Complexity for 1 D discrete time quantum walk circuits Aranya Bhattacharya 1 , Himanshu Sahu 2 , Ahmadullah Zahed 3 and Kallol Sen 4† 1,3 Center for High Energy Physics, Indian Institute of Science, C.V. Raman Avenue, Bangalore 560012, India 2 Dept. of Instrumentation & Applied Physics, Indian Institute of Sciences, C.V. Raman Avenue, Bengaluru 560012, India 4 ICTP-South American Instutute of Fundamental Research, IFT-UNESP (1º andar), Rua Dr. Bento Teobaldo Ferraz 271, Bloco 2 - Barra Funda 01140-070 São Paulo, SP Brazil † kallolmax@gmail.com Abstract We compute the complexity for the mixed state density operator derived from a one- dimensional discrete-time quantum walk (DTQW). The complexity is computed using a 2-qubit quantum circuit obtained from canonically purifying the mixed state. We demon- strate that the Nielson complexity for the unitary evolution oscillates around a mean circuit depth of k. Further, the complexity of the step-wise evolution operator grows cu- mulatively and linearly with the steps. From a quantum circuit perspective, this implies a succession of circuits of (near) constant depth to be applied to reach the final state. Contents 1 Introduction 2 2 1D Discrete Time Quantum Walk 4 3 Target Unitary Operator 6 4 Complexity 6 4.1 k = 1 8 4.2 k = 2 8 4.3 k = 3 9 4.4 Two definitions of complexity 9 4.4.1 Direct complexity 10 4.4.2 Stepwise complexity 11 4.5 Fermionic Hamiltonian in continuum limit 13 5 Quantum circuit 13 6 Discussions 14 7 Acknowledgements 16 A Details of generators 16 References 17 1 arXiv:2307.13450v1 [quant-ph] 25 Jul 2023