We consider the effects of plane-wave states scattering off finite graphs as an approach to implementing single-qubit unitary operations within the continuous-time quantum walk framework of universal quantum computation. Four semi-infinite tails are attached at arbitrary points of a given graph, representing the input and output registers of a single qubit. For a range of momentum eigenstates, we enumerate all of the graphs with up to $n=9$ vertices for which the scattering implements a single-qubit gate. As $n$ increases, the number of new unitary operations increases exponentially, and for $n>6$ the majority correspond to rotations about axes distributed roughly uniformly across the Bloch sphere. Rotations by both rational and irrational multiples of $\ensuremath{\pi}$ are found.
We consider the effects of plane-wave states scattering off finite graphs as an approach to implementing single-qubit unitary operations within the continuous-time quantum walk framework of universal quantum computation. Four semi-infinite tails are attached at arbitrary points of a given graph, representing the input and output registers of a single qubit. For a range of momentum eigenstates, we enumerate all of the graphs with up to n=9 vertices for which the scattering implements a single-qubit gate. As n increases, the number of new unitary operations increases exponentially, and for n>6 the majority correspond to rotations about axes distributed roughly uniformly across the Bloch sphere. Rotations by both rational and irrational multiples of \ensuremathπ are found.
Random walks are a powerful tool for the efficient implementation of algorithms in classical computation. Their quantum-mechanical analogues, called quantum walks, hold similar promise. Quantum walks provide a model of quantum computation that has recently been shown to be equivalent in power to the standard circuit model. As in the classical case, quantum walks take place on graphs and can undergo discrete or continuous evolution, though quantum evolution is unitary and therefore deterministic until a measurement is made. This thesis considers the usefulness of continuous-time quantum walks to quantum computation from the perspectives of both their fundamental power under various formulations, and their applicability in practical experiments. In one extant scheme, logical gates are effected by scattering processes. The results of an exhaustive search for single-qubit operations in this model are presented. It is shown that the number of distinct operations increases exponentially with the number of vertices in the scattering graph. A catalogue of all graphs on up to nine vertices that implement single-qubit unitaries at a specific set of momenta is included in an appendix. I develop a novel scheme for universal quantum computation called the discontinuous quantum walk, in which a continuous-time quantum walker takes discrete steps of evolution via perfect quantum state transfer through small `widget' graphs. The discontinuous quantum-walk scheme requires an exponentially sized graph, as do prior discrete and continuous schemes. To eliminate the inefficient vertex resource requirement, a computation scheme based on multiple discontinuous walkers is presented. In this model, n interacting walkers inhabiting a graph with 2n vertices can implement an arbitrary quantum computation on an input of length n, an exponential savings over previous universal quantum walk schemes. This is the first quantum walk scheme that allows for the application of quantum error correction. The many-particle quantum walk can be viewed as a single quantum walk undergoing perfect state transfer on a larger weighted graph, obtained via equitable partitioning. I extend this formalism to non-simple graphs. Examples of the application of equitable partitioning to the analysis of quantum walks and many-particle quantum systems are discussed.
In the present paper, the first in a series of two, we propose a model of universal quantum computation using a fermionic or bosonic multiparticle continuous-time quantum walk with two internal states (e.g., the spin-up and -down states of an electron). A dual-rail encoding is adopted to convert information: A single qubit is represented by the presence of a single quantum walker in either of the two parallel paths. We develop a roundabout gate that moves a walker from one path to the next, either clockwise or counterclockwise, depending on its internal state. It can be realized by a single-particle scattering on a directed weighted graph with the edge weights 1 and $\ifmmode\pm\else\textpm\fi{}i$. The roundabout gate also allows the spatial information of the quantum walker to be temporarily encoded in its internal states. The universal gates are constructed by appropriately combining several roundabout gates, some unitary gates that act on the internal states, and two-particle scatterings on straight paths. No ancilla qubits are required in our model. The computation is done by just passing quantum walkers through properly designed paths. Namely, there is no need for any time-dependent control. A physical implementation of quantum random access memory compatible with the present model will be considered in the second paper [R. Asaka et al., following paper, Phys. Rev. A 107, 022416 (2023)].
Any single-qubit unitary operation or quantum gate can be considered a rotation. Typical experimental implementations of single-qubit gates involve two or three fixed rotation axes, and up to three rotation steps. Here we show that, if the rotation axes can be tuned arbitrarily in a fixed plane, then two rotation steps are sufficient for implementing a single-qubit gate, and one rotation step is sufficient for implementing a state transformation. The results are relevant for "exchange-only" logical qubits encoded in three-spin blocks, which are important for universal quantum computation in decoherence free subsystems and subspaces.
Continuous-time quantum walk is one of the alternative approaches to quantum computation, where a universal set of quantum gates can be achieved by scattering a quantum walker on some specially-designed structures embedded in a sparse graph [Childs, Phys. Rev. Lett. 2009]. Recent advances in femtosecond laser-inscribed optical waveguides represent a promising physical platform for realizing this quantum-walk model of quantum computation. However, the major challenge is the problem of preparing a plane-wave initial state. Previously, the idea of quantum slide has been proposed and experimentally realized for demonstrating the working principle of NAND tree [Wang et al. Phy. Rev. Lett. 2020]. Here we show how quantum slide can be further applied to realize universal quantum computation, bypassing the plane-wave requirement. Specifically, we apply an external field to the perfect-state-transfer chain, which can generate a moving Gaussian wave packet with an arbitrary momentum. When the phase is properly tuned, the universal gate set in Childs' proposal can be realized in our scheme. Furthermore, we show that the gate fidelities increase with the length of the slide, and can reach unity asymptotically.
Continuous-time quantum walks (CTQWs) play a crucial role in quantum computing, especially for designing quantum algorithms. However, how to efficiently implement CTQWs is a challenging issue. In this paper, we study the implementation of CTQWs on sparse graphs, i.e., constructing efficient quantum circuits for implementing the unitary operator ${e}^{\ensuremath{-}iHt}$, where $H=\ensuremath{\gamma}A$ ($\ensuremath{\gamma}$ is a constant and A corresponds to the adjacency matrix of a graph). Our result is, for a d-sparse graph with N vertices and evolution time t, we can approximate ${e}^{\ensuremath{-}iHt}$ by a quantum circuit with gate complexity $({d}^{3}\ensuremath{\parallel}H\ensuremath{\parallel}tN{log}_{2}{(N))}^{1+o(1)}$, compared to the general Pauli decomposition, which scales like $(\ensuremath{\parallel}H\ensuremath{\parallel}t{N}^{4}{log}_{2}{(N))}^{1+o(1)}$. For sparse graphs, for instance, d=O(1), we obtain a noticeable improvement. Interestingly, our technique is related to graph decomposition. More specifically, we decompose the graph into a union of star graphs, and correspondingly, the Hamiltonian H can be represented as the sum of some Hamiltonians Hj, where each ${e}^{\ensuremath{-}i{H}_{j}t}$ is a CTQW on a star graph which can be implemented efficiently.
The continuous-time quantum walk is a quantum version of a random walk that evolves by Schrodinger's equation. With the Hamiltonian equal to the adjacency matrix of a sequence of graphs, called a dynamic graph, continuous-time quantum walks have been shown to implement quantum gates, including the T gate, Hadamard gate, and the Controlled{NOT gate. Since these gates make up a universal set of quantum gates, they can implement any other arbitrary quantum gate up to an arbitrary approximation. This process, however, can be tedious and ine cient. To alleviate this, we have developed a parameterized dynamic graph on which a continuous-time quantum walk can implement any arbitrary single-qubit quantum gate, and we have also extended this result to implement any two-qubit controlled-unitary quantum gate. These dynamic graphs have at most length three. Using these, we implemented Draper's quantum addition circuit, which is based on the quantum Fourier transform, using a continuous-time quantum walk on a dynamic graph.