Quantum Parallelism as a Tool for Ensemble Spin Dynamics Calculations

Abstract

Efficient simulations of quantum evolutions of spin-1/2 systems are relevant for ensemble quantum computation as well as in typical NMR experiments. We propose an efficient method to calculate the dynamics of an observable provided that the initial excitation is "local". It resorts a single entangled pure initial state built as a superposition, with random phases, of the pure elements that compose the mixture. This ensures self-averaging of any observable, drastically reducing the calculation time. The procedure is tested for two representative systems: a spin star (cluster with random long range interactions) and a spin ladder.PACS numbers: 03.67. Lx, 05.30.Ch, 75.40.Gb, 75.10.Pq One of the goals of quantum computers is the simulation of quantum systems. While quantum processing is ideally cast in terms of pure states, experimental realizations are a major challenge [1] met only for very small systems [2]. The alternative use of statistical mixtures of pure states, as the spin ensembles in standard NMR experiments [3,4,5,6], led to the development of the ensemble quantum computation (EQC) [7] that allowed useful algorithm optimizations [8,9]. Recently, EQC was experimentally implemented in a 12-qubits system [10] and much larger quantum registers [11] were prepared to assess its stability against decoherence. Hence, an efficient evaluation of ensemble evolutions is needed. Analytical solutions of ensemble dynamics, either from the integration of the Liouville von-Neumann equation [12] or the alternative Keldysh formalism [13,14] are limited to special cases (e.g. [15,16]). Thus, one has to resort to numerical solutions. One can describe an ensemble of M spins by using the 2 M × 2 M density matrix, but this quickly reaches a storage limit as M increases. Thus, a typical exact diagonalization method in a desktop computer slightly exceeds a dozen of qubits. Alternatively, the use of wave functions combined with the TrotterSuzuki decomposition [17,18] overcomes this limitation because it uses vectors of size 2 M . However, the average over individual evolutions of a large number of components of the ensemble takes a long time. Here, this limitation is overcome by profiting of quantum parallelism [19] to evaluate the ensemble dynamics of any observable evolved from a "local" initial condition. The idea is that when evaluated on single pure states that are a superposition of all the elements of the ensemble, these observables become self-averaging. This reinforces the suggestions that one can avoid ensemble or thermal averages by using a single pure state [20,21]. Ref.[20] considers a subsystem with the reduced density matrix derived from a pure state, where the subsystem is entangled with an environment which has a much bigger size. The resulting reduced density matrix describes the microcanonical ensemble without resorting to the equiprobability postulate of statistical mechanics. Following a similar inspiration, we focus on the non-equilibrium dynamics of any given observable. The key is that the initial non-eq...

Effects of the interplay between initial state and Hamiltonian on the thermalization of isolated quantum many-body systems

“…It has been shown that a state corresponding to a random superposition of Ising states manifests thermal features. Specifically, the results for local observables led to the same outcomes obtained with a mixed state at infinite temperature [35,53,54]. The state is constructed so that the probability amplitudes of the basis vectors have all the same absolute value, 1/ √ D, but differ by a random phase e i2πϕ , where ϕ is a uniformly distributed random variable in [0, 1).…”

Localization-delocalization transition in the dynamics of dipolar-coupled nuclear spins

“…The recent progress on the experimental control of cold atoms [5,7,15], trapped ions [16], Ryderberg atoms [17], polar molecules [18,19] and nitrogen-vacancy centers in diamond [20] in solid state systems leads to promising which we observe and control the All experiments are performed on a NMR spectrometer whose superco a strong magnetic field of 7 Tesla spins have the same resonance fr MHz and are subject to mutual di composing a 3D spin-coupling ne dipolar interaction scales with 1/ onance width of 7.9 kHz of the N the homogeneous broadening (Se of the sample). The system is lef temperature and the thermal equ ing density operator for the spins in the high-temperature limit as considering that the Zeeman inter than the dipolar one (ω z = 300 M Experimental method and quan spin-spin interaction Hamiltonian Zeeman rotating frame is tonian (ω z d ij ), since the effects of the non-sec terms are negligible [29].…”

“…However, experimentally addressing 3D manybody systems in a controlled manner poses severe experimental problems (5,8,14,16). Non-equilibrium dynamics of many-body systems has been investigated to provide complementary information about a large variety of situations but also remains challenging (18)(19)(20)(21)(22)(23)(24)(25)(26). Therefore, finding different experimental situations, new approaches and techniques for controlling and observing many-body dynamics can lead to new approaches for studying manybody physics.…”

Effective one-body dynamics in multiple-quantum NMR experiments

“…The control of interaction anisotropy, e.g., the switch from a dipolar to an XY (planar) interaction, provides a tool for enhancing the transfer of quantum information [11,12]. In particular, the interactions can be sequentially turned on and off to prune some branches in real space so that an excitation is directed to a desired target through a specific pathway [13].…”