Precision control of a quantum system requires accurate determination of the effective system Hamiltonian. We develop a method for estimating the Hamiltonian parameters for some unknown two-state system and providing uncertainty bounds on these parameters. This method requires only one measurement basis and the ability to initialise the system in some arbitrary state which is not an eigenstate of the Hamiltonian in question. The scaling of the uncertainty is studied for large numbers of measurements and found to be proportional to one on the square-root of the number of measurements.
Efficient Quantum Circuit Construction of Controlled Time-Evolution for Arbitrary Pauli-Sum Hamiltonians
Faculty of Engineering, Yokohama National University（横滨国立大学工学部）；Department of Applied Physics and Physico-Informatics, Keio University（庆应义塾大学应用物理·物理信息系）
We use concurrence as an entanglement measure and experimentally demonstrate the entanglement classification of arbitrary three-qubit pure states on a nuclear magnetic resonance (NMR) quantum information processor. Computing the concurrence experimentally under three different bipartitions, for an arbitrary three-qubit pure state, reveals the entanglement class of the state. The experiment involves measuring the expectation values of Pauli operators. This was achieved by mapping the desired expectation values onto the local $z$ magnetization of a single qubit. We tested the entanglement classi…
Identifying Hamiltonian of a quantum system is of vital importance for quantum information processing. In this Letter, we realized and benchmarked a quantum Hamiltonian identification algorithm recently proposed [Phys. Rev. Lett. \textbf{113}, 080401 (2014)]. we realized the algorithm on liquid nuclear magnetic resonance quantum information processor using two different working media with different forms of Hamiltonian. Our experiment realized the quantum identification algorithm based on free induction decay signals. We also showed how to process data obtained in practical experiment. We stud…
We consider systems where dynamical variables are the generators of the SU(2) group. A subset of these Hamiltonians is exactly solvable using the Bethe ansatz techniques. We show that Bethe ansatz equations are equivalent to polynomial relationships between the operator invariants, or equivalently, between eigenvalues of those invariants.
Synthesizing Arbitrary Non-Hermitian Hamiltonian with Stochastic Floquet Engineering
College of Science, NUDT（国防科技大学理学院）；Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University（天津大学理学院量子联合研究中心与物理系）；Department of Physics, Hunan Normal University（湖南师范大学物理系）
The Riemann hypothesis, one of the most important open problems in pure mathematics, implies the most profound secret of prime numbers. One of the most interesting approaches to solve this hypothesis is to connect the problem with the spectrum of the physical Hamiltonian of a quantum system. However, none of the proposed quantum Hamiltonians have been experimentally feasible.Here, we report the first experiment to identify the first non-trivial zeros of the Riemann zeta function and the first two zeros of Pólya's fake zeta function, using a novel Floquet method, through properly designed perio…
Hamiltonians describing the relativistic quantum dynamics of a particle with an arbitrary spin are shown to exhibit a supersymmetric structure when the even and odd elements of the Hamiltonian commute. For such supersymmetric Hamiltonians an exact Foldy-Wouthuysen transformation exits which brings it into a block-diagonal form separating the positive and negative energy subspaces. Here the supercharges transform between energy eigenstates of positive and negative energy. The relativistic dynamics of a charged particle in a magnetic field is considered for the case of a scalar (spin-zero) boson…
Ising spin Hamiltonians are often used to encode a computational problem in their ground states. Quantum Annealing (QA) computing searches for such a state by implementing a slow time-dependent evolution from an easy-to-prepare initial state to a low energy state of a target Ising Hamiltonian of quantum spins, $H_I$. Here, we point to the existence of an analytical solution for such a problem for an arbitrary $H_I$ beyond the adiabatic limit for QA. This solution provides insights into the accuracy of nonadiabatic computations. Our QA protocol in the pseudo-adiabatic regime leads to a monotoni…
We revisit the problem of characterizing band topology in dynamically-stable quadratic bosonic Hamiltonians that do not conserve particle number. We show this problem can be rigorously addressed by a smooth and local adiabatic mapping procedure to a particle number conserving Hamiltonian. In contrast to a generic fermionic pairing Hamiltonian, such a mapping can always be constructed for bosons. Our approach shows that particle non-conserving bosonic Hamiltonians can be classified using known approaches for fermionic models. It also provides a simple means for identifying and calculating appro…
A new procedure to diagonalize quadratic Hamiltonians is introduced. We show that one can find a unitary transformation such that the transformed quadratic Hamiltonian is diagonal but still written in terms of the original position and momentum observables. We give a general method to diagonalize an arbitrary quadratic Hamiltonian and derive a few of the simplest special cases in detail.
We prove that the entanglement entropy of any pure initial state of a bipartite bosonic quantum system grows linearly in time with respect to the dynamics induced by any unstable quadratic Hamiltonian. The growth rate does not depend on the initial state and is equal to the sum of certain Lyapunov exponents of the corresponding classical dynamics. This paper generalizes the findings of [Bianchi et al., JHEP 2018, 25 (2018)], which proves the same result in the special case of Gaussian initial states. Our proof is based on a recent generalization of the strong subadditivity of the von Neumann e…
We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to arbitrary spatial dimensions. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this f…
Certain real parameters of a Hamiltonian, when continued to complex values, can give rise to singular points called exceptional points ($EP$'s), where two or more eigenvalues coincide and the complexified Hamiltonian becomes non-diagonalizable. We show that for a generic $d$-dimensional topological superconductor / superfluid with a chiral symmetry, one can find $EP$'s associated with the chiral zero energy Majorana fermions bound to a topological defect / edge. Exploiting the chiral symmetry, we propose a formula for counting the number ($n$) of such chiral zero modes. We also establish the c…
We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form $D_{ω,λ}:=\begin{bmatrix}-\frac{λ+ω}{x}&-\partial_x \\ \partial_x & -\frac{λ-ω}{x}\end{bmatrix}$. We describe their closed realizations in the sense of the Hilbert space $L^2(\mathbb R_+,\mathbb C^2)$, allowing for complex values of the parameters $λ,ω$. In physical situations, $λ$ is proportional to the electric charge and $ω$ is related to the angular momentum. We focus on realizations of $D_{ω,λ}$ homogeneous of degree $-1$. They can be organized in a single holomorphic family of clos…
