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arXiv:1902.10681 · Physical Review A

Protocol of a quantum walk in circuit QED

Jiaqi Zhou, Ling Cai, Qi-Ping Su, Chui‐Ping Yang

Published Jul 25, 2019• 16 citations
arXivHTMLPDF

Abstract

Implementation of a discrete-time quantum walk (DTQW) with superconducting qubits is difficult since on-chip superconducting qubits cannot hop between lattice sites. We propose an efficient protocol for the implementation of DTQW in circuit quantum electrodynamics (QED), in which only $N+1$ qutrits, $N$ cavities, and other assistant devices are needed for an $N$-step DTQW. The operation of each DTQW step is very quick because only resonant processes are adopted. The numerical simulations show that high-similarity DTQW with a number of steps up to 20 is feasible with present-day circuit QED techniques. This protocol can help to study properties and applications of many-step DTQW in experiments, and can be extended to implement multidimensional DTQW in circuit QED or other quantum systems.

DOI
10.1103/physreva.100.012343

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Implementation of a discrete-time quantum walk (DTQW) with superconducting qubits is difficult since on-chip superconducting qubits cannot hop between lattice sites. We propose an efficient protocol for the implementation of DTQW in circuit quantum electrodynamics (QED), in which only N+1N+1N+1 qutrits, NNN cavities, and other assistant devices are needed for an NNN-step DTQW. The operation of each DTQW step is very quick because only resonant processes are adopted. The numerical simulations show that high-similarity DTQW with a number of steps up to 20 is feasible with present-day circuit QED techniques. This protocol can help to study properties and applications of many-step DTQW in experiments, and can be extended to implement multidimensional DTQW in circuit QED or other quantum systems.
Discrete-time quantum walks (DTQWs) have broad applications in quantum computation and simulation, particularly in universal quantum computing. Prior research in circuit quantum electrodynamics (QED) has focused on implementing DTQWs in one-dimensional chains. However, to achieve practical quantum tasks, it is crucial to extend DTQWs to multidimensional networks with diverse structures. Here, we propose a protocol for realizing DTQWs in multidimensional networks, which accommodate nodes with different numbers of neighbors. Using this protocol, we numerically simulate a Grover walk search algorithm in a cubic network with circuit QED. Our simulations, which consider decoherence, align well with theoretical predictions. Our universal protocol can be adapted for various quantum systems and paves the way for DTQWs in multidimensional networks, which has potential applications in quantum computing and simulation.
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.
Quantum walk serves as a versatile tool for universal quantum computing and algorithmic research. However, the implementation of discrete-time quantum walks (DTQWs) with superconducting circuits is still constrained by some limitations such as operation precision, circuit depth and connectivity. With improved hardware efficiency by using superconducting qutrits (three-level systems), we experimentally demonstrate a scalable DTQW in a superconducting circuit, observing the ballistic spreading of quantum walk in a qutrit chain. The usage of qutrits in our implementation allows hardware efficiently encoding of the walker position and the coin degree of freedom. By exploiting the flexibility and intrinsic symmetries of qutrit-based DTQWs, we successfully prepare two topological phases in the chain. For the first time, particle-hole-symmetry-protected edge states, bounded at the interface between these two topological phases, are observed in the superconducting platform. Measured parameter dependencies further validate the properties of edge states. The scalability and gate-control compatibility of the demonstrated DTQWs enable a versatile tool for superconducting quantum computing and quantum simulation.
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.
We show how a quantum walk can be implemented in a quantum quincunx created via superconducting circuit quantum electrodynamics QED and how interpolation from a quantum to a random walk is implemented by controllable decoherence using a two-resonator system. Direct control over the coin qubit is difficult to achieve in either cavity or circuit QED, but we show that a Hadamard coin flip can be effected via direct driving of the cavity, with the result that the walker jumps between circles in phase space, but still exhibits quantum walk behavior over 15 steps.
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.