We have implemented a Walsh-Hadamard gate, which performs a quantum Fourier transform, in a superconducting qutrit. The qutrit is encoded in the lowest three energy levels of a capacitively shunted flux device, operated at the optimal flux-symmetry point. We use an efficient decomposition of the Walsh-Hadamard gate into two unitaries, generated by off-diagonal and diagonal Hamiltonians respectively. The gate implementation utilizes simultaneous driving of all three transitions between the three pairs of energy levels of the qutrit, one of which is implemented with a two-photon process. The gate has a duration of 35 ns and an average fidelity over a representative set of states, including preparation and tomography errors, of 99.2%, characterized with quantum state tomography. Compensation of ac-Stark and Bloch-Siegert shifts is essential for reaching high gate fidelities.
A hybrid classical-quantum approach for the solution of nonlinear ordinary differential equations using Walsh-Hadamard basis functions is proposed. Central to this hybrid approach is the computation of the Walsh-Hadamard transform of arbitrary vectors, which is enabled in our framework using quantum Hadamard gates along with state preparation, shifting, scaling, and measurement operations. It is estimated that the proposed hybrid classical-quantum approach for the Walsh-Hadamard transform of an input vector of size N results in a considerably lower computational complexity (O(N) operations) co…
A set of generators of generalized Pauli matrices play a crucial role in quantum computation based on n level systems of an atom. In this paper we show how to construct them by making use of Rabi oscillations. We also construct the generalized Walsh-Hadamard matrix in the case of three level systems and present some related problems.
In this paper, we give a modified version of the qutrit ZX-calculus, by which we represent qutrit graph states as diagrams and prove that the qutrit version of local complementation property is true if and only if the qutrit Hadamard gate $H$ has an Euler decomposition into $4π/3$-green and red rotations. This paves the way for studying the completeness of qutrit ZX-calculus for qutrit stabilizer quantum mechanics.
A hybrid classical-quantum approach for evaluation of multi-dimensional Walsh-Hadamard transforms and its applications to quantum image processing are proposed. In this approach, multidimensional Walsh-Hadamard transforms are obtained using quantum Hadamard gates (along with state-preparation, shifting, scaling and measurement operations). The proposed approach for evaluation of multidimensional Walsh-Hadamard transform has a considerably lower computational complexity (involving $O(N^d)$ operations) in contrast to classical Fast Walsh-Hadamard transform (involving $O(N^d~\log_2 N^d)$ operatio…
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.
We introduce a dichromatic calculus (RG) for qutrit systems. We show that the decomposition of the qutrit Hadamard gate is non-unique and not derivable from the dichromatic calculus. As an application of the dichromatic calculus, we depict a quantum algorithm with a single qutrit. Since it is not easy to decompose an arbitrary d by d unitary matrix into Z and X phase gates when d > 2, the proof of the universality of qudit ZX calculus for quantum mechanics is far from trivial. We construct a counterexample to Ranchin's universality proof, and give another proof by Lie theory that the qudit ZX …
Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation $I_2\otimes V$ and $I_2\otimes W$. We show that $V$ and $W$ have no zero entry, and apply it to exclude examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is $\log_2 3$ ebits. …
Quantum state manipulation with gates based on geometric phases acquired during cyclic operations promises inherent fault-tolerance and resilience to local fluctuations in the control parameters. Here we create a general non-Abelian and non-adiabatic holonomic gate acting in the $(\ket{0},\ket{2})$ subspace of a three-level transmon fabricated in a fully coplanar design. Experimentally, this is realized by simultaneously coupling the first two transitions by microwave pulses with amplitudes and phases defined such that the condition of parallel transport is fulfilled. We demonstrate the creati…
Efficient Implementation of a Single-Qutrit Gate Set via Coherent Control
National Laboratory of Solid State Microstructures, School of Physics, Nanjing University（南京大学固体微结构物理学院）；Shishan Laboratory, Nanjing University（南京大学时山实验室）；Jiangsu Key Laboratory of Quantum Information Science and Technology, Nanjing University（南京大学江苏省量子信息科学与技术重点实验室）；Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China（中国科学技术大学量子信息与量子科技创新中心）；Hefei National Laboratory（合肥国家实验室）
Physically motivated decompositions of single-qutrit gates
School of Physics and Applied Physics, Southern Illinois University（南伊利诺伊大学物理与应用物理学院）；Advanced ICT Research Institute, National Institute of Information and Communications Technology（信息通信技术研究所先进ICT研究部，日本国立情报学研究所）；Research Institute for Science and Technology, Tokyo University of Science（东京理科大学科学技术研究院）；Department of Physics, University of the Basque Country UPV/EHU（巴斯克大学物理系）；IKERBASQUE Basque Foundation for Science（伊克尔巴斯科奎科学基金会）；EHU Quantum Center, University of the Basque Country UPV/EHU（巴斯克大学EHU量子中心）
According to the theoretical results, the quantum searching algorithm can be generalized by replacing the Walsh-Hadamard(W-H) transform by almost any quantum mechanical operation. We have implemented the generalized algorithm using nuclear magnetic resonance techniques with a solution of chloroform molecules. Experimental results show the good agreement between theory and experiment.
The improved quantum scheduling algorithm proposed by Grover has been generalized using the generalized quantum search algorithm, in which a unitary operator replaces the Walsh-Hadamard transform, and $π/2$ phase rotations replace the selective inversions, in order to make the quantum scheduling algorithm suitable for more cases. Our scheme is realized on a nuclear magnetic resonance (NMR) quantum computer. Experimental results show a good agreement between theory and experiment.
We explicitly derive a connection between quantum circuits utilising IBM's quantum gate set and multivariate quadratic polynomials over integers modulo 8. We demonstrate that the action of a quantum circuit over input qubits can be written as generalized Walsh-Hadamard transform. Here, we derive the polynomials corresponding to implementations of the Swap gate and Toffoli gate using IBM-Q gate set.
We report experimental realization of high-fidelity photonic quantum gates for frequency-encoded qubits and qutrits based on electro-optic modulation and Fourier-transform pulse shaping. Our frequency version of the Hadamard gate offers near-unity fidelity ($0.99998\pm0.00003$), requires only a single microwave drive tone for near-ideal performance, functions across the entire C-band (1530-1570 nm), and can operate concurrently on multiple qubits spaced as tightly as four frequency modes apart, with no observable degradation in the fidelity. For qutrits we implement a $3\times 3$ extension of …
We show that an n-th root of the Walsh-Hadamard transform (obtained from the Hadamard gate and a cyclic permutation of the qubits), together with two diagonal matrices, namely a local qubit-flip (for a fixed but arbitrary qubit) and a non-local phase-flip (for a fixed but arbitrary coefficient), can do universal quantum computation on n qubits. A quantum computation, making use of n qubits and based on these operations, is then a word of variable length, but whose letters are always taken from an alphabet of cardinality three. Therefore, in contrast with other universal sets, no choice of qubi…
We present a general methodology to obtain the basis of qudits which are admissible to Quantum Fourier Transform (QFT). We first study this method for qubits to characterize the ensemble that works for the Hadamard transformation (QFT for two dimension). In this regard we identify certain incompleteness in the result of Maitra and Parashar (IJQI, 2006). Next we characterize the ensemble of qutrits for which QFT is possible. Further, some theoretical results related to higher dimensions are also discussed.
Indian Institute of Technology Delhi（印度理工学院德里分校）；CHRIST (Deemed to be University)（基督（被视为大学）大学）；Wichita State University（威奇托州立大学）
Classical self-supervised networks suffer from convergence problems and reduced segmentation accuracy due to forceful termination. Qubits or bi-level quantum bits often describe quantum neural network models. In this article, a novel self-supervised shallow learning network model exploiting the sophisticated three-level qutrit-inspired quantum information system referred to as Quantum Fully Self-Supervised Neural Network (QFS-Net) is presented for automated segmentation of brain MR images. The QFS-Net model comprises a trinity of a layered structure of qutrits inter-connected through parametri…
In the paper (math-ph/0504049) Jarlskog gave an interesting simple parametrization to unitary matrices, which was essentially the canonical coordinate of the second kind in the Lie group theory (math-ph/0505047). In this paper we apply the method to a quantum computation based on multi-level system (qudit theory). Namely, by considering that the parametrization gives a complete set of modules in qudit theory, we construct the generalized Pauli matrices which play a central role in the theory and also make a comment on the exchange gate of two-qudit systems. Moreover we give an explicit constru…
A quantum computer directly manipulates information stored in the state of quantum mechanical systems. The available operations have many attractive features but also underly severe restrictions, which complicate the design of quantum algorithms. We present a divide-and-conquer approach to the design of various quantum algorithms. The class of algorithm includes many transforms which are well-known in classical signal processing applications. We show how fast quantum algorithms can be derived for the discrete Fourier transform, the Walsh-Hadamard transform, the Slant transform, and the Hartley…
We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM- and POVM-dynamical (quantum) entropies. They quantify the randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator. We study the class of chaotic unitaries, i.e., the ones of maximal entropy or, equivalently, such that they can be represented by suitably rescaled complex Hadamard matrices in some orthonormal bases. We provide…
We generalize quantum circuits for the Toffoli gate presented by Selinger and Jones for functionally controlled NOT gates, i.e., $X$ gates controlled by arbitrary $n$-variable Boolean functions. Our constructions target the gate set consisting of Clifford gates and single qubit rotations by arbitrary angles. Our constructions use the Walsh-Hadamard spectrum of Boolean functions and build on the work by Schuch and Siewert and Welch et al. We present quantum circuits for the case where the target qubit is in an arbitrary state as well as the special case where the target is in a known state. Add…
