Weakly measured while loops: peeking at quantum states

A while loop tests a termination condition on every iteration. On a quantum computer, such measurements perturb the evolution of the algorithm. We define a while loop primitive using weak measurements, offering a trade-off between the perturbation caused and the amount of information gained per iteration. This trade-off is adjusted with a parameter set by the programmer. We provide sufficient conditions that let us determine, with arbitrarily high probability, a worst-case estimate of the number of iterations the loop will run for. As an example, we solve Grover's search problem using a while loop and prove the quadratic quantum speed-up is maintained.

[1]  Seth Lloyd,et al.  Quantum feedback with weak measurements , 1999, quant-ph/9905064.

[2]  Jonathan Grattage.  A functional quantum programming language , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[3]  Prakash Panangaden,et al.  Quantum Alternation: Prospects and Problems , 2015, QPL.

[4]  Yuan Feng,et al.  Quantum loop programs , 2010, Acta Informatica.

[5]  Paulo E. M. F. Mendonca,et al.  Quantum Control of a Single Qubit , 2006, quant-ph/0608037.

[6]  Aleksandrs Belovs,et al.  Quantum Walks and Electric Networks , 2013, 1302.3143.

[7]  S. Popescu,et al.  The Halting Problem for Quantum Computers , 1998, quant-ph/9806054.

[8]  Yuan Feng,et al.  Alternation in Quantum Programming: From Superposition of Data to Superposition of Programs , 2014, ArXiv.

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

[10]  Juliana Kaizer Vizzotto,et al.  From Symmetric Pattern-Matching to Quantum Control , 2018, FoSSaCS.

[11]  Franco Nori,et al.  Quantum feedback: theory, experiments, and applications , 2014, 1407.8536.

[12]  K J Resch,et al.  Experimental feedback control of quantum systems using weak measurements. , 2009, Physical review letters.

[13]  Andris Ambainis,et al.  Quadratic speedup for finding marked vertices by Quantum walks , 2019, STOC.

[14]  Theodore J. Yoder,et al.  Fixed-point quantum search with an optimal number of queries. , 2014, Physical review letters.

[15]  Ari Mizel,et al.  Critically damped quantum search. , 2008, Physical review letters.

[16]  G. Brassard,et al.  Quantum Amplitude Amplification and Estimation , 2000, quant-ph/0005055.

[17]  Mingsheng Ying,et al.  Algorithmic analysis of termination problems for quantum programs , 2017, Proc. ACM Program. Lang..

[18]  Peter Selinger,et al.  Towards a quantum programming language , 2004, Mathematical Structures in Computer Science.

[19]  Kurt Jacobs,et al.  A straightforward introduction to continuous quantum measurement , 2006, quant-ph/0611067.

[20]  Mingsheng Ying,et al.  Foundations of Quantum Programming , 2016 .

[21]  Lov K. Grover,et al.  Fixed-point quantum search. , 2005, Physical review letters.

[22]  Simon J. Gay,et al.  Quantum Programming Languages Survey and Bibliography , 2006 .

[23]  Todd A. Brun,et al.  A simple model of quantum trajectories , 2002 .

[24]  Yuan Feng,et al.  Quantum Hoare logic with classical variables , 2020, ACM Transactions on Quantum Computing.

[25]  M. Szegedy,et al.  Quantum Walk Based Search Algorithms , 2008, TAMC.