Quantum Supremacy through the Quantum Approximate Optimization Algorithm

The Quantum Approximate Optimization Algorithm (QAOA) is designed to run on a gate model quantum computer and has shallow depth. It takes as input a combinatorial optimization problem and outputs a string that satisfies a high fraction of the maximum number of clauses that can be satisfied. For certain problems the lowest depth version of the QAOA has provable performance guarantees although there exist classical algorithms that have better guarantees. Here we argue that beyond its possible computational value the QAOA can exhibit a form of Quantum Supremacy in that, based on reasonable complexity theoretic assumptions, the output distribution of even the lowest depth version cannot be efficiently simulated on any classical device. We contrast this with the case of sampling from the output of a quantum computer running the Quantum Adiabatic Algorithm (QADI) with the restriction that the Hamiltonian that governs the evolution is gapped and stoquastic. Here we show that there is an oracle that would allow sampling from the QADI but even with this oracle, if one could efficiently classically sample from the output of the QAOA, the Polynomial Hierarchy would collapse. This suggests that the QAOA is an excellent candidate to run on near term quantum computers not only because it may be of use for optimization but also because of its potential as a route to establishing quantum supremacy.

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

[2]  Lane A. Hemaspaandra,et al.  Threshold Computation and Cryptographic Security , 1993, ISAAC.

[3]  H. Nishimori,et al.  Quantum annealing in the transverse Ising model , 1998, cond-mat/9804280.

[4]  Anthony Chefles,et al.  Unambiguous discrimination between linearly independent quantum states , 1998, quant-ph/9807022.

[5]  Jozef Gruska,et al.  Quantum Computing , 2008, Wiley Encyclopedia of Computer Science and Engineering.

[6]  Stephen A. Fenner,et al.  Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  M. Sipser,et al.  Quantum Computation by Adiabatic Evolution , 2000, quant-ph/0001106.

[8]  D. Aharonov.  A Simple Proof that Toffoli and Hadamard are Quantum Universal , 2003, quant-ph/0301040.

[9]  David P. DiVincenzo,et al.  Adaptive quantum computation, constant depth quantum circuits and arthur-merlin games , 2002, Quantum Inf. Comput..

[10]  Seth Lloyd,et al.  Adiabatic quantum computation is equivalent to standard quantum computation , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[11]  Scott Aaronson,et al.  Quantum computing, postselection, and probabilistic polynomial-time , 2004, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[12]  David P. DiVincenzo,et al.  The complexity of stoquastic local Hamiltonian problems , 2006, Quantum Inf. Comput..

[13]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[14]  R. Jozsa,et al.  Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[15]  Todd A. Brun,et al.  Quantum Computing , 2011, Computer Science, The Hardware, Software and Heart of It.

[16]  Scott Aaronson,et al.  The computational complexity of linear optics , 2010, STOC '11.

[17]  M. W. Johnson,et al.  Quantum annealing with manufactured spins , 2011, Nature.

[18]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[19]  P. Shor,et al.  Performance of the quantum adiabatic algorithm on random instances of two optimization problems on regular hypergraphs , 2012, 1208.3757.

[20]  John Preskill,et al.  Quantum computing and the entanglement frontier , 2012, 1203.5813.

[21]  M. W. Johnson,et al.  Thermally assisted quantum annealing of a 16-qubit problem , 2013, Nature Communications.

[22]  Scott Aaronson.  The Equivalence of Sampling and Searching , 2013, Theory of Computing Systems.

[23]  Matthew B. Hastings,et al.  Obstructions to classically simulating the quantum adiabatic algorithm , 2013, Quantum Inf. Comput..

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

[25]  M. W. Johnson,et al.  Entanglement in a Quantum Annealing Processor , 2014, 1401.3500.

[26]  Daniel A. Lidar,et al.  Evidence for quantum annealing with more than one hundred qubits , 2013, Nature Physics.

[27]  Daniel A. Lidar,et al.  Defining and detecting quantum speedup , 2014, Science.

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

[29]  Prasad Raghavendra,et al.  Beating the random assignment on constraint satisfaction problems of bounded degree , 2015, Electron. Colloquium Comput. Complex..

[30]  Ryan Babbush,et al.  What is the Computational Value of Finite Range Tunneling , 2015, 1512.02206.

[31]  Saeed Mehraban,et al.  Computational Complexity of Some Quantum Theories in $1+1$ Dimensions , 2015, ArXiv.

[32]  Daniel J. Brod,et al.  Complexity of simulating constant-depth BosonSampling , 2014, 1412.6788.

[33]  Aram Wettroth Harrow,et al.  Simulated Quantum Annealing Can Be Exponentially Faster Than Classical Simulated Annealing , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[34]  Vasil S. Denchev,et al.  Computational multiqubit tunnelling in programmable quantum annealers , 2015, Nature Communications.

[35]  Ashley Montanaro,et al.  Average-case complexity versus approximate simulation of commuting quantum computations , 2015, Physical review letters.