Showing papers on "Quantum computer published in 2010"

Quantum Computation and Quantum Information

Abstract: Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.

Quantum Computing

Abstract: Quantum mechanics---the theory describing the fundamental workings of nature---is famously counterintuitive: it predicts that a particle can be in two places at the same time, and that two remote particles can be inextricably and instantaneously linked These predictions have been the topic of intense metaphysical debate ever since the theory's inception early last century However, supreme predictive power combined with direct experimental observation of some of these unusual phenomena leave little doubt as to its fundamental correctness In fact, without quantum mechanics we could not explain the workings of a laser, nor indeed how a fridge magnet operates Over the last several decades quantum information science has emerged to seek answers to the question: can we gain some advantage by storing, transmitting and processing information encoded in systems that exhibit these unique quantum properties? Today it is understood that the answer is yes Many research groups around the world are working towards one of the most ambitious goals humankind has ever embarked upon: a quantum computer that promises to exponentially improve computational power for particular tasks A number of physical systems, spanning much of modern physics, are being developed for this task---ranging from single particles of light to superconducting circuits---and it is not yet clear which, if any, will ultimately prove successful Here we describe the latest developments for each of the leading approaches and explain what the major challenges are for the future

Generic new platform for topological quantum computation using semiconductor heterostructures.

Abstract: We show that a film of a semiconductor in which $s$-wave superconductivity and Zeeman splitting are induced by the proximity effect, supports zero-energy Majorana fermion modes in the ordinary vortex excitations Since time-reversal symmetry is explicitly broken, the edge of the film constitutes a chiral Majorana wire The heterostructure we propose---a semiconducting thin film sandwiched between an $s$-wave superconductor and a magnetic insulator---is a generic system which can be used as the platform for topological quantum computation by virtue of the existence of non-Abelian Majorana fermions

Majorana fermions in a tunable semiconductor device

Abstract: The experimental realization of Majorana fermions presents an important problem due to their non-Abelian nature and potential exploitation for topological quantum computation. Very recently Sau et al. [Phys. Rev. Lett. 104, 040502 (2010)] demonstrated that a topological superconducting phase supporting Majorana fermions can be realized using surprisingly conventional building blocks: a semiconductor quantum well coupled to an s-wave superconductor and a ferromagnetic insulator. Here we propose an alternative setup, wherein a topological superconducting phase is driven by applying an in-plane magnetic field to a (110)-grown semiconductor coupled only to an s-wave superconductor. This device offers a number of advantages, notably a simpler architecture and the ability to tune across a quantum phase transition into the topological superconducting state while still largely avoiding unwanted orbital effects. Experimental feasibility of both setups is discussed in some detail.

Optical quantum memory

Abstract: Quantum memory is important to quantum information processing in many ways: a synchronization device to match various processes within a quantum computer, an identity quantum gate that leaves any state unchanged, and a tool to convert heralded photons to photons-on-demand. In addition to quantum computing, quantum memory would be instrumental for the implementation of long-distance quantum communication using quantum repeaters. The importance of this basic quantum gate is exemplified by the multitude of optical quantum memory mechanisms being studied: optical delay lines, cavities, electromagnetically-induced transparency, photon-echo, and off-resonant Faraday interaction. Here we report on the state-of-the-art in the field of optical quantum memory, including criteria for successful quantum memory and current performance levels.

Demonstration of a Neutral Atom Controlled-NOT Quantum Gate

Abstract: We present the first demonstration of a CNOT gate between two individually addressed neutral atoms. Our implementation of the CNOT uses Rydberg blockade interactions between neutral atoms held in optical traps separated by $g8\text{ }\text{ }\ensuremath{\mu}\mathrm{m}$. Using two different gate protocols we measure CNOT fidelities of $F=0.73$ and 0.72 based on truth table probabilities. The gate was used to generate Bell states with fidelity $F=0.48\ifmmode\pm\else\textpm\fi{}0.06$. After correcting for atom loss we obtain an a posteriori entanglement fidelity of $F=0.58$.

Towards quantum chemistry on a quantum computer

Abstract: Exact first-principles calculations of molecular properties are currently intractable because their computational cost grows exponentially with both the number of atoms and basis set size. A solution is to move to a radically different model of computing by building a quantum computer, which is a device that uses quantum systems themselves to store and process data. Here we report the application of the latest photonic quantum computer technology to calculate properties of the smallest molecular system: the hydrogen molecule in a minimal basis. We calculate the complete energy spectrum to 20 bits of precision and discuss how the technique can be expanded to solve large-scale chemical problems that lie beyond the reach of modern supercomputers. These results represent an early practical step toward a powerful tool with a broad range of quantum-chemical applications.

Quantum Measurement and Control

Abstract: In the last two decades there has been an enormous progress in the experimental investigation of single quantum systems. This progress covers fields such as quantum optics, quantum computation, quantum cryptography, and quantum metrology, which are sometimes summarized as `quantum technologies'. A key issue there is entanglement, which can be considered as the characteristic feature of quantum theory. As disparate as these various fields maybe, they all have to deal with a quantum mechanical treatment of the measurement process and, in particular, the control process. Quantum control is, according to the authors, `control for which the design requires knowledge of quantum mechanics'. Quantum control situations in which measurements occur at important steps are called feedback (or feedforward) control of quantum systems and play a central role here. This book presents a comprehensive and accessible treatment of the theoretical tools that are needed to cope with these situations. It also provides the reader with the necessary background information about the experimental developments. The authors are both experts in this field to which they have made significant contributions. After an introduction to quantum measurement theory and a chapter on quantum parameter estimation, the central topic of open quantum systems is treated at some length. This chapter includes a derivation of master equations, the discussion of the Lindblad form, and decoherence – the irreversible emergence of classical properties through interaction with the environment. A separate chapter is devoted to the description of open systems by the method of quantum trajectories. Two chapters then deal with the central topic of quantum feedback control, while the last chapter gives a concise introduction to one of the central applications – quantum information. All sections contain a bunch of exercises which serve as a useful tool in learning the material. Especially helpful are also various separate boxes presenting important background material on topics such as the block representation or the feedback gain-bandwidth relation. The two appendices on quantum mechanics and phase-space and on stochastic differential equations serve the same purpose. As the authors emphasize, the book is aimed at physicists as well as control engineers who are already familiar with quantum mechanics. It takes an operational approach and presents all the material that is needed to follow research on quantum technologies. On the other hand, conceptual issues such as the relevance of the measurement process for the interpretation of quantum theory are neglected. Readers interested in them may wish to consult instead a textbook such as Decoherence and the Quantum-to-Classical Transition by Maximilian Schlosshauer. Although the present book does not contain applications to gravity, part of its content might become relevant for the physics of gravitational-wave detection and quantum gravity phenomenology. In this respect it should be of interest also for the readers of this journal.

Spin–orbit qubit in a semiconductor nanowire

Abstract: Motion of electrons can influence their spins through a fundamental effect called spin–orbit interaction This interaction provides a way to control spins electrically and thus lies at the foundation of spintronics Even at the level of single electrons, the spin–orbit interaction has proven promising for coherent spin rotations Here we implement a spin–orbit quantum bit (qubit) in an indium arsenide nanowire, where the spin–orbit interaction is so strong that spin and motion can no longer be separated In this regime, we realize fast qubit rotations and universal single-qubit control using only electric fields; the qubits are hosted in single-electron quantum dots that are individually addressable We enhance coherence by dynamically decoupling the qubits from the environment Nanowires offer various advantages for quantum computing: they can serve as one-dimensional templates for scalable qubit registers, and it is possible to vary the material even during wire growth Such flexibility can be used to design wires with suppressed decoherence and to push semiconductor qubit fidelities towards error correction levels Furthermore, electrical dots can be integrated with optical dots in p–n junction nanowires The coherence times achieved here are sufficient for the conversion of an electronic qubit into a photon, which can serve as a flying qubit for long-distance quantum communication

Single-Shot Readout of a Single Nuclear Spin

Abstract: Projective measurement of single electron and nuclear spins has evolved from a gedanken experiment to a problem relevant for applications in atomic-scale technologies like quantum computing Although several approaches allow for detection of a spin of single atoms and molecules, multiple repetitions of the experiment that are usually required for achieving a detectable signal obscure the intrinsic quantum nature of the spin's behavior We demonstrated single-shot, projective measurement of a single nuclear spin in diamond using a quantum nondemolition measurement scheme, which allows real-time observation of an individual nuclear spin's state in a room-temperature solid Such an ideal measurement is crucial for realization of, for example, quantum error correction protocols in a quantum register

Nonlocality and communication complexity

Abstract: Quantum information processing is the emerging field that defines and realizes computing devices that make use of quantum mechanical principles, like the superposition principle, entanglement, and interference. Until recently the common notion of computing was based on classical mechanics, and did not take into account all the possibilities that physically-realizable computing devices offer in principle. The field gained momentum after Peter Shor developed an efficient algorithm for factoring numbers, demonstrating the potential computing powers that quantum computing devices can unleash. In this review we study the information counterpart of computing. It was realized early on by Holevo, that quantum bits, the quantum mechanical counterpart of classical bits, cannot be used for efficient transformation of information, in the sense that arbitrary k-bit messages can not be compressed into messages of k − 1 qubits. The abstract form of the distributed computing setting is called communication complexity. It studies the amount of information, in terms of bits or in our case qubits, that two spatially separated computing devices need to exchange in order to perform some computational task. Surprisingly, quantum mechanics can be used to obtain dramatic advantages for such tasks. We review the area of quantum communication complexity, and show how it connects the foundational physics questions regarding non-locality with those of communication complexity studied in theoretical computer science. The first examples exhibiting the advantage of the use of qubits in distributed information-processing tasks were based on non-locality tests. However, by now the field has produced strong and interesting quantum protocols and algorithms of its own that demonstrate that entanglement, although it cannot be used to replace communication, can be used to reduce the communication exponentially. In turn, these new advances yield a new outlook on the foundations of physics, and could even yield new proposals for experiments that test the foundations of physics.

Quantum computing with defects

Abstract: Identifying and designing physical systems for use as qubits, the basic units of quantum information, are critical steps in the development of a quantum computer. Among the possibilities in the solid state, a defect in diamond known as the nitrogen-vacancy (NV-1) center stands out for its robustness—its quantum state can be initialized, manipulated, and measured with high fidelity at room temperature. Here we describe how to systematically identify other deep center defects with similar quantum-mechanical properties. We present a list of physical criteria that these centers and their hosts should meet and explain how these requirements can be used in conjunction with electronic structure theory to intelligently sort through candidate defect systems. To illustrate these points in detail, we compare electronic structure calculations of the NV-1 center in diamond with those of several deep centers in 4H silicon carbide (SiC). We then discuss the proposed criteria for similar defects in other tetrahedrally coordinated semiconductors.

Quantum control theory and applications: a survey

Abstract: This study presents a survey on quantum control theory and applications from a control systems perspective. Some of the basic concepts and main developments (including open-loop control and closed-loop control) in quantum control theory are reviewed. In the area of open-loop quantum control, the paper surveys the notion of controllability for quantum systems and presents several control design strategies including optimal control, Lyapunov-based methodologies, variable structure control and quantum incoherent control. In the area of closed-loop quantum control, this study reviews closed-loop learning control and several important issues related to quantum feedback control including quantum filtering, feedback stabilisation, linear-quadratic-Gaussian control and robust quantum control.

The Computational Complexity of Linear Optics.

Abstract: We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level Unfortunately, this result assumes an extremely accurate simulation Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A We present evidence for these conjectures, both of which seem interesting even apart from our application This paper does not assume knowledge of quantum optics Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists

Preparation and detection of a mechanical resonator near the ground state of motion

Abstract: Cold, macroscopic mechanical systems are expected to behave contrary to our usual classical understanding of reality; the most striking and counterintuitive predictions involve the existence of states in which the mechanical system is located in two places simultaneously. Various schemes have been proposed to generate and detect such states, and all require starting from mechanical states that are close to the lowest energy eigenstate, the mechanical ground state. Here we report the cooling of the motion of a radio-frequency nanomechanical resonator by parametric coupling to a driven, microwave-frequency superconducting resonator. Starting from a thermal occupation of 480 quanta, we have observed occupation factors as low as 3.8 ± 1.3 and expect the mechanical resonator to be found with probability 0.21 in the quantum ground state of motion. Further cooling is limited by random excitation of the microwave resonator and heating of the dissipative mechanical bath. This level of cooling is expected to make possible a series of fundamental quantum mechanical observations including direct measurement of the Heisenberg uncertainty principle and quantum entanglement with qubits.

Topological Quantum Computation

Abstract: The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones poly- nomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets are modeled by modular functors, opening a new possi- bility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like e −αl , where l is a length scale, and α is some posi- tive constant. In contrast, the "presumptive" qubit-model of quantum computation, which repairs errors combinatorically, requires a fantas- tically low initial error rate (about 10 −4 ) before computation can be stabilized. Quantum computation is a catch-all for several models of computation based on a theoretical ability to manufacture, manipulate and measure quan- tum states. In this context, there are three areas where remarkable algo- rithms have been found: searching a data base (15), abelian groups (factor- ing and discrete logarithm) (19 ,27), and simulating physical systems (5 ,21). To this list we may add a fourth class of algorithms which yield approximate,

Photons walking the line: A quantum walk with adjustable coin operations

Abstract: We present the first robust implementation of a coined quantum walk over five steps using only passive optical elements. By employing a fiber network loop we keep the amount of required resources constant as the walker's position Hilbert space is increased. We observed a non-Gaussian distribution of the walker's final position, thus characterizing a faster spread of the photon wave packet in comparison to the classical random walk. The walk is realized for many different coin settings and initial states, opening the way for the implementation of a quantum-walk-based search algorithm.

Almost all quantum states have nonclassical correlations

Abstract: Quantum discord quantifies nonclassical correlations in a quantum system including those not captured by entanglement. Thus, only states with zero discord exhibit strictly classical correlations. We prove that these states are negligible in the whole Hilbert space: typically a state picked out at random has positive discord and, given a state with zero discord, a generic arbitrarily small perturbation drives it to a positive-discord state. These results hold for any Hilbert-space dimension and have direct implications for quantum computation and for the foundations of the theory of open systems. In addition, we provide a simple necessary criterion for zero quantum discord. Finally, we show that, for almost all positive-discord states, an arbitrary Markovian evolution cannot lead to a sudden, permanent vanishing of discord.

Dynamics of a quantum phase transition and relaxation to a steady state

Abstract: We review recent theoretical work on two closely related issues: excitation of an isolated quantum condensed matter system driven adiabatically across a continuous quantum phase transition or a gapless phase, and apparent relaxation of an excited system after a sudden quench of a parameter in its Hamiltonian. Accordingly, the review is divided into two parts. The first part revolves around a quantum version of the Kibble–Zurek mechanism including also phenomena that go beyond this simple paradigm. What they have in common is that excitation of a gapless many-body system scales with a power of the driving rate. The second part attempts a systematic presentation of recent results and conjectures on apparent relaxation of a pure state of an isolated quantum many-body system after its excitation by a sudden quench. This research is motivated in part by recent experimental developments in the physics of ultracold atoms with potential applications in the adiabatic quantum state preparation and quantum computation.

Colloquium: Quantum networks with trapped ions

Abstract: Quantum computation and communication exploit the quantum properties of superposition and entanglement in order to perform tasks that may be impossible using classical means. In this Colloquium recent experimental and theoretical progress in the generation of entangled quantum networks based on the use of optical photons as carriers of information between fixed trapped atomic ion quantum memories are reviewed. Taken together, these quantum platforms offer a promising vision for the realization of a large-scale quantum network that could impact the future of communication and computation.

Chiral topological superconductor from the quantum Hall state

Abstract: The chiral topological superconductor in two dimensions has a full pairing gap in the bulk and a single chiral Majorana state at the edge. The vortex of the chiral superconducting state carries a Majorana zero mode which is responsible for the non-Abelian statistics of the vortices. Despite intensive searches, this superconducting state has not yet been identified in nature. In this paper, we consider a quantum Hall or a quantum anomalous Hall state near the plateau transition and in proximity to a fully gapped $s$-wave superconductor. We show that this hybrid system may realize the chiral topological superconductor state and propose several experimental methods for its observation.

Towards Quantum Chemistry on a Quantum Computer

Abstract: Exact first-principles calculations of molecular properties are currently intractable because their computational cost grows exponentially with both the number of atoms and basis set size. A solution is to move to a radically different model of computing by building a quantum computer, which is a device that uses quantum systems themselves to store and process data. Here we report the application of the latest photonic quantum computer technology to calculate properties of the smallest molecular system: the hydrogen molecule in a minimal basis. We calculate the complete energy spectrum to 20 bits of precision and discuss how the technique can be expanded to solve large-scale chemical problems that lie beyond the reach of modern supercomputers. These results represent an early practical step toward a powerful tool with a broad range of quantum-chemical applications.

The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation

Abstract: Leading experts from "The Physics of Quantum Information" network, initiated by the European Commission, bring together the most recent results from this emerging area of quantum technology. Written in a consistent style as a research monograph, the book introduces quantum cryptography, quantum teleportation, and quantum computation, considering both theory and newest experiments. Both scientists working in the field and advanced students will find a rich source of information on this exciting new area.

A quantum spin transducer based on nanoelectromechanical resonator arrays

Abstract: In a quantum computer, the data carriers (or qubits) must be well isolated from their environment to avoid information leakage. At the same time they have to interact with one another to process information. A proposed platform based on spin qubits connected through arrays of nanoelectromechanical resonators should be able to reconcile these conflicting requirements.

Experimental demonstration of a heralded entanglement source

Abstract: The heralded generation of entangled states is a long-standing goal in quantum information processing, because it is indispensable for a number of quantum protocols1,2. Polarization entangled photon pairs are usually generated through spontaneous parametric down-conversion3, but the emission is probabilistic. Their applications are generally accompanied by post-selection and destructive photon detection. Here, we report a source of entanglement generated in an event-ready manner by conditioned detection of auxiliary photons4. This scheme benefits from the stable and robust properties of spontaneous parametric down-conversion and requires only modest experimental efforts. It is flexible and allows the preparation efficiency to be significantly improved by using beamsplitters with different transmission ratios. We have achieved a fidelity better than 87% and a state preparation efficiency of 45% for the source. This could offer promise in essential photonics-based quantum information tasks, and particularly in enabling optical quantum computing by reducing dramatically the computational overhead5,6. An efficient source of entangled photons generated in an event-ready manner by conditioned detection of auxiliary photons is reported. A fidelity better than 87% and a state preparation efficiency of 45% are obtained. The scheme could offer promising applications in essential photonics-based quantum information tasks, and represents a particularly important development in the realization of optical quantum computing.

A trapped single ion inside a Bose-Einstein condensate.

Abstract: Improved control of the motional and internal quantum states of ultracold neutral atoms and ions has opened intriguing possibilities for quantum simulation and quantum computation. Many-body effects have been explored with hundreds of thousands of quantum-degenerate neutral atoms, and coherent light-matter interfaces have been built. Systems of single or a few trapped ions have been used to demonstrate universal quantum computing algorithms and to search for variations of fundamental constants in precision atomic clocks. Until now, atomic quantum gases and single trapped ions have been treated separately in experiments. Here we investigate whether they can be advantageously combined into one hybrid system, by exploring the immersion of a single trapped ion into a Bose-Einstein condensate of neutral atoms. We demonstrate independent control over the two components of the hybrid system, study the fundamental interaction processes and observe sympathetic cooling of the single ion by the condensate. Our experiment calls for further research into the possibility of using this technique for the continuous cooling of quantum computers. We also anticipate that it will lead to explorations of entanglement in hybrid quantum systems and to fundamental studies of the decoherence of a single, locally controlled impurity particle coupled to a quantum environment.

Towards high-speed optical quantum memories

Abstract: Quantum memories, capable of controllably storing and releasing a photon, are a crucial component for quantum computers1 and quantum communications2. To date, quantum memories3,4,5,6 have operated with bandwidths that limit data rates to megahertz. Here we report the coherent storage and retrieval of sub-nanosecond low-intensity light pulses with spectral bandwidths exceeding 1 GHz in caesium vapour. The novel memory interaction takes place through a far off-resonant two-photon transition in which the memory bandwidth is dynamically generated by a strong control field7,8. This should allow data rates more than 100 times greater than those of existing quantum memories. The memory works with a total efficiency of 15%, and its coherence is demonstrated through direct interference of the stored and retrieved pulses. Coherence times in hot atomic vapours are on the order of microseconds9, the expected storage time limit for this memory. Quantum memories for storing and releasing photons are required for quantum computers and quantum communications. So far, their operational bandwidths have limited data-rates to megahertz. Researchers now demonstrate coherent storage and retrieval of subnanosecond low-intensity light pulses with spectral bandwidths exceeding 1 GHz.

Non-Abelian states of matter

Abstract: Quantum mechanics classifies all elementary particles as either fermions or bosons, and this classification is crucial to the understanding of a variety of physical systems, such as lasers, metals and superconductors. In certain two-dimensional systems, interactions between electrons or atoms lead to the formation of quasiparticles that break the fermion-boson dichotomy. A particularly interesting alternative is offered by 'non-Abelian' states of matter, in which the presence of quasiparticles makes the ground state degenerate, and interchanges of identical quasiparticles shift the system between different ground states. Present experimental studies attempt to identify non-Abelian states in systems that manifest the fractional quantum Hall effect. If such states can be identified, they may become useful for quantum computation.

Experimental investigation of an eight-qubit unit cell in a superconducting optimization processor

Abstract: A superconducting chip containing a regular array of flux qubits, tunable interqubit inductive couplers, an XY-addressable readout system, on-chip programmable magnetic memory, and a sparse network of analog control lines has been studied. The architecture of the chip and the infrastructure used to control it were designed to facilitate the implementation of an adiabatic quantum optimization algorithm. The performance of an eight-qubit unit cell on this chip has been characterized by measuring its success in solving a large set of random Ising spin-glass problem instances as a function of temperature. The experimental data are consistent with the predictions of a quantum mechanical model of an eight-qubit system coupled to a thermal environment. These results highlight many of the key practical challenges that we have overcome and those that lie ahead in the quest to realize a functional large-scale adiabatic quantum information processor.

Topological quantum order: stability under local perturbations

Abstract: 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 H0, 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=H0+ϵV has well-defined spectral bands originating from low-lying eigenvalues of H0. These bands are separated from the rest of the spectra and from each other by a constant gap. The band originating from the smallest eigenvalue of H0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb–Robinson bound.