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arXiv:2402.01854 · Entropy

Efficient Implementation of Discrete-Time Quantum Walks on Quantum Computers

Luca Razzoli, Gabriele Cenedese, Maria Bondani, Giuliano Benenti

Published Apr 2, 2024• 16 citations
arXivHTMLPDF

Abstract

Quantum walks have proven to be a universal model for quantum computation and to provide speed-up in certain quantum algorithms. The discrete-time quantum walk (DTQW) model, among others, is one of the most suitable candidates for circuit implementation due to its discrete nature. Current implementations, however, are usually characterized by quantum circuits of large size and depth, which leads to a higher computational cost and severely limits the number of time steps that can be reliably implemented on current quantum computers. In this work, we propose an efficient and scalable quantum circuit implementing the DTQW on the 2n-cycle based on the diagonalization of the conditional shift operator. For t time steps of the DTQW, the proposed circuit requires only O(n2+nt) two-qubit gates compared to the O(n2t) of the current most efficient implementation based on quantum Fourier transforms. We test the proposed circuit on an IBM quantum device for a Hadamard DTQW on the 4-cycle and 8-cycle characterized by periodic dynamics and by recurrent generation of maximally entangled single-particle states. Experimental results are meaningful well beyond the regime of few time steps, paving the way for reliable implementation and use on quantum computers.

DOI
10.3390/e26040313

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Quantum walks have proven to be a universal model for quantum computation and to provide speed-up in certain quantum algorithms. The discrete-time quantum walk (DTQW) model, among others, is one of the most suitable candidates for circuit implementation due to its discrete nature. Current implementations, however, are usually characterized by quantum circuits of large size and depth, which leads to a higher computational cost and severely limits the number of time steps that can be reliably implemented on current quantum computers. In this work, we propose an efficient and scalable quantum circuit implementing the DTQW on the 2n-cycle based on the diagonalization of the conditional shift operator. For t time steps of the DTQW, the proposed circuit requires only O(n2+nt) two-qubit gates compared to the O(n2t) of the current most efficient implementation based on quantum Fourier transforms. We test the proposed circuit on an IBM quantum device for a Hadamard DTQW on the 4-cycle and 8-cycle characterized by periodic dynamics and by recurrent generation of maximally entangled single-particle states. Experimental results are meaningful well beyond the regime of few time steps, paving the way for reliable implementation and use on quantum computers.
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.
We present a generalized and efficient quantum circuit framework for implementing discrete-time quantum walks (DTQWs) on Cayley graphs of arbitrary dimension. Building on the Boundary QFT scheme of Razzoli et al., we introduce a systematic multi-stage decomposition of the shift operator for 1D Cayley graphs across three classes of generating sets: inverse-closed without involutions, inverse-closed with an involution, and non-inverse-closed. The decomposition hierarchically factorizes the QFT-diagonalized shift operator into structured block components, progressively reducing the control degree of the required rotation gates and replacing high-degree multi-qubit controlled operations with collections of lower-degree equivalents. We extend this construction to ddd-dimensional torus graphs and provide explicit circuit implementations for an 8-Cayley graph and a Z16×Z8\mathbb{Z}_{16} \times \mathbb{Z}_8Z16​×Z8​ torus graph as concrete illustrations. Gate complexity analysis using the linear CNOT scaling of Rosa et al. demonstrates that the decomposed implementation achieves a substantial reduction in upper-bound CNOT cost relative to the naive implementation within the regime k≤64k \leq 64k≤64 for inverse-closed graphs and k≤16k \leq 16k≤16 for non-inverse-closed graphs, where kkk denotes the degree of the generating set. Benchmarking further reveals that this efficiency gain is largely insensitive to the system size NNN, identifying kkk as the dominant resource parameter for the shift operator. These results provide a scalable and hardware-conscious pathway toward practical DTQW implementations on near-term quantum devices.
Quantum walks have wide applications in quantum information, such as universal quantum computation, so it is important to explore properties of quantum walks thoroughly. We propose a novel method to implement discrete-time quantum walks (DTQWs) using only a single qubit, in which both coin and walker are encoded in the two-dimensional state space of a single qubit, operations are realized using single-qubit gates only, and high-dimensional final states of DTQWs can be obtained naturally. With this "one-qubit" approach, DTQW experiments can be realized much more easily, compared with previous methods, in most quantum systems and all properties based on quantum states of DTQWs (such as quantum correlation and coherence) can be investigated. By this approach, we experimentally implement one-particle and two-particle DTQWs with seven steps using single photons. Furthermore, we systematically investigate quantum correlations and coherence (based on the full state of the coin and walker) of the DTQW systems with different initial states of the coin, which have not been obtained and studied in DTQW experiments. As an application, we also study the assisted distillation of quantum coherence using the full state of the two-particle DTQW from the experiment. The maximal increase in distillable coherence for high-dimensional mixed states is investigated for the first time by obtaining its upper and lower bounds. Our work opens a new door to implement DTQW experiments and to better explore properties of quantum walks.
The quantum circuit model is the most commonly used model for implementing quantum computers and quantum neural networks whose essential tasks are to realize certain unitary operations. The circuit model usually implements a desired unitary operation U(N)U(N)U(N) by a sequence of single-qubit and two-qubit unitary gates from a universal set. Although this certainly facilitates the experimentalists as they only need to prepare several different kinds of universal gates, the number of gates required to implement an arbitrary desired unitary operation is usually large. Hence, the efficiency in terms of the circuit depth or running time is not guaranteed. Here we propose an alternative approach; we use a simple discrete-time quantum walk (DTQW) on a cycle graph to model an arbitrary unitary operation U(N)U(N)U(N) without the need to decompose it into a sequence of gates of smaller sizes. Our model is essentially a quantum neural network based on DTQW. Firstly, it is universal as we show that any unitary operation U(N)U(N)U(N) can be realized via an appropriate choice of coin operators. Secondly, our DTQW-based neural network can be updated efficiently via a learning algorithm, i.e., a modified stochastic gradient descent algorithm adapted to our network. By training this network, one can promisingly find approximations to arbitrary desired unitary operations. With an additional measurement on the output, the DTQW-based neural network can also implement general positive-operator-valued measure (POVM) measurements. We show its capacity in implementing arbitrary two-outcome POVMs via numeric simulation. We further demonstrate that the network can be simplified and can overcome device noises during the training so that it becomes more friendly for laboratory implementations. Our work shows the capability of the DTQW-based neural network in quantum computation and its potential in laboratory implementations.
The discrete-time quantum walk (DTQW) with step-dependent (SD) scattering operator is implemented on quantum computer. The probabilities of different states, with their respective fidelities, are calculated. This is done by generalizing the coin with a rotation gate using the quantum gate model. The CNOT gates in the shift operator are replaced with the alternative to CNOT gates Rx( π). They are applied on a quantum device and a quantum simulator (QS). The fidelities varied around 50% and the probability distribution of SD-DTQW for the angle π/4 spread symmetrically, while the step-independent DTQW (SI-DTQW) tended to peak at one side. The symmetric probability distribution of SD-DTQW can help in better control of the walk on QS. In the case of angle π/2, the SI-DTQW spread equally across the states with four peaks, while the SD-DTQW spread with two peaks to one side.
Quantum walks are a promising framework for developing quantum algorithms and quantum simulations. They represent an important test case for the application of quantum computers. Here we present different forms of discrete-time quantum walks (DTQWs) and show their equivalence for physical realizations. Using an appropriate digital mapping of the position space on which a walker evolves to the multiqubit states of a quantum processor, we present different configurations of quantum circuits for the implementation of DTQWs in one-dimensional position space. We provide example circuits for a five-qubit processor and address scalability to higher dimensions as well as larger quantum processors.