It is known that SWAP operation for quantum systems of arbitrary dimension can be implemented using various qudit gates. In this paper, equivalence of some important qudit operations is demonstrated. Equivalence of the qudit operations is exploited to minimize the circuit complexity in the previously proposed qudit SWAP gate constructions. Further, constructions of qudit SWAP operation with minimum number of qudit gates are also proposed. Importantly, these circuit constructions of SWAP retain its properties like symmetry and simplicity.
It is known that SWAP operation for quantum systems of arbitrary dimension can be implemented using various qudit gates. In this paper, equivalence of some important qudit operations is demonstrated. Equivalence of the qudit operations is exploited to minimize the circuit complexity in the previously proposed qudit SWAP gate constructions. Further, constructions of qudit SWAP operation with minimum number of qudit gates are also proposed. Importantly, these circuit constructions of SWAP retain its properties like symmetry and simplicity.
The SWAP gate has become an integral feature of quantum circuit architectures and is designed to permute the states of two qubits through the use of the well-known controlled-NOT gate. We consider the question of whether a two-qudit quantum circuit composed entirely from instances of the generalised controlled-NOT gate can be constructed to permute the states of two qudits. Arguing via the signature of a permutation, we demonstrate the impossibility of such circuits for dimensions d≡3 (mod 4).
Quantum algorithms can be realized in the form of a quantum circuit. To map a quantum circuit for a specific quantum algorithm to quantum hardware, qubit mapping is an imperative technique based on the qubit topology. Due to the neighborhood constraint of qubit topology, the implementation of the quantum algorithm rightly, is essential for moving information around in a quantum computer. Swapping of qubits using a swap gate moves the quantum state between two qubits and solves the neighborhood constraint of qubit topology. Although, one needs to decompose the swap gate into three controlled-not gates to implement the swap gate efficiently, but unwillingly quantum cost with respect to the gate count and depth increases. In this paper, a formalism of moving quantum states without using swap operation is introduced. Moving quantum states through qubits have been attained with the adoption of temporary intermediate qudit states. This introduction of intermediate qudit states has exhibited a three times reduction in quantum cost with respect to the gate count and approximately two times reduction with respect to circuit depth compared to the state-of-the-art approach of the swap gate insertion. We also exhibit that the adoption of the intermediate qudit makes the approach sublimer than the existing works by obtaining a better fidelity. Furthermore, the proposed approach is generalized to any finite-dimensional quantum system.
We present explicit descriptions for the swap gate and the iswap gate in any arbitrary dimension d≥2, in terms of permutation matrices. Moreover, we unify these gates by introducing a more general gate xSWAP which includes SWAP and iSWAP for x=1 and x=i (i.e. −1), respectively. The higher dimensional xSWAP e.g., the swap and iswap gates for d>2 serve as quantum logic gates that operate on two d-level qudits. For d=2, it is well known that iSWAP unlike SWAP is universal for quantum computing. We will prove this fact for xSWAP in any dimension d, when x=±1. Our explicit representation of xSWAP by a permutation matrix facilitates the proof, greatly.
Circuit synthesis of multi-controlled gates is crucial for qudit (d-level) quantum computing. This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates. For synthesizing general (n−1)-controlled unitaries on n qudits, we reduce the controlled-increment (CINC) and generalized controlled-X (GCX) gate counts to O(n2), improving upon existing O(n2+log2d) CINC and O(n3) GCX bounds. For (n−1)-controlled special unitaries, this complexity is further reduced to O(n). Furthermore, we present a method for approximately synthesizing multi-controlled unitary gates with a linear complexity in the number of control qudits. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from n to m qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when d is prime. This enables all CINC-based circuits for various quantum operations to be converted into SUM-gate circuits while preserving the same asymptotic complexity. Finally, we establish a theoretical lower bound on the number of SUM and CINC gates required to synthesize general n-qudit unitaries.
Abstract Advances in quantum information science are closely related to our ability to prepare, manipulate, and measure the states of multiple quantum bits (qubits). To this end, quantum gates are the key components by which specific operations over individual and multiple qubits are implemented. In the context of quantum optics, qubits can be encoded in any degree of freedom of a multi-photon state, e.g. polarization, frequency or spatial configuration. The faithful reciprocal exchange of information between two qubits, a so-called SWAP, is one of the fundamental operations that underpin modern quantum architectures. Here, we present a novel method realizing multiple-qubit SWAP operations over several qubits, without relying on any other gates, in an integrated-photonic setting. In principle, our approach allows for the implementation of SWAP operations with unity efficiency regardless of the number of involved qubits. Further, we discuss how this approach can be applied to qudits. Our findings provide a powerful new functional element for the design of compact quantum-photonic circuitry.