We construct quantum gates entanglers for different classes of multipartite states. In particular we construct entangler operators for W and GHZ classes of multipartite states based on the construction of the concurrence classes. We also in detail discuss these two classes of the quantum gates entanglers for three-partite states.
We construct quantum gate entanglers for different classes of multipartite states based on definition of W and GHZ concurrence classes. First, we review the basic construction of concurrence classes based on orthogonal complement of a positive operator valued measure (POVM) on quantum phase. Then, we construct quantum gates entanglers for different classes of multi-qubit states. In particular, we show that these operators can entangle multipartite state if they satisfy some conditions for W and GHZ classes of states. Finally, we explicitly give the W class and GHZ classes of quantum gate entan…
We derive expressions for the invariant length element and measure for the simple compact Lie group SU(4) in a coordinate system particularly suitable for treating entanglement in quantum information processing. Using this metric, we compute the invariant volume of the space of two-qubit perfect entanglers. We find that this volume corresponds to more than 84% of the total invariant volume of the space of two-qubit gates. This same metric is also used to determine the effective target sizes that selected gates will present in any quantum-control procedure designed to implement them.
We construct the general gate operator for 2-players 3-strategies ELW games. It is shown that such a gate, implementing classical strategies, can be constructed out of the elements of Cartan subalgebra of $SU(3)$. The relation between the degree of entanglement and the structure of stability subgroup of initial state is analyzed.
Non-local properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix U equal to the Shannon entropy of the vector of singular values of the reshuffled matrix. Averaging the entropy over the Haar measure on U(N^2) we find its asymptotic behaviour. For two--qubit quantum gates we derive the induced probability distribution of the interaction content and show that the relative volume of the set of perfect entanglers reads 8/3 π\approx 0.85. We establish explicit conditions under which a giv…
Optimal control theory is a powerful tool for improving figures of merit in quantum information tasks. Finding the solution to any optimal control problem via numerical optimization depends crucially on the choice of the optimization functional. Here, we derive a functional that targets the full set of two-qubit perfect entanglers, gates capable of creating a maximally-entangled state out of some initial product state. The functional depends on easily-computable local invariants and uniquely determines when a gate evolves into a perfect entangler. Optimization with our functional is most usefu…
The difficulty of an optimization task in quantum information science depends on the proper mathematical expression of the physical target. Here we demonstrate the power of optimization functionals targeting an arbitrary perfect two-qubit entangler, creating a maximally-entangled state out of some initial product state. For two quantum information platforms of current interest, nitrogen vacancy centers in diamond and superconducting Josephson junctions, we show that an arbitrary perfect entangler can be reached faster and with higher fidelity than specific two-qubit gates or local equivalence …
The ability to reach a maximally entangled state from a separable one through the use of a two-qubit unitary operator is analyzed for mixed states. This extension from the known case of pure states shows that there are at least two families of gates which are able to give maximum entangling power for all values of purity. It is notable that one of this gates coincides with a maximum discording one. We give analytical proof that such gate is indeed perfect entangler at all purities and give numerical evidence for the existence of the second one. Further, we find that there are other gates, many…
A fundamental requirement in the circuit model of quantum information processing is the realization of fault-tolerant multi-qubit quantum gates with entangling capabilities. A key step towards this end is to achieve control of qubit states through geometric phases at very small spatial scales in an effective and feasible way. A spin-electric coupling present in antiferromagnetic triangular single-molecule magnets (SMMs) allows for manipulation of the spin (qubit) states with a great flexibility. Here, we establish an all-electrical two-qubit geometric phase shift gate acting on the four-fold g…
We construct a generalized controlled phased gate entangler for a multi-qubit state based on the geometrical structure of quantum systems. We also investigate relation between the generalized controlled phase construction of a quantum gate entangler and graph state for two-qubit and three-qubit states.
We present a Superconducting Planar ARchitecture for Quantum Simulations (SPARQS) intended to implement a scalable qubit layout for quantum simulators. To this end, we describe the iFREDKIN gate as a controlled entangler for the simulation of Fermionic systems that is advantageous if it can be directly implemented. Using optimal control, we show that and how this gate can be efficiently implemented in the SPARQS circuit, making it a promising platform and control scheme for quantum simulations. Such a quantum simulator can be built with current quantum technologies to advance the design of mol…
Producing and maintaining entanglement reside at the heart of the optimal construction of quan- tum operations and are fundamental issues in the realization of universal quantum computation. We here introduce a setup of spin qubits that allows for geometric implementation of entangling gates between the register qubits with any arbitrary entangling power. We show this by demon- strating a circuit through a spin chain, which performs universal nonadiabatic holonomic two-qubit entanglers. The proposed gates are all electric and geometric, which would help to realize fast and robust entangling ga…
As a class of multipartite entangled states, the multipartite concatenated GHZ (C-GHZ) states remain superior stability under the influence of decoherence. We propose two scalable experimental realization of the multiphoton C-GHZ states based on the entanglers of multiphoton GHZ state. Given a $km$-photon GHZ state as an input state, if $m$ is odd, one can create a $km$-photon C-GHZ state. Also, generally, we design a scheme to prepare $km$-photon C-GHZ states from $km$ single-photon states by using $k$ entanglers of $m$-photon GHZ state and $k$ $m$-control Toffoli gates.
The two-players $N$ strategies games quantized according to the Eisert-Lewenstein-Wilkens scheme (Phys. Rev. Lett. 83 (1999), 3077) are considered. Group theoretical methods are applied to the problem of finding a general form of gate operators (entanglers) under the assumption that the set of classical pure strategies is contained in the set of pure quantum ones. The role of the stability group of the initial state of the game is stressed. As an example, it is shown that the maximally entangled games do not admit nontrivial pure Nash strategies. The general arguments are supported by explicit…
The capacity of a quantum gate to produce entangled states on a bipartite system is quantified in terms of the entangling power. This quantity is defined as the average of the linear entropy of entanglement of the states produced after applying a quantum gate over the whole set of separable states. Here we focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit space, and calculate the entangling power in terms of the appropriate local-invariant. A geometric description of the local equivalence classes of gates is given in terms of the $\mathfrak{su}(3)$ Lie algebra root …
We study the quantum brachistochrone evolution for a system of two spins-$\frac{1}{2}$ described by an anisotropic Heisenberg Hamiltonian without $zx$, $zy$ interacting couplings in magnetic field directed along the z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by $\|\uparrow\uparrow\rangle$, $\|\downarrow\downarrow\rangle$ and $\|\uparrow\downarrow\rangle$, $\|\downarrow\uparrow\rangle$ separately and allows to consider the brachistochrone problem on each subspace separately. Using the evolution operator for this Hamiltonian we generate quantum gates, namely an …
Fourier-Geometric Circuit Design for Gate and Entanglement Placement in Quantum Neural Networks
Korea University（韩国大学）；Sookmyung Women’s University（淑明女子大学）；Purdue University（普渡大学）；Indiana University（印第安纳大学）；Seoul National University Hospital（首尔大学医院）
This study examines the possibility of finding perfect entanglers for a Hamiltonian which corresponds to several quantum information platforms of interest at the present time. However, in this study, we use a superconducting circuit that stands out from other quantum-computing devices, especially because Transmon qubits can be coupled via capacitors or microwave cavities, which enable us to combine high coherence, fast gates, and high flexibility in its design parameters. There are currently two factors limiting the performance of superconducting processors: timing mismatch and the limitation …
We study the minimum time to implement an arbitrary two-qubit gate in two heteronuclear spins systems. We give a systematic characterization of two-qubit gates based on the invariants of local equivalence. The quantum gates are classified into four classes, and for each class the analytical formula of the minimum time to implement the quantum gates is explicitly presented. For given quantum gates, by calculating the corresponding invariants one easily obtains the classes to which the quantum gates belong. In particular, we analyze the effect of global phases on the minimum time to implement th…
We present a collection of results concerning the structure of reversible gate classes over non-binary alphabets, including (1) a reversible gate class over non-binary alphabets that is not finitely generated (2) an explicit set of generators for the class of all gates, the class of all conservative gates, and a class of generalizations of the two (3) an embedding of the poset of reversible gate classes over an alphabet of size $k$ into that of an alphabet of size $k+1$ (4) a classification of gate classes containing the class of $(k-1,1)$-conservative gates, meaning gates that preserve the nu…
A general scheme is presented to decompose a $d$-by-$d$ unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in $d-1$ classes, where each class is isomorphic to the group of $2\times 2$ unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on $n$-qubit registe…
Quantum Optimal Control (QOC) enables the realization of accurate operations, such as quantum gates, and support the development of quantum technologies. To date, many QOC frameworks have been developed but those remain only naturally suited to optimize a single targeted operation at a time. We extend this concept to optimal control with a continuous family of targets, and demonstrate that an optimization based on neural networks can find families of time-dependent Hamiltonians that realize desired classes of quantum gates in minimal time.
A method for the implementation of a universal set of fault-tolerant logical gates is presented using homological product codes. In particular, it is shown that one can fault-tolerantly map between different encoded representations of a given logical state, enabling the application of different classes of transversal gates belonging to the underlying quantum codes. This allows for the circumvention of no-go results pertaining to universal sets of transversal gates and provides a general scheme for fault-tolerant computation while keeping the stabilizer generators of the code sparse.
Optimal control theory is a versatile tool that presents a route to significantly improving figures of merit for quantum information tasks. We combine it here with the geometric theory for local equivalence classes of two-qubit operations to derive an optimization algorithm that determines the best entangling two-qubit gate for a given physical setting. We demonstrate the power of this approach for trapped polar molecules and neutral atoms.
Reducing the circuit depth of quantum circuits is a crucial bottleneck to enabling quantum technology. This depth is inversely proportional to the number of available quantum gates that have been synthesised. Moreover, quantum gate synthesis and control problems exhibit a vast range of external parameter dependencies, both physical and application-specific. In this article we address the possibility of learning families of optimal control pulses which depend adaptively on various parameters, in order to obtain a global optimal mapping from the space of potential parameter values to the control…
We examine the following problem: given a collection of Clifford gates, describe the set of unitaries generated by circuits composed of those gates. Specifically, we allow the standard circuit operations of composition and tensor product, as well as ancillary workspace qubits as long as they start and end in states uncorrelated with the input, which rule out common "magic state injection" techniques that make Clifford circuits universal. We show that there are exactly 57 classes of Clifford unitaries and present a full classification characterizing the gate sets which generate them. This is th…
In this work we explore a correspondence between quantum circuits and low-degree polynomials over the finite field F_2. Any quantum circuit made up of Hadamard, Z, controlled-Z and controlled-controlled-Z gates gives rise to a degree-3 polynomial over F_2 such that calculating quantum circuit amplitudes is equivalent to counting zeroes of the corresponding polynomial. We exploit this connection, which is especially clean and simple for this particular gate set, in two directions. First, we give proofs of classical hardness results based on quantum circuit concepts. Second, we find efficient cl…
We propose a method to generate arbitrary symmetric states of N qubits, which can be easily associated with their entanglement classes. It is particularly suited to quantum optics systems like trapped ions or superconducting circuits. We encode each qubit in two metastable levels of the system and use a bosonic quantum bus for creating the states. The method is deterministic and relies on a sequence of selective unitary gates upon the qubits within the system coherence time.
Despite using a novel model of computation, quantum computers break down programs into elementary gates. Among such gates, entangling gates are the most expensive. In the context of fermionic simulations, we develop a suite of compilation and optimization techniques that massively reduce the entangling-gate counts. We exploit the well-studied non-quantum optimization algorithms to achieve up to 24\% savings over the state of the art for several small-molecule simulations, with no loss of accuracy or hidden costs. Our methodologies straightforwardly generalize to wider classes of near-term simu…
Hybrid Classical-Quantum Neural Networks for Multi-Characteristic Co-Optimization of Recessed-Gate AlGaN/GaN MIS-HEMTs
National Yang Ming Chiao Tung University（国立阳明交通大学）；NVIDIA Corp.（英伟达公司）；National Center for High-performance Computing, National Applied Research Laboratories（国家高速网络与计算中心，国家应用材料研究中心）
A proposal for a conditional definition of the Swap Gate
为了实现量子逻辑运算，量子计算（QC）要求其基本工具和概念必须遵守物理定律。QC中的一个基本概念是条件量子动力学\cite{barenco}，有时称为受控幺正操作\cite{kumar}，它由条件“If-Then”语句建立。最著名的例子是c-not门，它在计算基础上运行如下：\textit{If the control qubit is set to $% |1>$, then apply the single qubit quantum NOT gate on the target. Otherwise, if the control qubit is set to $|0>$, then the target qubit is unchanged}。这个门代表了条件量子动力学的范例，其中目标量子位的翻转以控制量子位的状态为条件。其他门也是以类似的条件演化方式定义的;例如，控制相位门\cite{paulo1}。然而，据我们所知，这种条件量子动力学尚未被用来定义SWAP门。在这里，我们提出了一个可能的条件定义，在If-Then语句的形式，在两个量子比特的情况下，构建一个交换门。这个定义提出了两类分类，这取决于必须经历条件量子动力学的量子比特的数量。
