Research Interests:
Research Interests:
Research Interests:
Distributed quantum computing with classical communications allows to relieve some of the limitations on the number of qubits and mitigate the noise in quantum computers. We give an algorithm that transforms a quantum circuit on a single... more
Distributed quantum computing with classical communications allows to relieve some of the limitations on the number of qubits and mitigate the noise in quantum computers. We give an algorithm that transforms a quantum circuit on a single processor to equivalent circuits on distributed processors. We address the quantum advantage of distributed circuits for the Grover search, Simon's and the Deutsch-Jozsa problems. In the case of Grover the quantum advantage of distributed computing remains the same, i.e. O(√(N)). In the case of Simon it remains exponential, but the complexity deteriorates from O(n) to O(n^2), where n = log_2(N). The distributed Deutsch-Jozsa deteriorates to being probabilistic but retains a quantum advantage over classical random sampling: A single quantum query gives the same error as O(n) random sampling. In section 5 we describe an experiment with the IBMQ5 machines that illustrates the advantages of distributed Grover search.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Electron-electron interactions have strong effects on the low-energy excitations of a one-dimensional metal. Luttinger liquid theory, which is supposed to describe this situation, predicts, among other things, that an injected electron... more
Electron-electron interactions have strong effects on the low-energy excitations of a one-dimensional metal. Luttinger liquid theory, which is supposed to describe this situation, predicts, among other things, that an injected electron will split into separate charge and spin excitations, which propagate at different velocities. We shall review some experiments and theoretical analyses where spin-charge separation can have manifest consequences, including discussion of the spin-incoherent regime, which can occur in low-density ...
Research Interests:
This paper completes the proof of the following statement, known as the Ten Martini Problem. The spectrum of the almost Mathieu operator (Hψ)(n) = ψ(n + 1) + ψ(n − 1) + 2λ cos(2ψ(θ + nα))ψ(n) in ℓ 2 (Z) is nowhere dense, provided that λ =... more
This paper completes the proof of the following statement, known as the Ten Martini Problem. The spectrum of the almost Mathieu operator (Hψ)(n) = ψ(n + 1) + ψ(n − 1) + 2λ cos(2ψ(θ + nα))ψ(n) in ℓ 2 (Z) is nowhere dense, provided that λ = 0 and α ∈ Q (in these exceptional cases, the potential is periodic and the spectrum of the operator is known not to be nowhere dense). This result was known due to earlier works for a large set of parameter values: periodic approximation works for Liouville α [J. Béllissard and B. Simon, J. Funct. Anal. 48 (1982), no. 3, 408–419; MR0678179 (84h:81019); M. D. Choi, G. A. Elliott and N. Yui, Invent. Math. 99
Research Interests:
Adiabatic evolutions with a gap condition have, under a range of circumstances, exponentially small tails that describe the leaking out of the spectral subspace. Adiabatic evolutions without a gap condition do not seem to have this... more
Adiabatic evolutions with a gap condition have, under a range of circumstances, exponentially small tails that describe the leaking out of the spectral subspace. Adiabatic evolutions without a gap condition do not seem to have this feature in general. This is a known fact for eigenvalue crossing. We show that this is also the case for eigenvalues at the threshold of the continuous spectrum by considering the Friedrichs model.
Research Interests:
Research Interests:
When basic tools of quantum information are applied to the quantum tomography data presented in [1], none of their devices appears to be a source of entangled photons. In a paper titled “A semiconductor quantum source of triggered... more
When basic tools of quantum information are applied to the quantum tomography data presented in [1], none of their devices appears to be a source of entangled photons. In a paper titled “A semiconductor quantum source of triggered entangled photon pairs ” Stevenson et. al. [1] claim to find evidence for the emission of polarization entangled photons from certain quantum dots. A density matrix completely specifies all properties of the quantum state [2]. Using quantum tomography [3], the authors of [1] construct the density matrices representing the polarization state of the photons emitted by each of their devices. We show that subjecting their quantum states to the basic definition of entanglement leads to the inevitable conclusion that none of their devices produced entangled light. In all the dots investigated in [1], quantum tomography yielded real density matrices of the form ρ = α 0 0 γ
Research Interests:
We provide a counterexample to the universal paramagnetism conjecture of Hogreve, Schrader and Seiler. The counterexample is based on the Bohm—Aharonov effect. Several years ago, one of us [11proved an inequal- (eq. (18) of ref. [41)that... more
We provide a counterexample to the universal paramagnetism conjecture of Hogreve, Schrader and Seiler. The counterexample is based on the Bohm—Aharonov effect. Several years ago, one of us [11proved an inequal- (eq. (18) of ref. [41)that ity expressingthe universal diamagnetic tendency of spinless bosons. For a single particle in external (local) Tr(exp(—j3H 2(a, V))) ~ ‘ Tr(exp(—~3H2(a = 0, V))) (6) electric potential V and magnetic potential a, the in- and, in particular, that equality can be expressed
Research Interests:
We announce three new rigorous results for the quantum mechanical hydrogen atom in constant magnetic field: (i) Borel summability of the small field perturbation series, (ii) detailed large field asymptotics, and (iii) non-degeneracy of... more
We announce three new rigorous results for the quantum mechanical hydrogen atom in constant magnetic field: (i) Borel summability of the small field perturbation series, (ii) detailed large field asymptotics, and (iii) non-degeneracy of the ground state ti~and a proof that it hasL~t~o = 0 for all values of the field. The weak field Zeeman effect [1] in simple atoms A =-4(r X B); B = (0,0, B) (2) was one of the earliest problems studied [2] in quantum mechanics. More recently, Ruderman [3] and Theorem 1. Let E~(O)be any negative eigenvalue of then others [4] discussed the analogous problem in the HydrogenHamiltonian H(0). Then there is an eigensuper-strong magnetic fields of the type encountered value [10] En(B) of H(B) for B small which is the Borel in neutron stars. It is perhaps surprising that any prob- sum [11,12] of the Rayleigh-Schrödinger perturbation lems remain open for such a well studied theory but coefficients for En [10]. there are some unresolved theoretical questions ...
Research Interests:
Abstract. We study the Lyaponov exponent yA(E) of (hu)(n) = u(n + 1) + u(n- l)+AV(n)u(n) in the limit as A +a where V is a suitable random potential. We prove that yA(E)-ln A as A +CO uniformly as E/A runs through compact sets. We also... more
Abstract. We study the Lyaponov exponent yA(E) of (hu)(n) = u(n + 1) + u(n- l)+AV(n)u(n) in the limit as A +a where V is a suitable random potential. We prove that yA(E)-ln A as A +CO uniformly as E/A runs through compact sets. We also describe a formal expansion (to order A-*) for random and almost periodic potentials. In this note, we study one-dimensional tight binding Hamiltonians h = ho + A V where (hou)(n) = u(n + 1) +u(n- 1). We are interested in the cases where V is either random or almost periodic. By random, we mean that V(n) is a family of identically distributed independent random variables with density P(y)dy where P is bounded with bounded support. In the random case, we will succeed in identifying the first few terms in the large A behaviour of the Lyaponov exponent. For the almost periodic case only a formal large A expansion is obtained. Explicitly, we let yA (E) be the Lyaponov exponent for h, i.e. where 1 yA(E) = lim-lnllM,,(u)...Ml(w)ll n+m n (1) The limit exists...
Incommensurate perturbations of classical orbits lead to an almost periodic Hill's operator whose spectrum, it is argued, is a Cantor set, but one with large Lebesgue measure. Applied to the rings of Saturn, this implies that the... more
Incommensurate perturbations of classical orbits lead to an almost periodic Hill's operator whose spectrum, it is argued, is a Cantor set, but one with large Lebesgue measure. Applied to the rings of Saturn, this implies that the complex groove structure in the rings approximates a Cantor set. The possible relevance of the sun in producing 'side gaps' which magnify the apparent gap size is also emphasized
Research Interests:
Research Interests:
Research Interests:
... An explicit expression for the resonance width is obtained. ... Formally, it has been shown that the Stark ladder states are solutions of the Schr6dinger equation in the one ... From a theo-retical viewpoint, the derivation of the... more
... An explicit expression for the resonance width is obtained. ... Formally, it has been shown that the Stark ladder states are solutions of the Schr6dinger equation in the one ... From a theo-retical viewpoint, the derivation of the ladder in-volves uncontrolled approximations.2 (It is not ...
Research Interests:
Research Interests:
Research Interests:
Asymptotic formulas are derived for the high-order perturbation coefficients for the hydrogen Zeeman Hamiltonian. The calculation of 100 coefficients in the series for the ground state are also reported. The method of calculation is based... more
Asymptotic formulas are derived for the high-order perturbation coefficients for the hydrogen Zeeman Hamiltonian. The calculation of 100 coefficients in the series for the ground state are also reported. The method of calculation is based on tilting the Hamiltonian by a generator of SO(2, 1) and the algebraization of the problem by means of the dynamical group SO(4,2).
Research Interests:
... March 1983 ALMOST PERIODIC SCHRt3DINGER OPERATORS II. THE INTEGRATED DENSITYOF STATES JOSEPH AVRON aND BARRY SIMON 1. Introduction. ... 369 Page 2. 370JOSEPH AVRON AND BARRY SIMON by standard approximation arguments. ...
Research Interests:
Research Interests:
Research Interests:
We derive a Diophantine equation for the Hall conductance of N interacting electrons moving on a torus. The equation holds for general background fields, including inhomogeneous magnetic fields, and random substrates, but is effective... more
We derive a Diophantine equation for the Hall conductance of N interacting electrons moving on a torus. The equation holds for general background fields, including inhomogeneous magnetic fields, and random substrates, but is effective when combined with symmetries. For example, together with translation invariance in one direction it determines the Hall conductance uniquely and constrains the degeneracy and crossings of
Research Interests:
Research Interests:
Research Interests:
We construct the manifold of zero-energy eigenstates for a nonrelativistic spin- 1/2 particle moving in a plane in an external magnetic field B(x-->)=n=0Nlambdan(x-->-c-->n), with {lambdan} and {c-->n} arbitrary reals and {kn}... more
We construct the manifold of zero-energy eigenstates for a nonrelativistic spin- 1/2 particle moving in a plane in an external magnetic field B(x-->)=n=0Nlambdan(x-->-c-->n), with {lambdan} and {c-->n} arbitrary reals and {kn} positive integers. For a given B the ground state is infinitely degenerate and the manifold of eigenfunctions is parametrized by a point in R2(2k+1). For such B's we prove paramagnetism with arbitrary external potential V(x-->).
Research Interests:
We discuss and extend an observation of Zinn-Justin that the double-well potential in one dimension and the anharmonic oscillator in two dimensions have coinciding perturbation expansions.
Research Interests:
Research Interests:
Let H0(B) denote the Hamiltonian of a free electron in a magnetic field B. Let V be a periodic potential. The authors show that if an interval (a,b) is not in the spectrum of H0(B0)+V for some B0, then it is not in the spectrum for all B... more
Let H0(B) denote the Hamiltonian of a free electron in a magnetic field B. Let V be a periodic potential. The authors show that if an interval (a,b) is not in the spectrum of H0(B0)+V for some B0, then it is not in the spectrum for all B sufficiently close to B0.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
It is shown that the integers found by Thouless et al. in the quantized Hall effect are the only quantized quantities associated with the energy bands. It is also proved that if two bands touch and then come apart as a parameter is... more
It is shown that the integers found by Thouless et al. in the quantized Hall effect are the only quantized quantities associated with the energy bands. It is also proved that if two bands touch and then come apart as a parameter is varied, then their individual integers (conductances) may ...