A common model of quantum computing is the gate model with binary basis states. Here, we consider the gate model of quantum computing with a non-binary radix resulting in more than two basis states to represent a quantum digit, or qudit. Quantum entanglement is an important phenomenon that is a critical component of quantum computation and communications algorithms. The generation and use of entanglement among radix-2 qubits is well-known and used often in quantum computing algorithms. Quantum entanglement exists in higher-radix systems as well although little is written regarding the generation of higher-radix entangled states. We provide background describing the feasibility of multiple-valued logic quantum systems and describe a new systematic method for generating maximally entangled states in quantum systems of dimension greater than two. This method is implemented in a synthesis algorithm that is described. Experimental results are included that demonstrate the transformations needed to create specific forms of maximally entangled quantum states.
A common model of quantum computing is the gate model with binary basis states. Here, we consider the gate model of quantum computing with a non-binary radix resulting in more than two basis states to represent a quantum digit, or qudit. Quantum entanglement is an important phenomenon that is a critical component of quantum computation and communications algorithms. The generation and use of entanglement among radix-2 qubits is well-known and used often in quantum computing algorithms. Quantum entanglement exists in higher-radix systems as well although little is written regarding the generation of higher-radix entangled states. We provide background describing the feasibility of multiple-valued logic quantum systems and describe a new systematic method for generating maximally entangled states in quantum systems of dimension greater than two. This method is implemented in a synthesis algorithm that is described. Experimental results are included that demonstrate the transformations needed to create specific forms of maximally entangled quantum states.
Entanglement is an important phenomenon that enables quantum information processing algorithms and quantum communications protocols. Although entangled quantum states are often described in radix-2, higher-radix qudits can become entangled as well. In this work, we both introduce partial entanglement, a concept that does not exist for radix-2 quantum systems, and differentiate between partial and maximal entanglement within non-binary quantum information processing systems. We also develop and present higher-radix maximal entanglement generator circuits that are analogous to the well-known Bell state generator for binary quantum systems. Because higher-dimensioned qudits can be subjected to entangling processes that result in either partially or maximally entangled states, we demonstrate how higher-radix qudit circuits can be composed to generate these varying degrees of partial quantum entanglement. Theoretical results are provided for the general case where the radix is greater than two, and specific results based on a nair of radix-4 audits are described.
Quantum information processing and communication techniques rely heavily upon entangled quantum states, and this dependence motivates the development of methods and systems to generate entanglement. Much research has been dedicated to state preparation for radix-2 qubits, and due to the pursuit of entangled states, the Bell state generator and its generalized forms where the number of entangled qubits is greater than two have been defined. In this work, we move beyond radix-2 and propose techniques for quantum state entanglement in high-dimensional systems through the generalization of the binary bipartite entanglement states. These higher-radix quantum informatic systems are composed of n quantum digits, or qudits, that are each mathematically characterized as elements of an r -dimensioned Hilbert vector space where r > 2. Consequently, the wave function is a time-dependent state vector of dimension r n . The generalization of the binary controlled-NOT to the controlled-modulo-addition gate, the concept of partial versus maximal entanglement, and architectures for generating higher-radix entangled states for the partial and maximal case are all presented.
Quantum computation established itself as a promising emerging technology and, hence, attracted considerable attention in the domain of computer-aided-design (CAD). However, quantum mechanical phenomena such as superposition, phase shifts, or entanglement lead to a logic model which poses serious challenges to the development of a proper design flow for quantum circuits. Consequently, researchers addressed synthesis of quantum circuits not as a single design step, but considered sub-tasks such as synthesis of Boolean components or synthesis of restricted subsets of quantum functionality. Generating a particularly desired quantum state is another of these sub-tasks. However, logic synthesis of quantum circuits accomplishing that has hardly been considered yet. In this work, we propose a generic method which automatically synthesizes a quantum circuit generating any desired quantum state from an initially given basis state. The proposed method allows for both, a theoretical determination of upper bounds as well as an experimental evaluation of the number of quantum gates needed for this important design step.
Most quantum computing architectures to date natively support multi-valued logic, albeit being typically operated in a binary fashion. Multi-valued, or qudit, quantum processors have access to much richer forms of quantum entanglement, which promise to significantly boost the performance and usefulness of quantum devices. However, much of the theory as well as corresponding design methods required for exploiting such hardware remain insufficient and generalizations from qubits are not straightforward. A particular challenge is the compilation of quantum circuits into sets of native qudit gates supported by state-of-the-art quantum hardware. In this work, we address this challenge by introducing a complete workflow for compiling any two-qudit unitary into an arbitrary native gate set. Case studies demonstrate the feasibility of both, the proposed approach as well as the corresponding implementation (which is freely available at github.com/cda-tum/qudit-entanglement-compilation).
Abstract In the circuit model of quantum computation, an entangling two-qubit gate and a set of single-qubit gates are used as universal gate set or basis gates for doing quantum computation. CNOT and iSWAP gates, the perfect entanglers that can create maximally entangled two-qubit states in one application, are broadly used entangling two-qubit basis gates in quantum computers. In this paper, we analyze the potentiality of B gate, an unexplored non-perfect entangler of the form, exp i π ( σ ˆ x ⨂ σ ˆ x ) 8 + i π ( σ ˆ y ⨂ σ ˆ y ) 16 , as an entangling two-qubit basis gate in quantum computers by studying its ability to generate other two-qubit gates. We derive a necessary condition for a two-qubit gate to be generated by n applications of B gate. Using this condition, we show that the gates that can be generated by two and three applications of B gate are contained in the 50% and 92.97% of the volume of Weyl chamber of two-qubit gates, respectively. We prove that two applications of B gate can generate both SWAP and SWAP
Recent research in generalizing quantum computation from 2-valued qudits to d-valued qudits has shown practical advantages for scaling up a quantum computer. A further generalization leads to quantum computing with hybrid qudits where two or more qudits have different finite dimensions. In both cases, a quantum computation is performed when a unitary evolution operator, acting as a quantum logic gate, transforms the state of qudits in a quantum system. Unitary operators can be represented by unitary matrices, and therefore for a quantum system consists of multiple qudits, a gate may be synthesized by matrix decomposition techniques such as QR factorization and the Cosine-sine Decomposition (CSD). In this article, we present a CSD based synthesis method for n qudit hybrid and d-valued quantum circuits.