J un 2 01 9 Average-Case Quantum Advantage with Shallow Circuits

Recently Bravyi, Gosset and König (Science 2018) proved an unconditional separation between the computational powers of small-depth quantum and classical circuits for a relation. In this paper we show a similar separation in the average-case setting that gives stronger evidence of the superiority of small-depth quantum computation: we construct a computational task that can be solved on all inputs by a quantum circuit of constant depth with bounded-fanin gates (a “shallow” quantum circuit) and show that any classical circuit with bounded-fanin gates solving this problem on a nonnegligible fraction of the inputs must have logarithmic depth. Our results are obtained by introducing a technique to create quantum states exhibiting global quantum correlations from any graph, via a construction that we call the extended graph. Similar results have been very recently (and independently) obtained by Coudron, Stark and Vidick (arXiv:1810.04233), and Bene Watts, Kothari, Schaeffer and Tal (STOC 2019).

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

[2]  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.

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

[4]  Ran Raz,et al.  Oracle separation of BQP and PH , 2019, Electron. Colloquium Comput. Complex..

[5]  J. Eisert,et al.  Multiparty entanglement in graph states , 2003, quant-ph/0307130.

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

[7]  Robert Spalek,et al.  Quantum Fan-out is Powerful , 2005, Theory Comput..

[8]  Andris Ambainis,et al.  Understanding Quantum Algorithms via Query Complexity , 2017, Proceedings of the International Congress of Mathematicians (ICM 2018).

[9]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[10]  Cristopher Moore,et al.  Counting, fanout and the complexity of quantum ACC , 2002, Quantum Inf. Comput..

[11]  Keisuke Fujii,et al.  On the hardness of classically simulating the one clean qubit model , 2013, Physical review letters.

[12]  Ashley Montanaro,et al.  Achieving quantum supremacy with sparse and noisy commuting quantum computations , 2016, 1610.01808.

[13]  Matthew Coudron,et al.  Trading Locality for Time: Certifiable Randomness from Low-Depth Circuits , 2018, Communications in Mathematical Physics.

[14]  François Le Gall,et al.  Quantum Advantage for the LOCAL Model in Distributed Computing , 2018, STACS.

[15]  Scott Aaronson,et al.  Complexity-Theoretic Foundations of Quantum Supremacy Experiments , 2016, CCC.

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

[17]  John Watrous,et al.  On the power of quantum finite state automata , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[18]  Umesh V. Vazirani,et al.  Quantum Complexity Theory , 1997, SIAM J. Comput..

[19]  Stefano Pironio,et al.  Modeling Pauli measurements on graph states with nearest-neighbor classical communication , 2007 .

[20]  Shuhei Tamate,et al.  Computational quantum-classical boundary of noisy commuting quantum circuits , 2016, Scientific Reports.

[21]  Adam Bouland,et al.  Quantum Supremacy and the Complexity of Random Circuit Sampling , 2018, ITCS.

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

[23]  Scott Aaronson,et al.  Bosonsampling is far from uniform , 2013, Quantum Inf. Comput..

[24]  Avishay Tal,et al.  Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits , 2019, STOC.

[25]  Yasuhiro Takahashi,et al.  Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits , 2011, 2013 IEEE Conference on Computational Complexity.