard gate with the single qubit path, the polarization X gate and two qubits C-NOT gate based on polarization and path. The detailed demonstrations of them are shown in Fig. 4(a) , Fig. 4(c) and Fig. 5(a) in Appendices A and B. In the following, we give the concrete implementation in Fig. 2(b). The Hadamard gate A on the input qubit $|b〉$ shown in Fig. 2(a) is realized by a BS A. The first C-Not gate B between input and register2 qubits is realized by putting one HWP B on one of the paths when the other path keeps unchanged. The second C-Not gate C between the register2 and register1 qubits is realized with two PBSs $C$ and $C^{\prime }$. So, we can obtain four paths encoded with $|0_a{0}_{r1}{1}_b〉$, $|0_a{1}_{r1}{1}_b〉$, $|0_a{1}_{r1}{0}_b〉$ and $|0_a{0}_{r1}{0}_b〉$, respectively. To implement the Hadamard gate D on the input qubit, we need to induce the two paths with the same states on ancilla and register1 qubits and the different states on input qubit into the BS $D$ and $D^{\prime }$. And the X-gate E on the polarization qubit register2 is realized by four HWPs $E$, $E^{\prime }$, $E^{″}$ and $E^{‴}$ with their optical axis oriented at ${45}^{\circ }$. The two sequential swap gates between the register1 and register2 qubits are omitted.

 

Fig. 3 The experimental results for solving linear equation $A|x〉=|b〉$. Orange and blue histograms represent the theoretically and the experimentally measured results. X1, X2 and X3 represent the input states for $|b_1〉$, $|b_2〉$ and $|b_3〉$, respectively. X, Y and Z represent the Pauli observables.

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Fig. 4 (a), (b) and (c) represent the single bit Hadamard gate based on the path, single bit Hadamard gate based on the polarization, and single bit X-gate based on the polarization, respectively. The BS is the beam splitter. $HWP@\theta$ is a half wave plate with the optical axis oriented at $\theta$. $|0〉/|1〉$ represents the state encoded with the path DoF. $|H〉/|V〉$ represents the polarization encoded with horizontal and vertical polarizations.

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Fig. 5 (a) shows the C-Not gate that the control and controlled bits are coded by the path and polarization DoFs, which is mainly realized by a PBS (polarizing sensing beam splitter). (b) shows the $C-H(\theta )$ gate that the control and controlled bits are both coded by the path. When the control bit is on the $|0〉$ state, the path remains unchanged by being reflected to the origin path with two M(mirror). When the control bit is on the $|1〉$ state, the path bit is transformed by a $H(\theta )$ gate with a BS. (c) shows the $C-H(\theta )$ gate that the control and controlled bits are coded by the polarization and path DoFs.

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The controlled rotation subroutine is realized by the devices labeled with F and G, which is in the middle region of Fig. 2(b). The $C-H(\pi /8)$ gate F is realized by the BS, $F$ and $F^{\prime }$, putting on the path with the state $|1〉$ on the register1 qubit and keeping the path with the state $|0〉$ on the register1 qubit unchanged. The $C-H(\theta )$ gate G is realized with four blue blocks $G$,$G^{\prime }$,$G^{″}$and $G^{‴}$. The detailed demonstration of these two control gates is shown in Appendix B.

In the subroutine of inverse phase estimation, the required two Hadamard gates are demonstrated in Figs. 4(a) and 4(b) in Appendix A. The concrete implementation of this subroutine is shown in the gray dotted box at the bottom of right corner in Fig. 2(b). The last two Hadamard gates on register1 and register2 are realized by the BS $J$ on the path qubit and the HWP $I$ with the optical axis oriented at ${22.5}^{\circ }$ on the polarization qubit.

After all these operations, we can obtain eight outputs which are encoded by the paths as $|0_a{0}_{r1}{0}_b〉$, $|0_a{0}_{r1}{1}_b〉$, $|0_a{1}_{r1}{0}_b〉$, $|0_a{1}_{r1}{1}_b〉$, $|1_a{0}_{r1}{0}_b〉$, $|1_a{0}_{r1}{1}_b〉$, $|1_a{1}_{r1}{0}_b〉$ and $|1_a{1}_{r1}{1}_b〉$. According to the circuit shown in Fig. 2(a), the Ancilla and Register1 qubit on the path DoF, $|1_a{0}_{r1}{0}_b〉$ and $|1_a{0}_{r1}{1}_b〉$, can be firstly post-selected. The post-selected paths are marked as Out1 and Out2 in Fig. 2(b). Then we induce these two paths into the PBS to accomplish the final post-selection on the polarization qubit register2. To characterize the expected state, we make measurement with Pauli observables $X$, $Y$and $Z$by using a Mach-Zehnder interferometer with a piezoelectric transducer to modify the relative phase between the two arms in the green box. The detailed description of the measurement on the DoF of path is given in Appendix C.

The experimental results are shown in Fig. 3. The blue and orange bars are experimental results and theoretical expectation values, respectively. To quantify the performance of our scheme, we calculate the fidelities between the theoretical and experimental results. According to [24], the fidelity between two pure states is defined as: $F(|{\psi }_1〉〈{\psi }_1|,|{\psi }_2〉〈{\psi }_2|)={\left|〈{\psi }_1|{\psi }_2〉\right|}^2$, where $|{\psi }_1〉$ and $|{\psi }_2〉$ are the theoretically and experimentally obtained state $|x〉$. They are about 0.991(4), 0.987(5) and 0.975(10) for $|b_1〉$, $|b_2〉$ and $|b_3〉$ respectively. These high fidelities indicate the good agreement between experimental results and theoretical predictions. The experimental error comes from the instability of the interferometer. The main challenge lies in keeping all the path lengths the same, which is maintained by tuning the length of the arms of the interferometer with a piezoelectric transducer installed on one Mirror of the MZ interferometer. And this structure is similar to the structure shown in Appendix C. Another effect that may lead to the experimental error is that a single gate is comprised of multiple optical elements, e.g. there are 4 separate objects performing the operation E or G on different paths. In other words, the unequal performance of the same optical elements may affect the experiment results. In fact, the mainly used optical elements are PBS and HWP. The performances of different PBSs on the polarized light are almost the same under current technical conditions. While the performances of HWPs is a little unequal. But this can be eliminated by choosing HWPs with similar performances in practical experiments.

The above experiment shows an example to solve linear equations with two variables. If such a method is generalized to the case to solve the linear equations with $2^n$variables, we can use the path DoF of $n$ photons to encode the input qubit, the polarization and path DoF of the same $n$ photons to encode the register and ancilla qubits. In such a case, the number of photons required to obtain the expected results is $n$, which is less than $3n+1$. So, our scheme of encoding qubit with multiple DoF of single photon can be used to simplify the realization of the HHL algorithm. Maybe, the integrated photonic chip would be a better choice for multiple path systems. In [25–27], it has shown that the wave plate operations for polarization in integrated chip can be realized. This means that our method can be performed on integrated photonic chip.

4. Conclusion and discussion

In summary, we have experimentally demonstrated an example of the HHL algorithm for solving a $2\times 2$ linear equations with the path and polarization DoFs of photons. The high fidelities have been obtained, which indicate the good performance efficiency of our scheme. In fact, the present method can be extended to solve any general linear equations, because the corresponding quantum circuit can be implemented using the quantum gates shown in Appendices A and B. Compared with the previous experimental implementations, our scheme requires less photons to solve the same problem because multiple DoFs of single photons have been used. Although the path and polarization DoFs of photons have only been used, such an idea is applicable to the general situation. For example, if the DoF of high-dimension orbital angular momentum is introduced to encode the qubits, a more optimized scheme may be obtained. It is expected that more complex quantum information processes can be solved using such an idea.

Appendix A single qubit quantum gate based on path and polarization DoFs

There are three types of single qubit quantum gate required in Fig. 2(a): single qubit Hadamard gate based on the path (Fig. 4(a)), single qubit Hadamard gate based on the polarization (Fig. 4(b)), single qubit X gate based on the polarization (Fig. 4(c)). Here we describe the concrete implementation process for these gates [28].

The single qubit Hadamard gate based on the path is realized by a 50/50 beam splitter with phase shifters at the input and output ports shown in Fig. 4(a). The phase shifter is realized by the microscope slide with adjustable effective thickness. The performance between the input path and output path is expressed as $\left(\begin{array}{c}{|0〉}^{\prime }\\ {|1〉}^{\prime }\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)\left(\begin{array}{c}|0〉\\ |1〉\end{array}\right)$. In the experiment, we use beam splitter to represent this Hadamard gate to simplify the expression. Figures 4(b) and 4(c) show the single qubit Hadamard gate and X-gate based on the polarization, which is realized by the HWP with the optical axis oriented at ${22.5}^{\circ }$and ${45}^{\circ }$respectively. The function of the HWP with the optical axis oriented at $\theta$ on the polarization is $\left(\begin{array}{cc}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array}\right)$.

Appendix B two qubit quantum gates based on the path and polarization DoFs

There are also three types of two qubit quantum gates required in Fig. 2(a): two qubit C-Not gate based on the polarization and path (Fig.5(a)), two qubit $C-H(\pi /8)$ gate based on the path and path (Fig.5(b)), and two qubit $C-H(\theta )$ gate based on the polarization and path (Fig.5(c)), where $H(\theta )=\left(\begin{array}{cc}\cos (2\theta )& \sin (2\theta )\\ \sin (2\theta )& -\cos (2\theta )\end{array}\right)$. The C-Not gate based on path and polarization is realized with a sensing beam splitter for polarization, which is shown in Fig. 5(a). It can be seen that if the input state is horizontally polarized, the state with the path qubit remains. If the input state is vertically polarized, the state with the path qubit will transform from $|0〉\to {|1〉}^{\prime }$ and $|1〉\to {|0〉}^{\prime }$. The $C-H(\pi /8)$ gate based on the path and path is shown in Fig. 5(b). The function of the $H(\pi /8)$ gate is equivalent to the Hadamard gate, which has been demonstrated above. So we keep the path unchanged when the state of the control bit is $|0〉$, which is realized with two mirrors (or even do nothing in the practical experiment). While the state of the control bit is $|1〉$, we use a beam splitter to perform the transformation of $H(\pi /8)$ on the controlled bit. As for the $C-H(\theta )$ gate based on the polarization and path, it is shown in Fig. 5(c). The function of the blue box corresponds to the $C-H(\theta )$ gate. When the input state is horizontally polarized, it will firstly appear at the upper arm. Then the output state on the path DoF remains the same to the input state. When the input state is vertically polarized, it will firstly appear at the lower arm. And the output state undergoes a $H(\theta )$ operation, which is realized by the green HWP with the optical axis oriented at $\theta$ and a PBS. The angle of the blue HWPs are setting as ${45}^{\circ }$.

Appendix C measurement method for the path DoF

In the following, the details of the measurement process shown in Fig. 2(b) is clarified. The measurement of the path DoF on $\left(|0〉,|1〉\right)$ basis is directly obtained because the post selected outputs at Out1 and Out2 are already on the eigenstates $|0〉$and $|1〉$. Thus, the measurement of Pauli observable $Z$ can be realized. The measurements of the path DoF on other Pauli observable $X$ and $Y$ are realized with the module shown in Fig. 6. It is a MZ interferometer with a piezoelectric transducer on one arm to modify the relative phase between the two arms. The function of the piezoelectric transducer on one arm is $PZ=\left(\begin{array}{cc}1& 0\\ 0& e^{i\theta }\end{array}\right)$ . The function of the beam splitter on the path DoF is $H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$. The whole function of the measurement part on the output state is $H\cdot PZ\cdot =\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& e^{i\theta }\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& e^{i\theta }\\ 1& -e^{i\theta }\end{array}\right)$. So the states at the two output ports of this module are $\frac{1}{\sqrt{2}}(|0〉+e^{i\theta }|1〉)$ and $\frac{1}{\sqrt{2}}(|0〉-e^{i\theta }|1〉)$. By setting $\theta =0^{\circ }$, the outputs are $\frac{1}{\sqrt{2}}(|0〉\pm |1〉)$, which are the eigenstates of Pauli observable X. Similarly, by setting $\theta ={90}^{\circ }$, the outputs are $\frac{1}{\sqrt{2}}(|0〉\pm i|1〉)$, which are the eigenstates of Pauli observable $Y$.

 

Fig. 6 The measurement method of the path DoF. The double sided arrow in the measurement module represents the phase adjustment.

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Funding

Ministry of Science and Technology of the People’s Republic of China Key Technologies R&D Program (2017YFA0303800); National Natural Science Foundation of China (11574031 and 61421001).

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