Deutsch-Jozsa and Bernstein-Vazirani algorithm using single-particle discrete-time quantum walk

TL;DR

The paper tackles efficient quantum implementations of the Deutsch-Jozsa and Bernstein-Vazirani algorithms using a single-particle discrete-time quantum walk (DTQW) on a photonic platform. It develops two DTQW-based schemes for each algorithm—one with an auxiliary qubit and one without—employing path and polarization encoding to realize the required oracles and a universal gate set, while preserving the algorithms’ characteristic speedups. Explicit optical architectures are provided, detailing waveplates, beam splitters, phase shifters, and polarization-dependent components, and a resource-analysis shows the auxiliary-qubit-free scheme incurs fewer components and gates. The work advances scalable, low-overhead photonic quantum computing via DTQW and substantiates the potential for universal quantum computation using single-particle quantum walks on integrated photonic platforms.

Abstract

The paper introduces an efficient implementation of the Deutsch-Jozsa and Bernstein-Vazirani algorithm using the single-particle discrete-time quantum walk. We also provide a detailed optical framework with specific optical components to achieve these implementations in the photonic quantum walk scheme by simultaneously exploiting both polarization and path degrees of freedom. These implementations demonstrate improved resource efficiency while maintaining the exponential speedup characteristic of both algorithms. This work contributes to the growing field of universal quantum computing using single particle discrete-time quantum walk.

Figures (7)

• Figure 1: Oracle implementations for all two-bit Boolean functions that are either constant or balanced given in Table \ref{['tab:two_bit_functions']}, by using $X$ and CNOT gates in standard circuit model with auxiliary qubit Qniverse. The $q_a$, $q_1$, and $q_2$ represents auxiliary, first working and second working qubit, respectively.
• Figure 2: Schematic illustration of the quantum walk based oracle implementations with auxiliary qubit for all two-bit boolean functions that are either constant or balanced given in Table \ref{['tab:two_bit_functions']}. Here, $\mathbb{I}$ represents the identity coin operator together with an identity shift operator identity shift operator ($\hat{C}( 0, 0, 0, 0) \otimes \hat{I}$).
• Figure 3: Schematic illustration of the photonic implementation of the operations shown in Fig. \ref{['fig:dj_oracle_two_qubit_qw']}. Here, the four path degrees ($|00\rangle,|10\rangle,|01\rangle, \text{and}~|11\rangle$) of freedom represent two working qubits and the polarization state of the photon repersents the auxiliary qubit.
• Figure 4: The photonic circuit for the photonic quantum walk based two-qubit Deutsch-Jozsa algorithm using auxiliary qubit for the function ($f(x_1,x_2)=0$). The four path degrees ($|00\rangle,|01\rangle,|11\rangle, \text{and}~|10\rangle$) of freedom represent two working qubits and the polarization state represents the auxiliary qubit. Here, (a), (b), (c), and (d) boxes represent $X$ gate on polarization state (auxiliary qubit), $H$ gate on all the qubits, oracle, and $H$ gate on two working qubits, respectively.
• Figure 5: Schematic illustration of the quantum walk based oracle implementations without auxiliary qubit for all two-bit boolean functions that are either constant or balanced given in Table \ref{['tab:two_bit_functions']}, by using identity operation $\mathbb{I}$ and three position dependent evolution operators ($\hat{O}_1, \hat{O}_2,\hat{O}_3$) as defined in \ref{['eq:operators_twoqubit_dj_usingauxiliary_qw']}.