We investigate the efficiency of Quantum Adiabatic Optimization when overcoming potential barriers to get from a local to a global minimum. Specifically we look at n qubit systems with symmetric cost functions f:{0, 1}^n->R where the ground state must tunnel through a potential barrier of width n^a and height n^b. By the quantum adiabatic theorem the time delay sufficient to ensure tunneling grows quadratically with the inverse spectral gap during this tunneling process. We analyze barrier sizes with 1/2 < a + b and a < 1/2 and show that the minimum gap scales polynomially as n^{1/2-a-b} when 2a+b < 1 and exponentially as n^{-b/2} exp(-C n^{(2a+b-1)/2} ) when 1 < 2a+b. Our proof uses elementary techniques and confirms and extends an unpublished folklore result by Goldstone, which used large spin and instanton methods. Parts of our result also refine recent results by Kong and Crosson and Jiang et al. about the exponential gap scaling.
We explore to what extent path-integral quantum Monte Carlo methods can efficiently simulate the tunneling behavior of quantum adiabatic optimization algorithms. Specifically we look at symmetric cost functions defined over n bits with a single potential barrier that a successful optimization algorithm will have to tunnel through. The height and width of this barrier depend on n, and by tuning these dependencies, we can make the optimization algorithm succeed or fail in polynomial time. In this article we compare the strength of quantum adiabatic tunneling with that of path-integral quantum Mo…
We investigate the Landau-Zener tunneling (LZT) of a self-interacting two-level system in which the coupling between the levels is nonreciprocal. In such a non-Hermitian system, when the energy bias between two levels is adjusted very slowly, i.e., in the adiabatic limit, we find that a quantum state can still closely follow the eigenstate solution until it encounters the exceptional points (EPs) at which two eigenvalues and their corresponding eigenvectors coalesce. In the absence of the nonlinear self-interaction, we can obtain explicit expressions for the eigenvectors and eigenvalues and an…
The use of adiabatic passage techniques to mediate particle transport through real space, rather than phase space is becoming an interesting possibility. We have investigated the properties of Coherent Tunneling Adiabatic Passage (CTAP) with alternating tunneling matrix elements. This geometry, not previously considered in the donor in silicon paradigm, provides an interesting route to long-range quantum transport. We introduce simplified coupling protocols, and transient eigenspectra as well as a realistic gate design for this transport protocol. Using a pairwise treatment of the tunnel coupl…
We model the evolution of a single atom in a cavity interacting with two lasers, one far off resonance which creates an optical potential lattice and one near resonance, which can interact with the atom. The atom may tunnel between sites in the lattice. We find the photon counting statistic $g^{(2)}(τ)$ depends on the tunneling rate. Additionally we derive methods to work with a Bose-Einstein Condensate in the same situation, in ther Josephson junction regime. It is shown that oscillations in $g^{(2)}(τ)$ are directly proportional to the tunneling coefficient. We work in the bad-cavity limit a…
By developing the preceding work on the fast forward of transient phenomena of quantum tunneling by Khujakulov and Nakamura (Phys. Rev. {\bf A 93}, 022101 (2016) ), we propose a scheme of the exact fast forward of adiabatic control of stationary tunneling states with use of the electromagnetic field. The idea allows the acceleration of both the amplitude and phase of wave functions throughout the fast-forward time range. The scheme realizes the fast-forward observation of the transport coefficients under the adiabatically-changing barrier with the fixed energy of an incoming particle. As typic…
We study the asymptotic behavior of the spectral gap of simple barrier tunneling problems, which are related to using quantum annealing to find the global optimum of cost functions defined over n bits. Specifically we look at the problem of having an n qubit system tunnel through a barrier of width and height proportional to n^a. We show that with these quantum annealing problems, the asymptotic, $n\to\infty$, behavior of the spectral gap does not accurately describe the behavior of the gap at finite n until extremely large values of n (n>10^{12}). We prove that this deficiency of the asymptot…
Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via quantum annealing (QA), especially for problems featuring a cost function with tall and thin barriers. We present and analyze several counterexamples from the class of perturbed Hamming-weight optimization problems with qubit permutation symmetry. We first show that, for these problems, the adiabatic dynamics that make tunneling possible should be understood not in terms of the cost function but rather the semi-classical potential arising from the spin-coherent path integral formalism. W…
We apply a quantum adiabatic evolution algorithm to a combinatorial optimization problem where the cost function depends entirely on the of the number of unit bits in a n-bit string (Hamming weight). The solution of the optimization problem is encoded as a ground state of the problem Hamiltonian H_p for the z-projection of a total spin-n/2. We show that tunneling barriers for the total spin can be completely suppressed during the algorithm if the initial Hamiltonian has its ground state extended in the space of the z-projections of the spin. This suppression takes place even if the cost functi…
We provide and analyze examples that counter the widely made claim that tunneling is needed for a quantum speedup in optimization problems. The examples belong to the class of perturbed Hamming-weight optimization problems. In one case, featuring a plateau in the cost function in Hamming weight space, we find that the adiabatic dynamics that make tunneling possible, while superior to simulated annealing, result in a slowdown compared to a diabatic cascade of avoided level-crossings. This, in turn, inspires a classical spin vector dynamics algorithm that is at least as efficient for the plateau…
A scheme based on Coherent Tunneling by Adiabatic Passage (CTAP) of exchange-only spin qubit quantum states in a linearly arranged double quantum dot chain is demonstrated. Logical states for the qubit are defined by adopting the spin state of three electrons confined in a double quantum dot. The possibility to obtain gate operations entirely with electrical manipulations makes this qubit a valuable architecture in the field of quantum computing for the implementation of quantum algorithms. The effect of the external control parameters as well as the effect of the dephasing on the coherent tun…
The measurement of the tunneling time in attosecond experiments, termed attoclock, triggered a hot debate about the tunneling time, the role of time in quantum mechanics, where the interaction with the laser pulse involves two regimes of a different character, the multiphoton and the tunneling (field-) ionization. In the adiabatic field calibration, one of us (O. K.) developed in earlier works a real tunneling time model and showed that the model fits well to the experimental data of Landsmann et al. (Optica {\bf 1}, 343 2014). In the present work, it is shown that the model explains the exper…
We study the enhancement of tunneling through a potential barrier $V(x)$ by a time-dependent electric field with special emphasis on pulse-shaped vector potentials such as $A_x(t)=A_0/\cosh^2(ωt)$. In addition to the known effects of pre-acceleration and potential deformation already present in the adiabatic regime, as well as energy mixing in analogy to the Franz-Keldysh effect in the non-adiabatic (impulse) regime, the pulse $A_x(t)$ can enhance tunneling by ``pushing'' part of the wave-function out of the rear end of the barrier. Besides the natural applications in condensed matter and atom…
A nonlinear Landau-Zener model was proposed recently to describe, among a number of applications, the nonadiabatic transition of a Bose-Einstein condensate between Bloch bands. Numerical analysis revealed a striking phenomenon that tunneling occurs even in the adiabatic limit as the nonlinear parameter $C$ is above a critical value equal to the gap $V$ of avoided crossing of the two levels. In this paper, we present analytical results that give quantitative account of the breakdown of adiabaticity by mapping this quantum nonlinear model into a classical Josephson Hamiltonian. In the critical r…
We derive an analog of the Landau-Zener adiabatic tunneling formula for an open, two-level system coupled to a memoryless, dephasing bath. The derivation rests on a geometric view of the spectral subspaces as adiabatic invariants.
We investigate the acceleration of an adiabatic process with the same survival probability of the ground state by sweeping a parameter nonlinearly, fast in the wide gap region and slow in the narrow gap region, as contrast to the usual linear sweeping. We find the expected acceleration in the Laudau-Zener tunneling model and in the adiabatic quantum computing model for factorizing the number N=21.
Coherent tunneling by adiabatic passage (CTAP) is a well-established technique for robust spatial transport of quantum particles in linear chains. Here we introduce two exactly-solvable models where the CTAP protocol can be extended to two-dimensional lattice geometries. Such bi-dimensional lattice models are synthesized from time-dependent second-quantization Hamiltonians, in which the bosonic field operators evolve adiabatically like in an ordinary three-level CTAP scheme thus ensuring adiabatic passage in Fock space.
The macroscopic tunneling of an optomechanical membrane is considered. A cavity mode which couples quadratically to the membranes position can create highly tunable adiabatic double-well potentials, which together with the high Q-factors of such membranes render the observation of macroscopic tunneling possible. A suitable, pulsed measurement scheme using a linearly coupled mode of the cavity for the verification of the effect is studied.
Adiabatic quantum computation, based on the adiabatic theorem, is a promising alternative to conventional quantum computation. The validity of an adiabatic algorithm depends on the existence of a nonzero energy gap between the ground and excited states. However, it is difficult to ascertain the exact value of the energy gap. In this paper, we put forward a theorem on the existence of nonzero energy gap for the Hamiltonians used in adiabatic quantum computation. It can help to effectively identify a large class of the Hamiltonians without energy-level crossing between the ground and excited sta…
The investigation presented by Doiron, Hache, and Winful [Phys. Rev. A 76, 023823 (2007)] is not valid for the tunneling process as claimed in the paper.
We analyze the performance of simulated quantum annealing (SQA) on an optimization problem for which simulated classical annealing (SA) is provably inefficient because of a high energy barrier. We present evidence that SQA can pass through this barrier to find the global minimum efficiently. This demonstrates the potential for SQA to inherit some of the advantages of quantum annealing (QA), since this problem has been previously shown to be efficiently solvable by quantum adiabatic optimization.
We show that it is possible to use a classical computer to efficiently simulate the adiabatic evolution of a quantum system in one dimension with a constant spectral gap, starting the adiabatic evolution from a known initial product state. The proof relies on a recently proven area law for such systems, implying the existence of a good matrix product representation of the ground state, combined with an appropriate algorithm to update the matrix product state as the Hamiltonian is changed. This implies that adiabatic evolution with such Hamiltonians is not useful for universal quantum computati…
Triple donor devices have the potential to exhibit adiabatic tunneling via the CTAP (Coherent Tunneling Adiabatic Passage) protocol which is a candidate transport mechanism for scalable quantum computing. We examine theoretically the statistics of dopant placement using counted ion implantation by employing an analytical treatment of CTAP transport properties under hydrogenic assumptions. We determine theoretical device yields for proof of concept devices for different implant energies. In particular, we determine a significant theoretical device yield (~80%) for 14keV phosphorus in silicon wi…
The parametrization of adiabatic paths is optimal when tunneling is minimized. Hamiltonian evolutions do not have unique optimizers. However, dephasing Lindblad evolutions do. The optimizers are simply characterized by an Euler-Lagrange equation and have a constant tunneling rate along the path irrespective of the gap. Application to quantum search algorithms recovers the Grover result for appropriate scaling of the dephasing. Dephasing rates that beat Grover imply hidden resources in Lindblad operators.
We introduce an algorithm to perform an optimal adiabatic evolution that operates without an apriori knowledge of the system spectrum. By probing the system gap locally, the algorithm maximizes the evolution speed, thus minimizing the total evolution time. We test the algorithm on the Landau-Zener transition and then apply it on the quantum adiabatic computation of 3-SAT: The result is compatible with an exponential speed-up for up to twenty qubits with respect to classical algorithms. We finally study a possible algorithm improvement by combining it with the quantum Zeno effect.
We use an adiabatic approximation in terms of instantaneous resonances to study the steady-state and time-dependent transport properties of interacting electrons in biased resonant tunneling heterostructures. This approach leads, in a natural way, to a transport model of large applicability consisting of reservoirs coupled to regions where the system is described by a nonlinear Schrödinger equation. From the mathematical point of view, this work is non-rigorous but may offer some fresh and interesting problems involving semiclassical approximation, adiabatic theory, non-linear Schrödinger equa…
We aim to give more insights on adiabatic evolution concerning the occurrence of anti-crossings and their link to the spectral minimum gap $Δ_{min}$. We study in detail adiabatic quantum computation applied to a specific combinatorial problem called weighted max $k$-clique. A clear intuition of the parametrization introduced by V. Choi is given which explains why the characterization isn't general enough. We show that the instantaneous vectors involved in the anti-crossing vary brutally through it making the instantaneous ground-state hard to follow during the evolution. This result leads to a…
Quantum annealing is guaranteed to find the ground state of optimization problems in the adiabatic limit. Recent work [Phys. Rev. X 6, 031010 (2016)] has found that for some barrier tunneling problems, quantum annealing can be run much faster than is adiabatically required. Specifically, an $n$-qubit optimization problem was presented for which a non-adiabatic, or diabatic, annealing algorithm requires only constant runtime, while an adiabatic annealing algorithm requires a runtime polynomial in $n$. Here we show that this non-adiabatic speed-up is a direct result of a specific symmetry in the…
The time or cost of simulating a quantum circuit by adiabatic evolution is determined by the spectral gap of the Hamiltonians involved in the simulation. In "standard" constructions based on Feynman's Hamiltonian, such a gap decreases polynomially with the number of gates in the circuit, L. Because a larger gap implies a smaller cost, we study the limits of spectral gap amplification in this context. We show that, under some assumptions on the ground states and the cost of evolving with the Hamiltonians (which apply to the standard constructions), an upper bound on the gap of order 1/L follows…
Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps $L$. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for…
An adiabatic approximation in terms of instantaneous resonances is developed to study the steady-state and time-dependent transport of interacting electrons in biased resonant tunneling heterostructures. The resulting model consists of quantum reservoirs coupled to regions where the system is described by nonlinear ordinary differential equations and has a general conceptual interest.
The tunneling Hamiltonian describes a particle transfer from one region to the other. While there is no particle storage in the tunneling region itself, it has associated certain amount of energy. We name the corresponding flux energy reactance since, like an electrical reactance, it manifests itself in time-dependent transport only. Noticeably, this quantity is crucial to reproduce the universal charge relaxation resistance for a single-channel quantum capacitor at low temperatures. We show that a conceptually simple experiment is capable of demonstrating the existence of the energy reactance…
We discuss the filtering of the vibrational states of a cold atom in an optical trap, by chaining this trap with two empty ones and controlling adiabatically the tunneling. Matter wave filtering is performed by selectively transferring the population of the highest populated vibrational state to the most distant trap while the population of the rest of the states remains in the initial trap. Analytical conditions for two-state filtering are derived and then applied to an arbitrary number of populated bound states. Realistic numerical simulations close to state-of-the-art experimental arrangeme…
We present a general theory for adiabatic evolution of quantum states as governed by the nonlinear Schrodinger equation, and provide examples of applications with a nonlinear tunneling model for Bose-Einstein condensates. Our theory not only spells out conditions for adiabatic evolution of eigenstates, but also characterizes the motion of non-eigenstates which cannot be obtained from the former in the absence of the superposition principle. We find that in the adiabatic evolution of non-eigenstates, the Aharonov-Anandan phases play the role of classical canonical actions.
We report time-resolved measurements of Landau-Zener tunneling of Bose-Einstein condensates in accelerated optical lattices, clearly resolving the step-like time dependence of the band populations. Using different experimental protocols we were able to measure the tunneling probability both in the adiabatic and in the diabatic bases of the system. We also experimentally determine the contribution of the momentum width of the Bose condensates to the width of the tunneling steps and discuss the implications for measuring the jump time in the Landau-Zener problem.
Metastability is a quintessential feature of first order quantum phase transitions, which is lost either by dynamical instability or by nucleating bubbles of a true vacuum through quantum tunneling. By considering a drive across the first order quantum phase transition in the quantum Ising chain in the presence of both transverse and longitudinal fields, we reveal multiple regions in the parameter space where the initial metastable state loses its metastability in successive stages. The mechanism responsible is found to be semi-degenerate resonant tunnelings to states with specific bubble size…
Cosmic Lockdown: When Decoherence Saves the Universe from Tunneling
University of Portsmouth（朴茨茅斯大学）；Syracuse University（雪城大学）；ENS, CNRS, Université PSL, Sorbonne Université, Université Paris Cité（法兰西高等师范学院，法国国家科学研究中心，巴黎文理研究大学，索邦大学，巴黎西岱大学）
By adiabatically manipulating tunneling amplitudes of cold atoms in a periodic potential with a multiple sublattice structure, we are able to coherently transfer atoms from a sublattice to another without populating the intermediate sublattice, which can be regarded as a spatial analogue of stimulated Raman adiabatic passage. A key is the existence of dark eigenstates forming a flat band in a Lieb-type optical lattice. We also successfully observe a matter-wave analogue of Autler-Townes doublet using the same setup. This work shed light on a novel kind of coherent control of cold atoms in opti…
A gapped quantum system that is adiabatically perturbed remains approximately in its eigenstate after the evolution. We prove that, for constant gap, general quantum processes that approximately prepare the final eigenstate require a minimum time proportional to the ratio of the length of the eigenstate path to the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We also give a necessary condition for the adiabatic approximation that depends on local properties of the path, which is appropriate when the gap varies.
We present a detailed analysis of the Landau-Zener problem for an interacting Bose-Einstein condensate in a time-varying double-well trap, especially focussing on the relation between the full many-particle problem and the mean-field approximation. Due to the nonlinear self-interaction a dynamical instability occurs, which leads to a breakdown of adiabaticity condition and thus fundamentally alters the dynamics. It is shown that essentially all features of the Landau-Zener problem including the depletion of the condensate mode can be already understood within a semiclassical phase space pictur…
We derive an adiabatic theory for a stochastic differential equation, $ \varepsilon\, \mathrm{d} X(s) = L_1(s) X(s)\, \mathrm{d} s + \sqrt{\varepsilon} L_2(s) X(s) \, \mathrm{d} B_s, $ under a condition that instantaneous stationary states of $L_1(s)$ are also stationary states of $L_2(s)$. We use our results to derive the full statistics of tunneling for a driven stochastic Schrödinger equation describing a dephasing process.
We study the tunneling time of Dirac fermions in graphene magnetic barrier through an electrostatic potential and a mass term. This latter generates an energy gap in the spectrum and therefore affects the proprieties of tunneling of the system. For clarification, we first start by deriving the eigenspinors solutions of Dirac equation and second connect them to the incident, reflected and transmitted beam waves. This connection allows us to obtain the corresponding phases shifts and consequently compute the group delay time in transmission and reflection. Our numerical results show that the gro…
We investigate the influence of dissipation on one- and two-qubit rotations in coupled semiconductor quantum dots, using a (pseudo) spin-boson model with adiabatically varying parameters. For weak dissipation, we solve a master equation, compare with direct perturbation theory, and derive an expression for the `fidelity loss' during a simple operation that adiabatically moves an electron between two coupled dots. We discuss the possibility of visualizing coherent quantum oscillations in electron `pump' currents, combining quantum adiabaticity and Coulomb blockade. In two-qubit spin-swap operat…
We propose a two-qubit optically controlled phase gate in quantum dot molecules via adiabatic passage and hole tunneling. Our proposal combines the merits of the current generation of vertically stacked self-assembled InAs quantum dots and adiabatic passage. The simulation shows an implementation of the gate with a fidelity exceeding 0.98.
We study a linear ramp of the nearest-neighbor tunneling rate in the Bose-Hubbard model driving the system from the Mott insulator state into the superfluid phase. We employ the truncated Wigner approximation to simulate linear quenches of a uniform system in 1,2, and 3 dimensions, and in a harmonic trap in 3 dimensions. In all these setups the excitation energy decays like one over third root of the quench time. The -1/3 scaling arises from an impulse-adiabatic approximation - a variant of the Kibble-Zurek mechanism - describing a crossover from non-adiabatic to adiabatic evolution when the s…
We show that adiabatic evolution of a low-dimensional lattice of quantum spins with a spectral gap can be simulated efficiently. In particular, we show that as long as the spectral gap ΔE between the ground state and the first excited state is any constant independent of n, the total number of spins, then the ground-state expectation values of local operators, such as correlation functions, can be computed using polynomial space and time resources. Our results also imply that the local ground-state properties of any two spin models in the same quantum phase can be efficiently obtained from eac…
We study the corrections to adiabatic dynamics of two coupled quantum dot spin-qubits, each dot singly occupied with an electron, in the context of a quantum computing operation. Tunneling causes double occupancy at the conclusion of an operation and constitutes a processing error. We model the gate operation with an effective two-level system, where non-adiabatic transitions correspond to double occupancy. The model is integrable and possesses three independent parameters. We confirm the accuracy of Dykhne's formula, a nonperturbative estimate of transitions, and discuss physically intuitive …
We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.
