Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.19151

Quantum Circuits for Quantum Spatial Search on ddd-Dimensional Lattices

Rei Sato

quant-phSubmitted Jul 21, 2026• Announced Jul 22, 2026• 0 citations
arXivHTMLPDF

Abstract

We propose an explicit quantum circuit for quantum spatial search based on discrete-time quantum walks on $d$-dimensional lattices. In this algorithm, the flip-flop shift operator moves the walker to a neighboring site along the selected spatial direction and reverses the corresponding direction label after the move. By encoding each pair of opposite directions so that they differ only in the least significant qubit of the coin register, we implement the shift using coin-controlled modular increment and decrement operations on the position registers, together with a single $X$ gate that reverses the direction label. We verify that the proposed circuits reproduce the theoretical dynamics on two- and three-dimensional periodic lattices. We further extend the circuit construction to systems with position-dependent shift rules, such as non-periodic boundaries, and validate this extension on a two-dimensional lattice. The resource analysis shows logarithmic growth of the circuit width. For the two-dimensional lattice, the synthesized depth is consistent with $O(\sqrt{N}(\log N)^{3/2})$, while the three-dimensional depth empirically follows an $O(\sqrt{N})$ dependence over the investigated range. Under a CX-gate depolarizing noise model, CX-optimized circuits exhibit improved noise robustness. These results provide a practical framework for implementing quantum spatial search on regular lattices and extending it to defective and other irregular lattice structures.

Comments
9 pages, 7 figures

Related papers

Generalized Efficient Quantum Circuit Implementation of Discrete-Time Quantum Walks on Cayley Graphs

Seoyoon Kang

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 $d$-dimensional torus graphs and provide explicit circuit implementations for an 8-Cayley graph and a $\mathbb{Z}_{16} \times \mathbb{Z}_8$ 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 $k \leq 64$ for inverse-closed graphs and $k \leq 16$ for non-inverse-closed graphs, where $k$ denotes the degree of the generating set. Benchmarking further reveals that this efficiency gain is largely insensitive to the system size $N$, identifying $k$ 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.

quant-pharXiv:2608.25136•Aug 27, 2026•0 citations

Gate-level Implementation and Resource Analysis of Lackadaisical Quantum Walk Search

Amit Saha, Debanjan Kola, Nishanka Das +1 more

Lackadaisical quantum walks (LQW) extend discrete-time quantum walks (DTQW) by introducing weighted self-loops, enabling improved spatial-search performance through controlled localization of the walker. Despite substantial theoretical progress, practical gate-level implementations suitable for quantum hardware remain largely unexplored, limiting evaluation under realistic architectural constraints, noise, and resource requirements. In this work, we present a gate-level implementation framework for lackadaisical quantum walk search. The proposed construction encodes the position and coin spaces into qubit registers, and realizes the walk dynamics through oracle, coin, and flip-flop shift operations. We validate the circuit by reproducing the expected search behavior for single and multiple marked vertices and by analyzing the effect of the self-loop weight on the success probability. We further evaluate the implementation under realistic noisy settings using superconducting hardware's noise models. Logical resource analysis shows that, for grids ranging from $8\times8$ to $64\times64$, the transpiled gate count increases from $3.63\times10^{5}$ to $4.38\times10^{6}$ and the circuit depth from $2.13\times10^{5}$ to $2.56\times10^{6}$. Finally, fault-tolerant resource estimates based on a surface-code model using the Microsoft Quantum Resource Estimator demonstrate the substantial space-time trade-off associated with magic-state production. The results establish a practical circuit-level pathway for implementing the LQW search and provide a basis for evaluating its performance.

quant-pharXiv:2608.17136•Aug 19, 2026•0 citations

Coined Quantum Walks on Complex Networks for Quantum Computers

Rei Sato

We propose a quantum circuit design for implementing coined quantum walks on complex networks. In complex networks, the coin and shift operators depend on the varying degrees of the nodes, which makes circuit construction more challenging than for regular networks. To address this issue, we use a dual-register encoding to enable a simplified shift operator and reduces the resource overhead. We implement the circuit using Qmod, a high-level quantum programming language, and evaluated the performance through numerical simulations on Erdős-Rényi, Watts-Strogatz, and Barabási-Albert models. The results show that the circuit depth scales as approximately $N^{1.9}$ regardless of the network topology. Furthermore, we execute the proposed circuits on the ibm\_torino superconducting quantum processor for Watts-Strogatz models with $N=4$ and $N=8$. The experiments show that hardware-aware optimization slightly improved the variation distance and Hellinger fidelity for the larger network, whereas connectivity constraints imposed overhead for the smaller one. These results indicate that while current NISQ devices are limited to small-scale validations, the polynomial scaling of our framework makes it suitable for larger-scale implementations in the fault-tolerant quantum computing era.

quant-pharXiv:2512.16400•Quantum Information Processing•Dec 19, 2025•0 citations

Search on a fractal lattice using a quantum random walk

Apoorva D. Patel, K. S. Raghunathan

The spatial search problem on regular lattice structures in integer number of dimensions $d\ensuremath{\ge}2$ has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here, we investigate the spatial search problem on fractals of noninteger dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.

arXiv:1203.3950•Physical Review A•Jul 27, 2012•31 citations

Robust implementation of discrete-time quantum walks in any finite-dimensional quantum system

Biswayan Nandi, Sandipan Singha, Ankan Datta +2 more

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.

arXiv:2408.00530•Modern Physics Letters A•Mar 29, 2025•0 citations

Toward Optimal Circuit Depth for Geometrically Local Hamiltonian Simulation

Zheyu Shen, Yusen Wu, Penghui Yao +2 more

Understanding the dynamics of interacting quantum systems is a central goal of quantum simulation. For geometrically local Hamiltonians, locality suggests that digital simulation should retain much of the parallelism of the underlying dynamics. For a target accuracy $ε$, we construct a nearest-neighbor quantum circuit that simulates the dynamics of an $n$-qubit, time-independent, finite-range Hamiltonian with normalized local interaction strength on a fixed-dimensional lattice for time $T\ge1$, with circuit depth $O(T\log(nT/ε))$ and gate count $O(nT\log(nT/ε))$. This reduces the polylogarithmic overhead of Haah, Hastings, Kothari, and Low to a single logarithm. On logarithmic-scale blocks from the Lieb--Robinson decomposition, a shallow but highly accurate implementation is essential for reducing the circuit depth. Our key observation is that spatial localization also controls the cost of high precision. In each block, the error of a constant-order product formula is compensated for by a correction, yielding exponential accuracy. The correction can be implemented with depth proportional to the block diameter, introducing no additional asymptotic depth scale beyond that required by spatial localization. Furthermore, we prove an unconditional circuit-depth lower bound for uniform XXZ Hamiltonians: at fixed evolution time $T$ in the stated regime and sufficiently small fixed error $ε$, any nearest-neighbor quantum circuit simulating these dynamics requires depth $Ω(\log n/\log\log n)$. This matches our $O(\log n)$ upper bound up to a doubly logarithmic factor, establishing near-optimal circuit depth for lattice Hamiltonian simulation in this regime. Together, these results clarify the role of locality and precision in determining the circuit complexity of quantum dynamics and suggest broader principles for spatially constrained quantum computation.

quant-pharXiv:2610.01839•Oct 2, 2026•0 citations
We propose an explicit quantum circuit for quantum spatial search based on discrete-time quantum walks on ddd-dimensional lattices. In this algorithm, the flip-flop shift operator moves the walker to a neighboring site along the selected spatial direction and reverses the corresponding direction label after the move. By encoding each pair of opposite directions so that they differ only in the least significant qubit of the coin register, we implement the shift using coin-controlled modular increment and decrement operations on the position registers, together with a single XXX gate that reverses the direction label. We verify that the proposed circuits reproduce the theoretical dynamics on two- and three-dimensional periodic lattices. We further extend the circuit construction to systems with position-dependent shift rules, such as non-periodic boundaries, and validate this extension on a two-dimensional lattice. The resource analysis shows logarithmic growth of the circuit width. For the two-dimensional lattice, the synthesized depth is consistent with O(N(log⁡N)3/2)O(\sqrt{N}(\log N)^{3/2})O(N​(logN)3/2), while the three-dimensional depth empirically follows an O(N)O(\sqrt{N})O(N​) dependence over the investigated range. Under a CX-gate depolarizing noise model, CX-optimized circuits exhibit improved noise robustness. These results provide a practical framework for implementing quantum spatial search on regular lattices and extending it to defective and other irregular lattice structures.
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.
Lackadaisical quantum walks (LQW) extend discrete-time quantum walks (DTQW) by introducing weighted self-loops, enabling improved spatial-search performance through controlled localization of the walker. Despite substantial theoretical progress, practical gate-level implementations suitable for quantum hardware remain largely unexplored, limiting evaluation under realistic architectural constraints, noise, and resource requirements. In this work, we present a gate-level implementation framework for lackadaisical quantum walk search. The proposed construction encodes the position and coin spaces into qubit registers, and realizes the walk dynamics through oracle, coin, and flip-flop shift operations. We validate the circuit by reproducing the expected search behavior for single and multiple marked vertices and by analyzing the effect of the self-loop weight on the success probability. We further evaluate the implementation under realistic noisy settings using superconducting hardware's noise models. Logical resource analysis shows that, for grids ranging from 8×88\times88×8 to 64×6464\times6464×64, the transpiled gate count increases from 3.63×1053.63\times10^{5}3.63×105 to 4.38×1064.38\times10^{6}4.38×106 and the circuit depth from 2.13×1052.13\times10^{5}2.13×105 to 2.56×1062.56\times10^{6}2.56×106. Finally, fault-tolerant resource estimates based on a surface-code model using the Microsoft Quantum Resource Estimator demonstrate the substantial space-time trade-off associated with magic-state production. The results establish a practical circuit-level pathway for implementing the LQW search and provide a basis for evaluating its performance.
We propose a quantum circuit design for implementing coined quantum walks on complex networks. In complex networks, the coin and shift operators depend on the varying degrees of the nodes, which makes circuit construction more challenging than for regular networks. To address this issue, we use a dual-register encoding to enable a simplified shift operator and reduces the resource overhead. We implement the circuit using Qmod, a high-level quantum programming language, and evaluated the performance through numerical simulations on Erdős-Rényi, Watts-Strogatz, and Barabási-Albert models. The results show that the circuit depth scales as approximately N1.9N^{1.9}N1.9 regardless of the network topology. Furthermore, we execute the proposed circuits on the ibm\_torino superconducting quantum processor for Watts-Strogatz models with N=4N=4N=4 and N=8N=8N=8. The experiments show that hardware-aware optimization slightly improved the variation distance and Hellinger fidelity for the larger network, whereas connectivity constraints imposed overhead for the smaller one. These results indicate that while current NISQ devices are limited to small-scale validations, the polynomial scaling of our framework makes it suitable for larger-scale implementations in the fault-tolerant quantum computing era.
The spatial search problem on regular lattice structures in integer number of dimensions $d\ensuremath{\ge}2$ has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here, we investigate the spatial search problem on fractals of noninteger dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.
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.
Understanding the dynamics of interacting quantum systems is a central goal of quantum simulation. For geometrically local Hamiltonians, locality suggests that digital simulation should retain much of the parallelism of the underlying dynamics. For a target accuracy εεε, we construct a nearest-neighbor quantum circuit that simulates the dynamics of an nnn-qubit, time-independent, finite-range Hamiltonian with normalized local interaction strength on a fixed-dimensional lattice for time T≥1T\ge1T≥1, with circuit depth O(Tlog⁡(nT/ε))O(T\log(nT/ε))O(Tlog(nT/ε)) and gate count O(nTlog⁡(nT/ε))O(nT\log(nT/ε))O(nTlog(nT/ε)). This reduces the polylogarithmic overhead of Haah, Hastings, Kothari, and Low to a single logarithm. On logarithmic-scale blocks from the Lieb--Robinson decomposition, a shallow but highly accurate implementation is essential for reducing the circuit depth. Our key observation is that spatial localization also controls the cost of high precision. In each block, the error of a constant-order product formula is compensated for by a correction, yielding exponential accuracy. The correction can be implemented with depth proportional to the block diameter, introducing no additional asymptotic depth scale beyond that required by spatial localization. Furthermore, we prove an unconditional circuit-depth lower bound for uniform XXZ Hamiltonians: at fixed evolution time TTT in the stated regime and sufficiently small fixed error εεε, any nearest-neighbor quantum circuit simulating these dynamics requires depth Ω(log⁡n/log⁡log⁡n)Ω(\log n/\log\log n)Ω(logn/loglogn). This matches our O(log⁡n)O(\log n)O(logn) upper bound up to a doubly logarithmic factor, establishing near-optimal circuit depth for lattice Hamiltonian simulation in this regime. Together, these results clarify the role of locality and precision in determining the circuit complexity of quantum dynamics and suggest broader principles for spatially constrained quantum computation.