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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.

This paper concerns the construction of k-uniform maximally mixed multipartite states by using the formalism of phase states for finite dimensional systems (qudits). The k-uniform states are a special kind of entangled (n)-qudits states, such that after tracing out arbitrary (n$-$k) subsystems, the remaining (k) subsystems are maximally mixed. We recall some basic elements about unitary phase operators of a multi-qudit system and we give the corresponding separable density matrices. Evolved density matrices arise when qudits of the multipartite system are allowed to interact via an Hamiltonian of Heisenberg type. The expressions of maximally mixed states are explicitly derived from multipartite evolved phase states and their properties are discussed.

Complementarity relations in composite bipartite quantum systems of arbitrary dimensions are proposed. Genuine bipartite quantum properties whose information content is quantified by the generalized concurrence mutually exclude the single-partite properties of the subsystems. The single-partite properties are determined by generalized predictabilities and visibilities, i.e. by the traditional concepts of wave-particle duality and define two complementary realities. These properties combined together are complementary to the generalized I-concurrence which quantifies genuine quantum correlations in n ⊗ m-dimensional bipartite systems. An important feature of the proposed quantitative complementarity relation is its upper bound which is given by a dimensional-dependent factor that exceeds unity in case of qudits with dimensionality d > 2. In d > 2-dimensional systems the information content can exceed one bit of information.

We develop a comprehensive approach of stabilizer codes and provide a scheme generating equientangled basis interpolating between the product basis and maximally entangled basis. The key ingredient is the theory of phase states for finite-dimensional systems (qudits). In this respect, we derive entangled phase states for a multiqudit system whose dynamics is governed by a two-qudit interaction Hamiltonian. We construct the stabilizer codes for this family of entangled phase states. The stabilizer phase states are defined as the common eigenvectors of the stabilizer group generators which are explicitly specified. Furthermore, we construct equally entangled bases from bipartite as well as multipartite entangled qudit phase states.

We review recent theoretical and experimental works where are proposed and demonstrated how to use photon pairs created by spontaneous parametric down-conversion to generate entangled states of D-dimensional quantum systems, or qudits. This is the first demonstration of high-dimensional entanglement based on the intrinsic transverse momentum entanglement of the type-II down-converted photons. The qudit space is defined by an aperture made up of an opaque screen with D slits (paths), placed in the arms of the twin photons. By manipulating the pump beam profile we can prepare different entangled states of these possible paths. We focus our attention on an important case for applications in quantum information: the maximally entangled states. Experimental results for qudits with D=4 and D=8 are shown and measuring a two-photon conditional interference, we also demonstrate the nonclassical character of the correlations.

The description of qudits in a formalism based on a generalized variant of Weyl–Heisenberg algebras is discussed. The unitary phase operators for a multi-qudit system and the corresponding phase states (the eigenstates of the phase operator) are constructed. We discuss the dynamics of multi-qudit phase states governed by a generalized Hamiltonian involving one- and two-body interactions which offer a remarkable connection between phase states, generalized graph states and the mutually unbiased bases. The entangled phase states are shown to possess the following properties simultaneously, namely the mutually unbiasedness of phase states resulting from the one-body generalized oscillator Hamiltonian and the entanglement properties of generalized graph states resulting from the two-body interaction Hamiltonian.

We report an experiment to generate and propagate two entangled spatial qudits, -dimensional quantum systems, using spontaneous parametric down-conversion. The manipulation, via pump beam, of the transverse spatial correlation between the photon pairs is explored. Inserting apertures with -slits in the arms of the down-converted photons, we associate the qudit space with the possible paths followed by each photon. Experimental results for qudits with and 8 are shown. We demonstrate that the generated states cannot be classically correlated. We also show the propagation of entangled states of spatial qudits. Their free-space distribution is performed at the laboratory scale and the propagated states maintain a high fidelity with their original form.

The quantum Fourier transform (QFT) is a key ingredient of several quantum algorithms and a qudit-specific implementation of the QFT is hence an important step toward the realization of qudit-based quantum computers. This work develops a circuit decomposition of the QFT for hybrid qudits based on generalized Hadamard and generalized controlled-phase gates, which can be implemented using selective rotations in NMR. We experimentally implement the hybrid qudit QFT on an NMR quantum emulator, which uses four qubits to emulate a single qutrit coupled to two qubits.

Quantum computation strongly relies on the realization, manipulation, and control of qubits. A central method for realizing qubits is by creating a double-well potential system with a significant gap between the first two eigenvalues and the rest. In this work, we first revisit the theoretical grounds underlying the double-well qubit dynamics, then proceed to suggest novel extensions of these principles to a triple-well qutrit with periodic boundary conditions, followed by a general $d$-well analysis of qudits. These analyses are based on representations of the special unitary groups SU$(d)$ which expose the systems’ symmetry and employ them for performing computations. We conclude with a few notes on coherence and scalability of $d$-well systems.

We derive absolutely maximally entangled (AME) states from phase states for a multi-qudit system whose dynamics is governed by a two-qudit interaction Hamiltonian of Heisenberg type. AME states are characterized by being maximally entangled for all bipartitions of the multi-qudit system and present absolute multipartite entanglement. The key ingredient of this approach is the theory of phase states for finite-dimensional systems (qudits). We define further the unitary phase operators of $N$-qudit systems and we give next the corresponding separable phase states. Using a qudit–qudit Hamiltonian acting as entangling operator on separable phase states, we generate entangled phase states. Finally, from the labeled entangled phase states, we derive the absolutely maximally entangled states.

Most prior semi-quantum key distribution (SQKD) protocols are designed in the context of qubits. In this paper, we proposed an SQKD protocol using ($d$+1) quantum states in two mutually unbiased bases in the context of qudits without invoking the classical party’s measurement capability. It not only decreases the quantum states sent by quantum party by half, but also improves the efficiency of key rate than qubits do. To prove the security of present protocol, we first illustrate that the present protocol is completely robust, and then prove its unconditional security from information theoretical perspective by computing its key rate and noise tolerance against cloning-based individual attack. The unconditional security proof demonstrates that the higher dimension of quantum system is, the higher key rate can be obtained and the higher noise tolerance is allowed in depolarization channel.