Metric Structure of the Space of Two-Qubit Gates, Perfect Entanglers and Quantum Control

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.Comment: 27 pages, 5 figure

A geometric theory of non-local two-qubit operations

We study non-local two-qubit operations from a geometric perspective. By applying a Cartan decomposition to su(4), we find that the geometric structure of non-local gates is a 3-Torus. We derive the invariants for local transformations, and connect these local invariants to the coordinates of the 3-Torus. Since different points on the 3-Torus may correspond to the same local equivalence class, we use the Weyl group theory to reduce the symmetry. We show that the local equivalence classes of two-qubit gates are in one-to-one correspondence with the points in a tetrahedron except on the base. We then study the properties of perfect entanglers, that is, the two-qubit operations that can generate maximally entangled states from some initially separable states. We provide criteria to determine whether a given two-qubit gate is a perfect entangler and establish a geometric description of perfect entanglers by making use of the tetrahedral representation of non-local gates. We find that exactly half the non-local gates are perfect entanglers. We also investigate the non-local operations generated by a given Hamiltonian. We first study the gates that can be directly generated by a Hamiltonian. Then we explicitly construct a quantum circuit that contains at most three non-local gates generated by a two-body interaction Hamiltonian, together with at most four local gates generated by single qubit terms. We prove that such a quantum circuit can simulate any arbitrary two-qubit gate exactly, and hence it provides an efficient implementation of universal quantum computation and simulation.Comment: 22 pages, 6 figure

Universal quantum Controlled-NOT gate

An investigation of an optimal universal unitary Controlled-NOT gate that performs a specific operation on two unknown states of qubits taken from a great circle of the Bloch sphere is presented. The deep analogy between the optimal universal C-NOT gate and the `equatorial' quantum cloning machine (QCM) is shown. In addition, possible applications of the universal C-NOT gate are briefly discussed.Comment: 18 reference

Deterministic Generation of Multipartite Entanglement via Causal Activation in the Quantum Internet

Entanglement represents ``\textit{the}'' key resource for several applications of quantum information processing, ranging from quantum communications to distributed quantum computing. Despite its fundamental importance, deterministic generation of maximally entangled qubits represents an on-going open problem. Here, we design a novel generation scheme exhibiting two attractive features, namely, i) deterministically generating different classes -- namely, GHZ-like, W-like and graph states -- of genuinely multipartite entangled states, ii) without requiring any direct interaction between the qubits. Indeed, the only necessary condition is the possibility of coherently controlling -- according to the indefinite causal order framework -- the causal order among the unitaries acting on the qubits. Through the paper, we analyze and derive the conditions on the unitaries for deterministic generation, and we provide examples for unitaries practical implementation. We conclude the paper by discussing the scalability of the proposed scheme to higher dimensional genuine multipartite entanglement (GME) states and by introducing some possible applications of the proposal for quantum networks

Optimizing entangling quantum gates for physical systems

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.Comment: extended version; Phys. Rev. A (2011

Krotov: A Python implementation of Krotov's method for quantum optimal control

We present a new open-source Python package, krotov, implementing the quantum optimal control method of that name. It allows to determine time-dependent external fields for a wide range of quantum control problems, including state-to-state transfer, quantum gate implementation and optimization towards an arbitrary perfect entangler. Krotov's method compares to other gradient-based optimization methods such as gradient-ascent and guarantees monotonic convergence for approximately time-continuous control fields. The user-friendly interface allows for combination with other Python packages, and thus high-level customization

Quantum Optimal Control via Semi-Automatic Differentiation

We develop a framework of "semi-automatic differentiation" that combines existing gradient-based methods of quantum optimal control with automatic differentiation. The approach allows to optimize practically any computable functional and is implemented in two open source Julia packages, GRAPE.jl and Krotov.jl, part of the QuantumControl.jl framework. Our method is based on formally rewriting the optimization functional in terms of propagated states, overlaps with target states, or quantum gates. An analytical application of the chain rule then allows to separate the time propagation and the evaluation of the functional when calculating the gradient. The former can be evaluated with great efficiency via a modified GRAPE scheme. The latter is evaluated with automatic differentiation, but with a profoundly reduced complexity compared to the time propagation. Thus, our approach eliminates the prohibitive memory and runtime overhead normally associated with automatic differentiation and facilitates further advancement in quantum control by enabling the direct optimization of non-analytic functionals for quantum information and quantum metrology, especially in open quantum systems. We illustrate and benchmark the use of semi-automatic differentiation for the optimization of perfectly entangling quantum gates on superconducting qubits coupled via a shared transmission line. This includes the first direct optimization of the non-analytic gate concurrence.Comment: 30 pages, 2 figures, 2 tables. Accepted in Quantu

Quantum Instruction Set Design for Performance

A quantum instruction set is where quantum hardware and software meet. We develop new characterization and compilation techniques for non-Clifford gates to accurately evaluate different quantum instruction set designs. We specifically apply them to our fluxonium processor that supports mainstream instruction $\mathrm{iSWAP}$ by calibrating and characterizing its square root $\mathrm{SQiSW}$. We measure a gate fidelity of up to $99.72\%$ with an average of $99.31\%$ and realize Haar random two-qubit gates using $\mathrm{SQiSW}$ with an average fidelity of $96.38\%$. This is an average error reduction of $41\%$ for the former and a $50\%$ reduction for the latter compared to using $\mathrm{iSWAP}$ on the same processor. This shows designing the quantum instruction set consisting of $\mathrm{SQiSW}$ and single-qubit gates on such platforms leads to a performance boost at almost no cost.Comment: 2 figures in main text and 21 figures in Supplementary Materials. This manuscript subsumes version 1 with significant improvements such as experimental demonstration and materials presentatio