We show how to map a given n-qubit target Hamiltonian with bounded-strength k-body interactions onto a simulator Hamiltonian with two-body interactions, such that the ground-state energy of the target and the simulator Hamiltonians are the same up to an extensive error O(epsilon n) for arbitrary small epsilon. The strength of interactions in the simulator Hamiltonian depends on epsilon and k but does not depend on n. We accomplish this reduction using a new way of deriving an effective low-energy Hamiltonian which relies on the Schrieffer-Wolff transformation of many-body physics.
In single Hilbert space, Pauli's well-known theorem implies that the existence of a self-adjoint time operator canonically conjugate to a given Hamiltonian signifies that the time operator and the Hamiltonian possess completely continuous spectra spanning the entire real line. Thus the conclusion that there exists no self-adjoint time operator conjugate to a semibounded or discrete Hamiltonian despite some well-known illustrative counterexamples. In this paper we evaluate Pauli's theorem against the single Hilbert space formulation of quantum mechanics, and consequently show the consistency of…
In this paper, we present a protocol to engineer upper-bounded and sliced Jaynes-Cummings and anti-Jaynes-Cummings Hamiltonians in cavity quantum electrodynamics. In the upper-bounded Hamiltonians, the atom-field interaction is confined to a subspace of Fock states ranging from $\left\vert 0\right\rangle $ up to $\left\vert 4\right\rangle $, while in the sliced interaction the Fock subspace ranges from $\left\vert M\right\rangle $ up to $\left\vert M+4\right\rangle $. We also show how to build upper-bounded and sliced Liouvillians irrespective of engineering Hamiltonians. The upper-bounded and…
We address the task of switching off the Hamiltonian of a system by removing all internal and system-environment couplings. We propose dynamical decoupling schemes, that use only bounded-strength controls, for quantum many-body systems with local system Hamiltonians and local environmental couplings. To do so, we introduce the combinatorial concept of balanced-cycle orthogonal arrays (BOAs) and show how to construct them from classical error-correcting codes. The derived decoupling schemes may be useful as a primitive for more complex schemes, e.g., for Hamiltonian simulation. For the case of …
We propose dynamical control schemes for Hamiltonian simulation in many-body quantum systems that avoid instantaneous control operations and rely solely on realistic bounded-strength control Hamiltonians. Each simulation protocol consists of periodic repetitions of a basic control block, constructed as a suitable modification of an "Eulerian decoupling cycle," that would otherwise implement a trivial (zero) target Hamiltonian. For an open quantum system coupled to an uncontrollable environment, our approach may be employed to engineer an effective evolution that simulates a target Hamiltonian …
We consider the task of approximating the ground state energy of two-local quantum Hamiltonians on bounded-degree graphs. Most existing algorithms optimize the energy over the set of product states. Here we describe a family of shallow quantum circuits that can be used to improve the approximation ratio achieved by a given product state. The algorithm takes as input an $n$-qubit product state $|v\rangle$ with mean energy $e_0=\langle v|H|v\rangle$ and variance $\mathrm{Var}=\langle v|(H-e_0)^2|v\rangle$, and outputs a state with an energy that is lower than $e_0$ by an amount proportional to $…
We systematically formulate a hierarchy of isospectral Hamiltonians in one-dimensional supersymmetric quantum mechanics on an interval and on a circle, in which two successive Hamiltonians form N=2 supersymmetry. We find that boundary conditions compatible with supersymmetry are severely restricted. In the case of an interval, a hierarchy of, at most, three isospectral Hamiltonians is possible with unique boundary conditions, while in the case of a circle an infinite tower of isospectral Hamiltonians can be constructed with two-parameter family of boundary conditions.
This article investigates the stability of the ground state subspace of a canonical parent Hamiltonian of a Matrix product state against local perturbations. We prove that the spectral gap of such a Hamiltonian remains stable under weak local perturbations even in the thermodynamic limit, where the entire perturbation might not be bounded. Our discussion is based on preceding work by D.A. Yarotsky that develops a perturbation theory for relatively bounded quantum perturbation of classical Hamiltonians. We exploit a renormalization procedure, which on large scale transforms the parent Hamiltoni…
The process of measuring a two-level quantum system was examined by applying Hamiltonian formalism. For the measurement of an observable that does not commute with the system Hamiltonian, a non-trivial relationship among the strength of interaction, the time interval of the process, and the accuracy of the measurement was obtained. Particularly, to achieve an error-free measurement of such an observable, a condition stating that the interaction Hamiltonian does not commute with the system Hamiltonian needs to be satisfied.
We propose new Lieb-Robinson bounds (bounds on the speed of propagation of information in quantum systems) with an explicit dependence on the interaction strengths of the Hamiltonian. For systems with more than two interactions it is found that the Lieb-Robinson speed is not always algebraic in the interaction strengths. We consider Hamiltonians with any finite number of bounded operators and also a certain class of unbounded operators. We obtain bounds and propagation speeds for quantum systems on lattices and also general graphs possessing a kind of homogeneity and isotropy. One area for whi…
Beyond Integrability Preserving Renormalization-Group Protocol in Non-Hermitian Hamiltonians with Time-Dependent Interaction Strengths
Condensed Matter Theory Center, Department of Physics, University of Maryland（马里兰大学物理系凝聚态理论中心）；Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校物理与天文学系）
We investigate relations between spectral properties of a single-centre point-interaction Hamiltonian describing a particle confined to a bounded domain $Ω\subset\mathbb{R}^{d},\: d=2,3$, with Dirichlet boundary, and the geometry of $Ω$. For this class of operators Krein's formula yields an explicit representation of the resolvent in terms of the integral kernel of the unperturbed one, $(-Δ_Ω^{D}+z) ^{-1}$. We use a moving plane analysis to characterize the behaviour of the ground-state energy of the Hamiltonian with respect to the point-interaction position and the shape of $Ω$, in particular…
We show that the eigenvectors of the PT-symmetric imaginary cubic oscillator are complete, but do not form a Riesz basis. This results in the existence of a bounded metric operator having intrinsic singularity reflected in the inevitable unboundedness of the inverse. Moreover, the existence of non-trivial pseudospectrum is observed. In other words, there is no quantum-mechanical Hamiltonian associated with it via bounded and boundedly invertible similarity transformations. These results open new directions in physical interpretation of PT-symmetric models with intrinsically singular metric, si…
The task of decoupling, i.e., removing unwanted interactions in a system Hamiltonian and/or couplings with an environment (decoherence), plays an important role in controlling quantum systems. There are many efficient decoupling schemes based on combinatorial concepts like orthogonal arrays, difference schemes and Hadamard matrices. So far these (combinatorial) decoupling schemes have relied on the ability to effect sequences of instantaneous, arbitrarily strong control Hamiltonians (bang-bang controls). To overcome the shortcomings of bang-bang control Viola and Knill proposed a method called…
Towards Nonperturbative Solution of Quantum Dynamics : A Hamiltonian Mean Field Approximation Scheme with Perturbation Theory for Arbitray Strength of Interaction
We study a scheme of quantum simulator for two-dimensional xy-model Hamiltonian. Previously the quantum simulator for a coupled cavity array spin model has been explored, but the coupling strength is fixed by the system parameters. In the present scheme several cavity resonators can be coupled with each other simultaneously via an ancilla qubit. In the two-dimensional Kagome lattice of the resonators the hopping of resonator photonic modes gives rise to the tight-binding Hamiltonian which in turn can be transformed to the quantum xy-model Hamiltonian. We employ the transmon as an ancilla qubit…
Entanglement in the ground state of a many-body quantum system may arise when the local terms in the system Hamiltonian fail to commute with the interaction terms in the Hamiltonian. We quantify this phenomenon, demonstrating an analogy between ground-state entanglement and the phenomenon of frustration in spin systems. In particular, we prove that the amount of ground-state entanglement is bounded above by a measure of the extent to which interactions frustrate the local terms in the Hamiltonian. As a corollary, we show that the amount of ground-state entanglement is bounded above by a ratio …
We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H_0 we prove that there exists a constant threshold ε>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions the perturbed Hamiltonian H=H_0+εV has well-defined spectral bands originating from O(1) smallest eigenvalues of H_0. The…
We demonstrate how using two-qubit composite rotations a high fidelity controlled-NOT (CNOT) gate can be constructed, even when the strength of the interaction between qubits is not accurately known. We focus on the exchange interaction oscillation in silicon based solid-state architectures with a Heisenberg Hamiltonian. This method easily applies to a general two-qubit Hamiltonian. We show how the robust CNOT gate can achieve a very high fidelity when a single application of the composite rotations is combined with a modest level of Hamiltonian characterisation. Operating the robust CNOT gate…
We consider the Hamiltonian for a charged particle in a harmonic potential in the presence of a magnetic field. The most symmetric case depends on one parameter, the variation of which leads from a spectrum bounded from below to an unbounded spectrum. At the transition point the spectrum is bounded from below but each eigenvalue has infinite multiplicity. The algebraic method proves to be a remarkable tool for the analysis of this quadratic Hamiltonian.
In recent years there has been a growing interest in treating many-body systems as Bell scenarios, where lattice sites play the role of distant parties and only near-neighbor statistics are accessible. We investigate contextuality arising from three Bell scenarios in infinite, translation-invariant 1D models: nearest-neighbor with two dichotomic observables per site; nearest- and next-to-nearest neighbor with two dichotomic observables per site and nearest-neighbor with three dichotomic observables per site. For the first scenario, we give strong evidence that it cannot exhibit contextuality, …
The dynamics of a quantum system can be simulated using a quantum computer by breaking down the unitary into a quantum circuit of one and two qubit gates. The most established methods are the Trotter-Suzuki decompositions, for which rigorous bounds on the circuit size depend on the number of terms $L$ in the system Hamiltonian and the size of the largest term in the Hamiltonian $Λ$. Consequently, Trotter-Suzuki is only practical for sparse Hamiltonians. Trotter-Suzuki is a deterministic compiler but it was recently shown that randomised compiling offers lower overheads. Here we present and ana…
Bounded-Error Quantum Simulation via Hamiltonian and Lindbladian Learning
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；University of Innsbruck（因斯布鲁克大学）；Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Forschungszentrum Jülich GmbH（于利希研究中心有限公司）；SISSA – International School for Advanced Studies（国际高等研究院）
This thesis focuses on the mathematical description and application of nonlinear cavity optomechanical systems. The first part is concerned with solving the dynamics of the standard nonlinear optomechanical Hamiltonian with an additional time-dependent mechanical displacement and single-mode squeezing term. The solution is based on identifying a Lie algebra that generates the time-evolution of the system, which reduces the problem to considering a finite set of coupled ordinary differential equations of real functions. The second part applies the solutions of the extended optomechanical Hamilt…
Generic non-Markovian quantum processes have infinitely long memory, implying an exact description that grows exponentially in complexity with observation time. Here, we present a finite memory ansatz that approximates (or recovers) the true process with errors bounded by the strength of the non-Markovian memory. The introduced memory strength is an operational quantity and depends on the way the process is probed. Remarkably, the recovery error is bounded by the smallest memory strength over all possible probing methods. This allows for an unambiguous and efficient description of non-Markovia…
We consider whether or not Hamiltonians which are sums of commuting projectors have "trivial" ground states which can be constructed by a local quantum circuit of bounded depth and range acting on a product state. While the toric code only has nontrivial ground states, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states. We define an "interaction complex" for a Hamiltonian, generalizing the interaction graph, and we show that if this complex can be continuously mapped to a 1-complex using a map with bounded diameter of pre-images then the Hamilto…
Learning Many-Body Hamiltonians Using a Local Probe
Joint Center for Quantum Information and Computer Science, University of Maryland（量子信息与计算机科学联合中心，马里兰大学）；Department of Computer Science, University of Maryland（计算机科学系，马里兰大学）；Joint Quantum Institute, NIST/University of Maryland（联合量子研究所，美国国家标准与技术研究院/马里兰大学）
We developed a novel perturbative expansion based on the replica trick for the Floquet Hamiltonian governing the dynamics of periodically kicked systems where the kick strength is the small parameter. The expansion is formally equivalent to an infinite resummation of the Baker-Campbell-Hausdorff series in the un-driven (non-perturbed) Hamiltonian, while considering terms up to a finite order in the kick strength. As an application of the replica expansion, we analyze an Ising spin 1/2 chain periodically kicked with magnetic field of strength $h$, which has both longitudinal and transverse comp…
Perturbative gadgets are used to construct a quantum Hamiltonian whose low-energy subspace approximates a given quantum $k$-body Hamiltonian up to an absolute error $ε$. Typically, gadget constructions involve terms with large interaction strengths of order $\text{poly}(ε^{-1})$. Here we present a 2-body gadget construction and prove that it approximates a target many-body Hamiltonian of interaction strength $γ= O(1)$ up to absolute error $ε\llγ$ using interactions of strength $O(ε)$ instead of the usual inverse polynomial in $ε$. A key component in our proof is a new condition for the converg…
Istanbul Medeniyet University（伊斯坦布尔梅德尼耶大学）；Purdue University（普渡大学）
In this paper, we present a method for the Hamiltonian simulation in the context of eigenvalue estimation problems which improves earlier results dealing with Hamiltonian simulation through the truncated Taylor series. In particular, we present a fixed-quantum circuit design for the simulation of the Hamiltonian dynamics, $H(t)$, through the truncated Taylor series method described by Berry et al. \cite{berry2015simulating}. The circuit is general and can be used to simulate any given matrix in the phase estimation algorithm by only changing the angle values of the quantum gates implementing t…
We investigate one-dimensional harmonically trapped two-component systems for repulsive interaction strengths ranging from the non-interacting to the strongly interacting regime for Fermi-Fermi mixtures. A new and powerful mapping between the interaction strength parameters from a continuous Hamiltonian and a discrete lattice Hamiltonian is derived. As an example, we show that this mapping does not depend neither on the state of the system nor on the number of particles. Energies, density profiles and correlation functions are obtained both numerically (DMRG and Exact diagonalization) and anal…
