### Solving Linear Equations on Scalable Superconducting Quantum Computing Chip

**From left to right: Chao-Yang Lu, Jian-Wei Pan, Xiaobo Zhu, H. Wang, Ming-Cheng Chen**

**Authors: Ming-Cheng Chen**

^{1}, H. Wang^{2}, Xiaobo Zhu^{1,3}, Chao-Yang Lu^{1}, Jian-Wei Pan^{1}

**Affiliation:**

^{1}Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China,

^{2}Department of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China,

^{3}Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.

**One-sentence summary:***Quantum solver of linear system is achieved on scalable superconducting quantum computing chip.*

The most ambitious target of quantum computing is to provide both high-efficiency and useful quantum software for killer applications. After decades of intense research on quantum computing, several quantum algorithms are found to demonstrate speedup over their classical counterparts, such as quantum simulation of molecular or condensed system [1], Grover search on unstructured database [2], Shor's period finding to crack RSA cryptography [3], matrix inversion to solve linear systems [4] and sampling from a hard distribution [5, 6]. Among them, sampling from a hard distribution is the most radical example to show the pure quantum computing power beyond the reach of any modern conventional computer and achieve "quantum supremacy" in the near-term [7].

However, the high-efficiency quantum supremacy algorithms have not been found to have practical applications yet. And the Grover search and Shor's period finding algorithms are limited to specific applications. On the contrary, the linear equations quantum algorithm can be applied to almost all areas of science and engineering and recently it finds fascinating applications in data science as a basic subroutine, for instance, in quantum data fitting [8] and quantum support vector machine [9].

Matrix inversion quantum algorithm to solve linear systems [4] is proposed by Harrow, Hassidim and Lloyd (HHL) in 2009 to estimate some features of the solution with exponential speedup. The algorithm uses the celebrated quantum phase estimation technology to force the computation to work at the eigen-basis of system matrix and reduce the matrix inversion to simple eigenvalue reciprocal. A compiled version of the HHL algorithm was previously demonstrated with linear optics [10, 11] and liquid NMR [12] quantum computing platforms, however, both of which are considered not easily scalable to a large number of qubits. Recently, we report the new implementation of the HHL algorithm on solid superconducting quantum circuit system, which is deterministic and easy scalable to large scale.

We run a nontrivial instance of smallest 2×2 system on superconducting circuit chip with four X-shape transmon qubits [13] and tens of one- and two-qubit quantum gates. The chip was fabricated on a sapphire substrate and used aluminum material to define superconducting qubits, resonators and transmission lines. With careful calibration, the single-qubit rotating gates were estimated to be of 98% fidelity and two-qubit entangling gates were of above 95% fidelity. Figure 1 and Figure 2 illustrate the quantum chip and the working quantum circuits, respectively.

**Fig. 1: False color photomicrograph of the superconducting quantum circuit for solving 2×2 linear equations. Shown are the four X-shape transmon qubits, marked from Q1 to Q4, and their corresponding readout resonators.**

**Fig. 2: (**

*click on image to view with higher resolution*) Compiled quantum circuits for solving 2×2 linear equations with four qubits. There are three subroutines and more than 15 gates as indicated.The quantum solver was tested by 18 different input vectors and the corresponding output solution vectors were characterized using quantum state tomography. In our 2*2 instance, the output qubit was measured along X, Y and Z axes of the Bloch sphere, respectively. The estimated quantum state fidelities ranged from 84.0% to 92.3%. The collected data was further used to infer the quantum process matrix of the solver and yielded the process fidelity of 83.7%. Figure 3 and Figure 4 show the experimental quantum state fidelity distribution and the quantum process matrix, respectively.

**Fig. 3: Experimental quantum state fidelity distribution of the output states corresponding to the 18 input states.**

**Fig. 4: The real parts of the experimental quantum process matrix (bars with color) and the ideal quantum process matrix (black frames). All imaginary components (data not shown) of quantum process are measured to be no higher than 0.015 in magnitude.**

These experimental results indicate that our superconducting quantum linear solver for 2*2 system have successfully operated. To scale the solver for more complicated instance with high solution accuracy, further improvement of device design and fabrication to increase quantum bit coherent time and optimization of quantum control pulses to reduce the operating time and error rate are necessary. In superconducting quantum circuit platform, there have been vast efforts devoted to scale the circuit complexity and quality, which have extended the qubit coherent time 5~6 orders of magnitude [14] manipulated up to 10 qubits [15] in the past decades, and we can expect the continuous progress in next decade to reach a mature level.

**References:**

**[1]**I. M. Georgescu, S. Ashhab, Franco Nori, "Quantum simulation". Reviews of Modern Physics, 86, 153 (2014). Abstract.

**[2]**P. Shor, “Algorithms for quantum computation: discrete logarithms and factoring” in Proc. 35th Annu. Symp. on the Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society Press, Los Alamitos, California, 1994), p. 124–134. Abstract.

**[3]**Lov K. Grover, "A Fast Quantum Mechanical Algorithm for Database Search", Proceedings of 28th Annual ACM Symposium on Theory of Computing, pp. 212-219 (1996). Abstract.

**[4]**Aram W. Harrow, Avinatan Hassidim, Seth Lloyd, “Quantum algorithm for linear systems of equations”. Physical Review Letters, 103, 150502 (2009). Abstract.

**[5]**A. P. Lund, Michael J. Bremner, T. C. Ralph, "Quantum sampling problems, BosonSampling and quantum supremacy." NPJ Quantum Information, 3:15 (2017). Abstract.

**[6]**Hui Wang, Yu He, Yu-Huai Li, Zu-En Su, Bo Li, He-Liang Huang, Xing Ding, Ming-Cheng Chen, Chang Liu, Jian Qin, Jin-Peng Li, Yu-Ming He, Christian Schneider, Martin Kamp, Cheng-Zhi Peng, Sven Höfling, Chao-Yang Lu, Jian-Wei Pan, "High-efficiency multiphoton boson sampling". Nature Photonics 11, 361 (2017). Abstract.

**[7]**John Preskill, "Quantum computing and the entanglement frontier". arXiv:1203.5813 (2012).

**[8]**Nathan Wiebe, Daniel Braun, Seth Lloyd, "Quantum algorithm for data fitting". Physical review letters 109, 050505 (2012). Abstract.

**[9]**Patrick Rebentrost, Masoud Mohseni, Seth Lloyd, "Quantum support vector machine for big data classification". Physical review letters 113, 130503 (2014). Abstract.

**[10]**X.-D. Cai, C. Weedbrook, Z.-E. Su, M.-C. Chen, Mile Gu, M.-J. Zhu, Li Li, Nai-Le Liu, Chao-Yang Lu, Jian-Wei Pan, "Experimental quantum computing to solve systems of linear equations". Physical review letters 110, 230501 (2013). Abstract.

**[11]**Stefanie Barz, Ivan Kassal, Martin Ringbauer, Yannick Ole Lipp, Borivoje Dakić, Alán Aspuru-Guzik, Philip Walther, "A two-qubit photonic quantum processor and its application to solving systems of linear equations". Scientific reports 4, 6115 (2014). Abstract.

**[12]**Jian Pan, Yudong Cao, Xiwei Yao, Zhaokai Li, Chenyong Ju, Hongwei Chen, Xinhua Peng, Sabre Kais, Jiangfeng Du, "Experimental realization of quantum algorithm for solving linear systems of equations". Physical Review A 89, 022313 (2004). Abstract.

**[13]**Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer, Alexandre Blais, M. H. Devoret, S. M. Girvin, R. J. Schoelkopf, "Charge-insensitive qubit design derived from the Cooper pair box". Physical Review A 76, 042319 (2007). Abstract.

**[14]**M. H. Devoret, R. J. Schoelkopf1, "Superconducting circuits for quantum information: an outlook". Science, 339, 1169 (2013). Abstract.

**[15]**Chao Song, Kai Xu, Wuxin Liu, Chuiping Yang, Shi-Biao Zheng, Hui Deng, Qiwei Xie, Keqiang Huang, Qiujiang Guo, Libo Zhang, Pengfei Zhang, Da Xu, Dongning Zheng, Xiaobo Zhu, H. Wang, Y.-A. Chen, C.-Y. Lu, Siyuan Han, J.-W. Pan, "10-qubit entanglement and parallel logic operations with a superconducting circuit". arXiv:1703.10302 (2017).