A quantum algorithm for linear differential equations with layerwise parameterized quantum circuits

Junxiang Xiao1, Jingwei Wen1, Zengrong Zhou2, Ling Qian3, Zhiguo Huang3, Shijie Wei2, Gui‐Lu Long2
1State Key Laboratory of Low-dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
2Beijing Academy of Quantum Information Sciences, Beijing, 100193, China
3China Mobile (Suzhou) Software Technology Company Limited, Suzhou, 215163, China

Tóm tắt

AbstractSolving linear differential equations is a common problem in almost all fields of science and engineering. Here, we present a variational algorithm with shallow circuits for solving such a problem: given an $$N \times N$$ N × N matrix $${\varvec{A}}$$ A , an N-dimensional vector $$\varvec{b}$$ b , and an initial vector $$\varvec{x}(0)$$ x ( 0 ) , how to obtain the solution vector $$\varvec{x}(T)$$ x ( T ) at time T according to the constraint $$\textrm{d}\varvec{x}(t)/\textrm{d} t = {\varvec{A}}\varvec{x}(t) + \varvec{b}$$ d x ( t ) / d t = A x ( t ) + b . The core idea of the algorithm is to encode the equations into a ground state problem of the Hamiltonian, which is solved via hybrid quantum-classical methods with high fidelities. Compared with the previous works, our algorithm requires the least qubit resources and can restore the entire evolutionary process. In particular, we show its application in simulating the evolution of harmonic oscillators and dynamics of non-Hermitian systems with $$\mathcal{P}\mathcal{T}$$ P T -symmetry. Our algorithm framework provides a key technique for solving so many important problems whose essence is the solution of linear differential equations.

Từ khóa


Tài liệu tham khảo

J.J. Sakurai, Modern Quantum Mechanics, ed. by S. F. Tuan (Addison-Wesley Publishing Company, Boston, U.S., 1994)

C.I. Trombley, M.L. Ekiel-Jeżewska, Basic Concepts of Stokes Flows (Springer International Publishing, Cham, 2019), pp. 35–50. https://doi.org/10.1007/978-3-030-23370-9_2

J.D. Jackson, Classical electrodynamics (Wiley, New York, 1998)

M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010). https://doi.org/10.1017/CBO9780511976667

D.R. Simon, On the power of quantum computation. SIAM J. Comput. 26(5), 1474–1483 (1997). https://doi.org/10.1137/S0097539796298637

G.L. Long, General quantum interference principle and duality computer. Commun. Theor. Phys. 45(5), 825–844 (2006). https://doi.org/10.1088/0253-6102/45/5/013

G.L. Long, Duality quantum computing and duality quantum information processing. Int. J. Theor. Phys. 50(4), 1305–1318 (2011). https://doi.org/10.1007/s10773-010-0603-z

S. Lloyd, Universal quantum simulators. Science 273(5278), 1073–1078 (1996)

X. Qiang, T. Loke, A. Montanaro, K. Aungskunsiri, X. Zhou, J.L. O’Brien, J.B. Wang, J.C.F. Matthews, Efficient quantum walk on a quantum processor. Nat. Commun. 7(1), 11511 (2016). https://doi.org/10.1038/ncomms11511

L.K. Grover, Quantum computers can search arbitrarily large databases by a single query. Phys. Rev. Lett. 79, 4709–4712 (1997). https://doi.org/10.1103/PhysRevLett.79.4709

G.L. Long, Grover algorithm with zero theoretical failure rate. Phys. Rev. A 64, 022307 (2001). https://doi.org/10.1103/PhysRevA.64.022307

D. Coppersmith, An approximate Fourier transform useful in quantum factoring, Tech. Rep. RC–19642 (IBM Research Division, New York, 1994).

P. Shor, in Proceedings 35th Annual Symposium on Foundations of Computer Science. Algorithms for quantum computation: discrete logarithms and factoring (1994), pp. 124–134. https://doi.org/10.1109/SFCS.1994.365700

P.W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997). https://doi.org/10.1137/S0097539795293172

K. Kim, M.S. Chang, S. Korenblit, R. Islam, E.E. Edwards, J.K. Freericks, G.D. Lin, L.M. Duan, C. Monroe, Quantum simulation of frustrated ising spins with trapped ions. Nature 465(7298), 590–593 (2010). https://doi.org/10.1038/nature09071

X. Zhang, K. Zhang, Y. Shen, S. Zhang, J.N. Zhang, M.H. Yung, J. Casanova, J.S. Pedernales, L. Lamata, E. Solano, K. Kim, Experimental quantum simulation of fermion-antifermion scattering via boson exchange in a trapped ion. Nat. Commun. 9(1), 195 (2018). https://doi.org/10.1038/s41467-017-02507-y

J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, R. Han, Experimental implementation of the quantum random-walk algorithm. Phys. Rev. A 67, 042316 (2003). https://doi.org/10.1103/PhysRevA.67.042316

H. Tang, X.F. Lin, Z. Feng, J.Y. Chen, J. Gao, K. Sun, C.Y. Wang, P.C. Lai, X.Y. Xu, Y. Wang, L.F. Qiao, A.L. Yang, X.M. Jin, Experimental two-dimensional quantum walk on a photonic chip. Sci. Adv. 4(5), eaat3174 (2018). https://doi.org/10.1126/sciadv.aat3174

I.L. Chuang, N. Gershenfeld, M. Kubinec, Experimental implementation of fast quantum searching. Phys. Rev. Lett. 80, 3408–3411 (1998). https://doi.org/10.1103/PhysRevLett.80.3408

J.A. Jones, M. Mosca, R.H. Hansen, Implementation of a quantum search algorithm on a quantum computer. Nature 393(6683), 344–346 (1998). https://doi.org/10.1038/30687

P.G. Kwiat, J.R. Mitchell, P.D.D. Schwindt, A.G. White, Grover’s search algorithm: an optical approach. J. Mod. Opt. 47(2–3), 257–266 (2000). https://doi.org/10.1080/09500340008244040

J.L. Dodd, T.C. Ralph, G.J. Milburn, Experimental requirements for grover’s algorithm in optical quantum computation. Phys. Rev. A 68, 042328 (2003). https://doi.org/10.1103/PhysRevA.68.042328

A. Mandviwalla, K. Ohshiro, B. Ji, in 2018 IEEE International Conference on Big Data (Big Data). Implementing grover’s algorithm on the ibm quantum computers (2018), pp. 2531–2537. https://doi.org/10.1109/BigData.2018.8622457

C.Y. Lu, D.E. Browne, T. Yang, J.W. Pan, Demonstration of a compiled version of shor’s quantum factoring algorithm using photonic qubits. Phys. Rev. Lett. 99, 250504 (2007). https://doi.org/10.1103/PhysRevLett.99.250504

B.P. Lanyon, T.J. Weinhold, N.K. Langford, M. Barbieri, D.F.V. James, A. Gilchrist, A.G. White, Experimental demonstration of a compiled version of shor’s algorithm with quantum entanglement. Phys. Rev. Lett. 99, 250505 (2007). https://doi.org/10.1103/PhysRevLett.99.250505

E. Martín-López, A. Laing, T. Lawson, R. Alvarez, X.Q. Zhou, J.L. O’Brien, Experimental realization of shor’s quantum factoring algorithm using qubit recycling. Nat. Photonics 6(11), 773–776 (2012). https://doi.org/10.1038/nphoton.2012.259

B. Lu, L. Liu, J.Y. Song, K. Wen, C. Wang, Recent progress on coherent computation based on quantum squeezing. AAPPS Bull. 33(1), 7 (2023). https://doi.org/10.1007/s43673-023-00077-4

F. Zhang, J. Xing, X. Hu, X. Pan, G. Long, Coupling-selective quantum optimal control in weak-coupling nv-13c system. AAPPS Bull. 33(1), 2 (2023). https://doi.org/10.1007/s43673-022-00072-1

A.W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150502 (2009). https://doi.org/10.1103/PhysRevLett.103.150502

S. Wei, Z. Zhou, D. Ruan, G. Long, in 2017 IEEE 85th Vehicular Technology Conference (VTC Spring). Realization of the algorithm for system of linear equations in duality quantum computing (2017), pp. 1–4. https://doi.org/10.1109/VTCSpring.2017.8108698

S. Barz, I. Kassal, M. Ringbauer, Y.O. Lipp, B. Dakić, A. Aspuru-Guzik, P. Walther, A two-qubit photonic quantum processor and its application to solving systems of linear equations. Sci. Rep. 4(1), 6115 (2014). https://doi.org/10.1038/srep06115

J. Pan, Y. Cao, X. Yao, Z. Li, C. Ju, H. Chen, X. Peng, S. Kais, J. Du, Experimental realization of quantum algorithm for solving linear systems of equations. Phys. Rev. A 89, 022313 (2014). https://doi.org/10.1103/PhysRevA.89.022313

X.D. Cai, C. Weedbrook, Z.E. Su, M.C. Chen, M. Gu, M.J. Zhu, L. Li, N.L. Liu, C.Y. Lu, J.W. Pan, Experimental quantum computing to solve systems of linear equations. Phys. Rev. Lett. 110, 230501 (2013). https://doi.org/10.1103/PhysRevLett.110.230501

J. Wen, X. Kong, S. Wei, B. Wang, T. Xin, G. Long, Experimental realization of quantum algorithms for a linear system inspired by adiabatic quantum computing. Phys. Rev. A 99, 012320 (2019). https://doi.org/10.1103/PhysRevA.99.012320

D.W. Berry, High-order quantum algorithm for solving linear differential equations. J. Phys. A Math. Theor. 47(10), 105301 (2014). https://doi.org/10.1088/1751-8113/47/10/105301

D.W. Berry, A.M. Childs, A. Ostrander, G. Wang, Quantum algorithm for linear differential equations with exponentially improved dependence on precision. Commun. Math. Phys. 356(3), 1057–1081 (2017). https://doi.org/10.1007/s00220-017-3002-y

T. Xin, S. Wei, J. Cui, J. Xiao, I.n. Arrazola, L. Lamata, X. Kong, D. Lu, E. Solano, G. Long, Quantum algorithm for solving linear differential equations: theory and experiment. Phys. Rev. A 101, 032307 (2020). https://doi.org/10.1103/PhysRevA.101.032307

J.W.Z. Lau, K.H. Lim, H. Shrotriya, L.C. Kwek, Nisq computing: where are we and where do we go? AAPPS Bull. 32(1), 27 (2022). https://doi.org/10.1007/s43673-022-00058-z

A. Peruzzo, J. McClean, P. Shadbolt, M.H. Yung, X.Q. Zhou, P.J. Love, A. Aspuru-Guzik, J.L. O’Brien, A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5(1), 4213 (2014). https://doi.org/10.1038/ncomms5213

J.R. McClean, J. Romero, R. Babbush, A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18(2), 023023 (2016). https://doi.org/10.1088/1367-2630/18/2/023023

A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J.M. Chow, J.M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549(7671), 242–246 (2017). https://doi.org/10.1038/nature23879

I.G. Ryabinkin, T.C. Yen, S.N. Genin, A.F. Izmaylov, Qubit coupled cluster method: A systematic approach to quantum chemistry on a quantum computer. J Chem. Theory Comput. 14(12), 6317–6326 (2018). https://doi.org/10.1021/acs.jctc.8b00932

X. Xu, J. Sun, S. Endo, Y. Li, S.C. Benjamin, X. Yuan, Variational algorithms for linear algebra. Sci. Bull. 66, 2181 (2021). https://doi.org/10.1016/j.scib.2021.06.023

C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, P.J. Coles, Variational quantum linear solver. Quantum 7, 1188 (2023). https://doi.org/10.22331/q-2023-11-22-1188

K. Mitarai, M. Negoro, M. Kitagawa, K. Fujii, Quantum circuit learning. Phys. Rev. A 98, 032309 (2018). https://doi.org/10.1103/PhysRevA.98.032309

A. Skolik, J.R. McClean, M. Mohseni, P. van der Smagt, M. Leib, Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3 (2021). https://doi.org/10.1007/s42484-020-00036-4

S. Wei, Y. Chen, Z. Zhou, G. Long, A quantum convolutional neural network on nisq devices. AAPPS Bull. 32(1), 2 (2022). https://doi.org/10.1007/s43673-021-00030-3

S. Endo, J. Sun, Y. Li, S.C. Benjamin, X. Yuan, Variational quantum simulation of general processes. Phys. Rev. Lett. 125, 010501 (2020). https://doi.org/10.1103/PhysRevLett.125.010501

H.L. Liu, Y.S. Wu, L.C. Wan, S.J. Pan, S.J. Qin, F. Gao, Q.Y. Wen, Variational quantum algorithm for the poisson equation. Phys. Rev. A 104, 022418 (2021). https://doi.org/10.1103/PhysRevA.104.022418

P. García-Molina, J. Rodríguez-Mediavilla, J.J. García-Ripoll, Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations. Phys. Rev. A 105, 012433 (2022). https://link.aps.org/doi/10.1103/PhysRevA.105.012433

“Numerical differential equation methods,” in Numerical methods for ordinary differential equations (Wiley, Chichester, 2008) Chap. 2, pp. 51–135

R.D. Subaşı, D. Somma, Orsucci, Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing. Phys. Rev. Lett. 122, 060504 (2019). https://doi.org/10.1103/PhysRevLett.122.060504

M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, N. Killoran, Evaluating analytic gradients on quantum hardware. Phys. Rev. A 99, 032331 (2019). https://doi.org/10.1103/PhysRevA.99.032331

J. Li, X. Yang, X. Peng, C.P. Sun, Hybrid quantum-classical approach to quantum optimal control. Phys. Rev. Lett. 118, 150503 (2017). https://doi.org/10.1103/PhysRevLett.118.150503

T. Fösel, M.Y. Niu, F. Marquardt, L. Li. Quantum circuit optimization with deep reinforcement learning (2021). arXiv:2103.07585 [quant-ph]

V.V. Konotop, J. Yang, D.A. Zezyulin, Nonlinear waves in $$\cal{PT}$$-symmetric systems. Rev. Mod. Phys. 88, 035002 (2016). https://doi.org/10.1103/RevModPhys.88.035002

T.J. Milburn, J. Doppler, C.A. Holmes, S. Portolan, S. Rotter, P. Rabl, General description of quasiadiabatic dynamical phenomena near exceptional points. Phys. Rev. A 92, 052124 (2015). https://doi.org/10.1103/PhysRevA.92.052124

Y.C. Lee, M.H. Hsieh, S.T. Flammia, R.K. Lee, Local $$\cal{P} \cal{T}$$ symmetry violates the no-signaling principle. Phys. Rev. Lett. 112, 130404 (2014). https://doi.org/10.1103/PhysRevLett.112.130404

A. Guo, G.J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G.A. Siviloglou, D.N. Christodoulides, Observation of $$\cal{P} \cal{T}$$-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009). https://doi.org/10.1103/PhysRevLett.103.093902

K. Ding, G. Ma, Z.Q. Zhang, C.T. Chan, Experimental demonstration of an anisotropic exceptional point. Phys. Rev. Lett. 121, 085702 (2018). https://doi.org/10.1103/PhysRevLett.121.085702

S.L. Chen, G.Y. Chen, Y.N. Chen, Increase of entanglement by local $$\cal{PT}$$-symmetric operations. Phys. Rev. A 90, 054301 (2014). https://doi.org/10.1103/PhysRevA.90.054301

J. Wen, C. Zheng, X. Kong, S. Wei, T. Xin, G. Long, Experimental demonstration of a digital quantum simulation of a general $$\cal{PT}$$-symmetric system. Phys. Rev. A 99, 062122 (2019). https://doi.org/10.1103/PhysRevA.99.062122

J. Wen, C. Zheng, Z. Ye, T. Xin, G. Long, Stable states with nonzero entropy under broken $$\cal{PT}$$ symmetry. Phys. Rev. Res. 3, 013256 (2021). https://doi.org/10.1103/PhysRevResearch.3.013256

W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998). https://doi.org/10.1103/PhysRevLett.80.2245

S. McArdle, T. Jones, S. Endo, Y. Li, S.C. Benjamin, X. Yuan, Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Inf. 5(1), 75 (2019). https://doi.org/10.1038/s41534-019-0187-2

R.H. Byrd, P. Lu, J. Nocedal, C. Zhu, A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16(5), 1190–1208 (1995). https://doi.org/10.1137/0916069

J. Nocedal, C. Zhu, R. Byrd, P. Lu, Algorithm 778: L-bfgs-b, fortran routines for large scale bound constrained optimization. ACM Trans. Math. Softw. 23(4), 550–560 (1997)