Beating classical heuristics for the binary paint shop problem with the quantum approximate optimization algorithm.

The binary paint shop problem (BPSP) is an APX-hard optimization problem of the automotive industry. In this work, we show how to use the Quantum Approximate Optimization Algorithm (QAOA) to find solutions of the BPSP and demonstrate that QAOA with constant depth is able to beat classical heuristics on average in the infinite size limit $n\rightarrow\infty$. For the BPSP, it is known that no classical algorithm can exist which approximates the problem in polynomial runtime. We introduce a BPSP instance which is hard to solve with QAOA, and numerically investigate its performance and discuss QAOA's ability to generate approximate solutions. We complete our studies by running first experiments of small-sized instances on a trapped-ion quantum computer through AWS Braket.

[1]  Keisuke Fujii,et al.  Quantum circuit learning , 2018, Physical Review A.

[2]  Seth Lloyd,et al.  Continuous-variable quantum neural networks , 2018, Physical Review Research.

[3]  Andrew Lucas,et al.  Ising formulations of many NP problems , 2013, Front. Physics.

[4]  Martin Leib,et al.  Forbidden subspaces for level-1 quantum approximate optimization algorithm and instantaneous quantum polynomial circuits , 2020 .

[5]  M. B. Hastings,et al.  Classical and quantum bounded depth approximation algorithms , 2019, Quantum Inf. Comput..

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

[7]  Aniruddha Bapat,et al.  Bang-bang control as a design principle for classical and quantum optimization algorithms , 2018, Quantum Inf. Comput..

[8]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[9]  Winfried Hochstättler,et al.  Some Results on a Paint Shop Problem for Words , 2001, Electron. Notes Discret. Math..

[10]  W. W. Ho,et al.  Efficient variational simulation of non-trivial quantum states , 2018, SciPost Physics.

[11]  Sabrina Hong,et al.  Demonstration of universal parametric entangling gates on a multi-qubit lattice , 2017, Science Advances.

[12]  Robert Koenig,et al.  Obstacles to Variational Quantum Optimization from Symmetry Protection. , 2019, Physical review letters.

[13]  Hans De Raedt,et al.  Benchmarking the quantum approximate optimization algorithm , 2020, Quantum Inf. Process..

[14]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

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

[16]  David Gamarnik,et al.  The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: A Typical Case , 2020, ArXiv.

[17]  David Gosset,et al.  Classical algorithms for quantum mean values , 2019, Nature Physics.

[18]  Stephan Dominique Andres,et al.  Some heuristics for the binary paint shop problem and their expected number of colour changes , 2011, J. Discrete Algorithms.

[19]  Gavin E. Crooks,et al.  Performance of the Quantum Approximate Optimization Algorithm on the Maximum Cut Problem , 2018, 1811.08419.

[20]  M. Leib,et al.  Comparison of QAOA with Quantum and Simulated Annealing , 2019, 1901.01903.

[21]  Martin Leib,et al.  Training the quantum approximate optimization algorithm without access to a quantum processing unit , 2019, Quantum Science and Technology.

[22]  J. D. Wong-Campos,et al.  Benchmarking an 11-qubit quantum computer , 2019, Nature Communications.

[23]  W. L. Tan,et al.  Quantum Approximate Optimization with a Trapped-Ion Quantum Simulator , 2019, 1906.02700.

[24]  S. Debnath,et al.  Demonstration of a small programmable quantum computer with atomic qubits , 2016, Nature.

[25]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[26]  E. Farhi,et al.  A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem , 2014, 1412.6062.

[27]  Alán Aspuru-Guzik,et al.  A variational eigenvalue solver on a photonic quantum processor , 2013, Nature Communications.

[28]  Bertrand Neveu,et al.  Computing solutions of the paintshop-necklace problem , 2012, Comput. Oper. Res..

[29]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[30]  Dmitri Maslov,et al.  Basic circuit compilation techniques for an ion-trap quantum machine , 2016, ArXiv.

[31]  L. DiCarlo,et al.  Demonstration of two-qubit algorithms with a superconducting quantum processor , 2009, Nature.

[32]  F. Brandão,et al.  For Fixed Control Parameters the Quantum Approximate Optimization Algorithm's Objective Function Value Concentrates for Typical Instances , 2018, 1812.04170.

[33]  E. Rieffel,et al.  Near-optimal quantum circuit for Grover's unstructured search using a transverse field , 2017, 1702.02577.

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

[35]  Masoud Mohseni,et al.  Quantum approximate optimization of non-planar graph problems on a planar superconducting processor , 2020, 2004.04197.

[36]  Baruch Schieber,et al.  The Approximability of the Binary Paintshop Problem , 2013, APPROX-RANDOM.

[37]  Winfried Hochstättler,et al.  Complexity results on a paint shop problem , 2004, Discret. Appl. Math..

[38]  A. Harrow,et al.  Quantum Supremacy through the Quantum Approximate Optimization Algorithm , 2016, 1602.07674.