Robust Implementation of Discrete-time Quantum Walks in Any Finite-dimensional Quantum System

TL;DR

We address efficient, ancilla-free DTQW implementation in finite-dimensional quantum systems. The authors introduce an Enhanced Increment-Decrement (EID) approach in binary qubit registers, using separate even and odd shift unitaries to halve circuit depth and gate count while preserving universality, and they extend the scheme to higher-dimensional qudits via intermediate-qudit decomposition. They provide mathematical formulations for $U_{even}$ and $U_{odd}$ and demonstrate 3- and 4-qubit DTQWs with experimental IBM Qiskit simulations, showing improved resource metrics and higher success probabilities compared with naive methods. They further generalize Toffoli decompositions to N-controlled gates using intermediate qudits (dimensions $d$ and $d+1$ or $d+1$ and $d+2$), enabling scalable, ancilla-free decompositions and enabling DTQW on higher-dimensional lattices. The work culminates in a practical, scalable DTQW framework with open-source code and potential impact on quantum simulations and quantum algorithms.

Abstract

Research has shown that quantum walks can accelerate certain quantum algorithms and act as a universal paradigm for quantum processing. The discrete-time quantum walk (DTQW) model, owing to its discrete nature, stands out as one of the most suitable choices for circuit implementation. Nevertheless, most current implementations are characterized by extensive, multi-layered quantum circuits, leading to higher computational expenses and a notable decrease in the number of confidently executable time steps on current quantum computers. Since quantum computers are not scalable enough in this NISQ era, we also must confine ourselves to the ancilla-free frontier zone. Therefore, in this paper, we have successfully cut down the circuit cost concerning gate count and circuit depth by half through our proposed methodology in qubit systems as compared to the state-of-the-art increment-decrement approach. Furthermore, for the engineering excellence of our proposed approach, we implement DTQW in any finite-dimensional quantum system with akin efficiency. To ensure an efficient implementation of quantum walks without requiring ancilla, we have incorporated an intermediate qudit technique for decomposing multi-qubit gates. Experimental outcomes hold significance far beyond the realm of just a few time steps, laying the groundwork for dependable implementation and utilization on quantum computers.

Figures (18)

• Figure 1: 2-controlled Toffoli decomposition using Clifford+T gate set
• Figure 2: The circuit on the left, is the increment circuit and the circuit on the right is the decrement circuit. After a coin toss, both circuits need to be employed. We need an increment for a rightwards shift and a decrement for a leftwards shift.
• Figure 3: Naive circuit for a single step of DTQW.
• Figure 4: Circuit implementing DTQW in a 3-qubit system. The circuit on the left is for when the walker is in some even position state and the circuit on the right is for when the walker is in some odd position state.
• Figure 5: Generalized circuit diagram of our proposed enhanced increment decrement quantum walk for $n$ qubits. The circuit on the left is to be used when the walker is in some even position state and the circuit on the right is to be used in case the walker on some odd position state.