Quantum Algorithms for Near-term Devices

Here we discuss quantum algorithms for the so-called k-local Hamiltonian problem. This is one of the problems that is QMA-complete which, roughly speaking, is the NP-complete analogue for a quantum computer. We discuss exact methods, both classical and quantum, as well as approximate algorithms, more tailored to existing NISQ (noisy, intermediate-scale quantum) devices. We present a roadmap towards building an algorithmic cooling procedure in a NISQ device and we conclude observing some of the challenges that quantum software engineering might encounter in the future.

[1]  Yun-Feng Xiao,et al.  Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics. , 2013, Physical review letters.

[2]  Andrew M. Childs,et al.  Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision , 2015, SIAM J. Comput..

[3]  SCOTT AARONSON,et al.  On the Classical Hardness of Spoofing Linear Cross-Entropy Benchmarking , 2019, ArXiv.

[4]  Ying Li,et al.  Efficient Variational Quantum Simulator Incorporating Active Error Minimization , 2016, 1611.09301.

[5]  M. A. Rol,et al.  Experimental error mitigation via symmetry verification in a variational quantum eigensolver , 2019, Physical Review A.

[6]  John A. Gunnels,et al.  Leveraging Secondary Storage to Simulate Deep 54-qubit Sycamore Circuits , 2019, 1910.09534.

[7]  R. Barends,et al.  Superconducting quantum circuits at the surface code threshold for fault tolerance , 2014, Nature.

[8]  J. Biamonte,et al.  Realizable Hamiltonians for Universal Adiabatic Quantum Computers , 2007, 0704.1287.

[9]  Kristan Temme,et al.  Error Mitigation for Short-Depth Quantum Circuits. , 2016, Physical review letters.

[10]  Leo Zhou,et al.  Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices , 2018, Physical Review X.

[11]  P. Zoller,et al.  Preparation of entangled states by quantum Markov processes , 2008, 0803.1463.

[12]  Catherine C. McGeoch,et al.  Benchmarking a quantum annealing processor with the time-to-target metric , 2015, 1508.05087.

[13]  E. Knill.  Quantum computing with realistically noisy devices , 2005, Nature.

[14]  Guang-Can Guo,et al.  Correction: Corrigendum: Demon-like algorithmic quantum cooling and its realization with quantum optics , 2012, Nature Photonics.

[15]  Tyson Jones,et al.  Quantum compilation and circuit optimisation via energy dissipation , 2018 .

[16]  Craig Gidney,et al.  How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits , 2019, Quantum.

[17]  F. Verstraete,et al.  Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems , 2008, 0907.2796.

[18]  J. Ignacio Cirac,et al.  Faster ground state preparation and high-precision ground energy estimation with fewer qubits , 2017, Journal of Mathematical Physics.

[19]  M. Troyer,et al.  Elucidating reaction mechanisms on quantum computers , 2016, Proceedings of the National Academy of Sciences.

[20]  J. Gambetta,et al.  Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets , 2017, Nature.

[21]  Matthias Troyer,et al.  Solving the quantum many-body problem with artificial neural networks , 2016, Science.

[22]  S. Lloyd,et al.  Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors , 1998, quant-ph/9807070.

[23]  Marcus Huber,et al.  Unifying Paradigms of Quantum Refrigeration: A Universal and Attainable Bound on Cooling. , 2019, Physical review letters.

[24]  Simone Severini,et al.  Quantum machine learning: a classical perspective , 2017, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[25]  Andrew M. Childs,et al.  Simulating Hamiltonian dynamics with a truncated Taylor series. , 2014, Physical review letters.

[26]  M. Hastings,et al.  An area law for one-dimensional quantum systems , 2007, 0705.2024.

[27]  Robert König,et al.  Quantum advantage with shallow circuits , 2017, Science.

[28]  L. Lamata,et al.  Cooling of many-body systems via selective interactions , 2018, Physical Review A.

[29]  M. Hastings,et al.  Gate count estimates for performing quantum chemistry on small quantum computers , 2013, 1312.1695.

[30]  Lei Wang,et al.  Variational quantum eigensolver with fewer qubits , 2019, Physical Review Research.

[31]  Hans-J. Briegel,et al.  Machine learning \& artificial intelligence in the quantum domain , 2017, ArXiv.

[32]  Keisuke Fujii,et al.  Generalization of the Output of a Variational Quantum Eigensolver by Parameter Interpolation with a Low-depth Ansatz , 2018, Physical Review Applied.

[33]  Andrew M. Childs,et al.  Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[34]  F. Verstraete,et al.  Computational complexity of interacting electrons and fundamental limitations of density functional theory , 2007, 0712.0483.

[35]  Alexei Y. Kitaev,et al.  Quantum measurements and the Abelian Stabilizer Problem , 1995, Electron. Colloquium Comput. Complex..

[36]  Marco Tomamichel,et al.  Quantum Advantage with Noisy Shallow Circuits in 3D , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[37]  F. Verstraete,et al.  Matrix product states represent ground states faithfully , 2005, cond-mat/0505140.

[38]  R. Feynman.  Simulating physics with computers , 1999 .

[39]  Travis S. Humble,et al.  Quantum supremacy using a programmable superconducting processor , 2019, Nature.

[40]  Rupak Biswas,et al.  A flexible high-performance simulator for the verification and benchmarking of quantum circuits implemented on real hardware , 2018 .

[41]  S. C. Benjamin,et al.  Fault-tolerant protection of near-term trapped-ion topological qubits under realistic noise sources , 2018, Physical Review A.

[42]  Simon C. Benjamin,et al.  One-dimensional quantum computing with a ‘segmented chain’ is feasible with today’s gate fidelities , 2017, npj Quantum Information.

[43]  F. Verstraete,et al.  Quantum computation and quantum-state engineering driven by dissipation , 2009 .

[44]  John A. Gunnels,et al.  Breaking the 49-Qubit Barrier in the Simulation of Quantum Circuits , 2017, 1710.05867.

[45]  White,et al.  Density-matrix algorithms for quantum renormalization groups. , 1993, Physical review. B, Condensed matter.

[46]  M. Lukin,et al.  Probing many-body dynamics on a 51-atom quantum simulator , 2017, Nature.

[47]  E. Farhi,et al.  A Quantum Approximate Optimization Algorithm , 2014, 1411.4028.

[48]  Andrew S. Dzurak,et al.  Fidelity benchmarks for two-qubit gates in silicon , 2018, Nature.

[49]  S. Matsuura,et al.  VanQver: the variational and adiabatically navigated quantum eigensolver , 2018, New Journal of Physics.

[50]  P. Prelovšek,et al.  Lanczos method for the calculation of finite-temperature quantities in correlated systems. , 1994, Physical review. B, Condensed matter.

[51]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[52]  P. Zoller,et al.  Self-verifying variational quantum simulation of lattice models , 2018, Nature.

[53]  M. Ruskai,et al.  Bounds for the adiabatic approximation with applications to quantum computation , 2006, quant-ph/0603175.

[54]  H. Neven,et al.  Characterizing quantum supremacy in near-term devices , 2016, Nature Physics.

[55]  I. Chuang,et al.  Hamiltonian Simulation by Qubitization , 2016, Quantum.

[56]  A. Wallraff,et al.  Exploring Interacting Quantum Many-Body Systems by Experimentally Creating Continuous Matrix Product States in Superconducting Circuits , 2015, 1508.06471.

[57]  F. Verstraete,et al.  Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions , 2004, cond-mat/0407066.

[58]  Ievgeniia Oshurko.  Quantum Machine Learning , 2020, Quantum Computing.

[59]  M. Lukin,et al.  Quantum optimization of maximum independent set using Rydberg atom arrays , 2018, Science.

[60]  A. Garcia-Saez,et al.  Addressing hard classical problems with Adiabatically Assisted Variational Quantum Eigensolvers , 2018, 1806.02287.