Using an NMR quantum computer, we experimentally simulate the quantum phase transition of a Heisenberg spin chain. The Hamiltonian is generated by a multiple pulse sequence, the nuclear spin system is prepared in its (pseudo-pure) ground state and the effective Hamiltonian varied in such a way that the Heisenberg chain is taken from a product state to an entangled state and finally to a different product state.
The evolution of entanglement in a 3-spin chain with nearest-neighbor Heisenberg-XY interactions for different initial states is investigated here. In an NMR experimental implementation, we generate multipartite entangled states starting from initial separable pseudo-pure states by simulating nearest-neighbor XY interactions in a 3-spin linear chain of nuclear spin qubits. For simulating XY interactions, we follow algebraic method of Zhang et al. [Phys. Rev. A 72, 012331 (2005)]. Bell state between end qubits has been generated by using only the unitary evolution of the XY Hamiltonian. For gen…
The propagation of excitation along a one-dimensional chain of atoms is simulated by means of NMR. The physical system used as an analog quantum computer is a nucleus of 133-Cs (spin 7/2) in a liquid crystalline matrix. The Hamiltonian of migration is simulated by using a special 7-frequency pulse, and the dynamics is monitored by following the transfer of population from one of the 8 spin energy levels to the other.
We have developed a concrete quantum simulation scheme and experimentally simulated a pairing model on an NMR quantum computer. The design of our experiment includes choosing an appropriate initial state in order to make our scheme scalable in near future, and the accomplishment of our experiment makes use of twice Fourier transforms so that our method is applicable to other physical models. Our results show that the experimental simulation can give the spectrum of the simulated Hamiltonian. Consequently, the potential power of a quantum computer on the simulation of complex physical systems i…
Extensions of average Hamiltonian theory to quantum computation permit the design of arbitrary Hamiltonians, allowing rotations throughout a large Hilbert space. In this way, the kinematics and dynamics of any quantum system may be simulated by a quantum computer. A basis mapping between the systems dictates the average Hamiltonian in the quantum computer needed to implement the desired Hamiltonian in the simulated system. The flexibility of the procedure is illustrated with NMR on 13-C labelled Alanine by creating the non-physical Hamiltonian ZZZ corresponding to a three body interaction.
Utilizing the highly correlated quantum NOON states of particles, we have implemented a proof-of-principle quantum thermometer using the NMR technique for measuring the variation of local magnetic field with the temperature variation. The system used was the star-topology system of hexafluorophosphate molecules and the thermometer showed a sensitivity of $85nT/ ^o C$. Using the hexafluorophosphate and the trimethylphosphite spin systems, we have quantified the advantage of the quantum protocol over the classical one for measuring magnetic field. The quantum protocol showed the best performance…
The three- spin chain with Heisenberg XY- interaction is simulated in a three- qubit nuclear magnetic resonance (NMR) quantum computer. The evolution caused by the XY- interaction is decomposed into a series of single- spin rotations and the $J$- coupling evolutions between the neighboring spins. The perfect state transfer (PST) algorithm proposed by M. Christandl et al [Phys. Rev. Lett, 92, 187902(2004)] is realized in the XY- chain.
Transport of quantum information in linear spin chains has been the subject of much theoretical work. Experimental studies by nuclear spin systems in solid-state by NMR (a natural implementation of such models) is complicated since the dipolar Hamiltonian is not solely comprised of nearest-neighbor XY-Heisenberg couplings. We present here a similarity transformation between the XY-Heisenberg Hamiltonian and the grade raising Hamiltonian, an interaction which is achievable with the collective control provided by radio-frequency pulses in NMR. Not only does this second Hamiltonian allows us to s…
Geometric phases play a central role in a variety of quantum phenomena, especially in condensed matter physics. Recently, it was shown that this fundamental concept exhibits a connection to quantum phase transitions where the system undergoes a qualitative change in the ground state when a control parameter in its Hamiltonian is varied. Here we report the first experimental study using the geometric phase as a topological test of quantum transitions of the ground state in a Heisenberg XY spin model. Using NMR interferometry, we measure the geometric phase for different adiabatic circuits that …
Recent technological advances may lead to the development of small scale quantum computers capable of solving problems that cannot be tackled with classical computers. A limited number of algorithms has been proposed and their relevance to real world problems is a subject of active investigation. Analysis of many-body quantum system is particularly challenging for classical computers due to the exponential scaling of Hilbert space dimension with the number of particles. Hence, solving problems relevant to chemistry and condensed matter physics are expected to be the first successful applicatio…
Feynman, in 1982, proposed the idea of using a quantum simulator to perform quantum simulations. A quantum simulator is basically a controllable quantum system that can mimic the dynamics of other quantum systems we wish to study. In this thesis we investigate some aspects of quantum simulation using both unitary and nonunitary controls. In the unitary part, we experimentally simulated the phenomenon of quantum many-body freezing, and also developed a general protocol to decompose an arbitrary unitary operator. In the nonunitary part, we studied phase decoherence by artificially inducing it. W…
Quantum computing in terms of geometric phases, i.e. Berry or Aharonov-Anandan phases, is fault-tolerant to a certain degree. We examine its implementation based on Zeeman coupling with a rotating field and isotropic Heisenberg interaction, which describe NMR and can also be realized in quantum dots and cold atoms. Using a novel physical representation of the qubit basis states, we construct $π/8$ and Hadamard gates based on Berry and Aharonov-Anandan phases. For two interacting qubits in a rotating field, we find that it is always impossible to construct a two-qubit gate based on Berry phases…
Motivated by some recent results of quantum control theory, we discuss the feasibility of local operator control in arrays of interacting qubits modeled as isotropic Heisenberg spin chains. Acting on one of the end spins, we aim at finding piecewise-constant control pulses that lead to optimal fidelities for a chosen set of quantum gates. We analyze the robustness of the obtained results f or the gate fidelities to random errors in the control fields, finding that with faster switching between piecewise-constant controls the system is less susceptible to these errors. The observed behavior fal…
We propose an iterative algorithm to simulate the dynamics generated by any $n$-qubit Hamiltonian. The simulation entails decomposing the unitary time evolution operator $U$ (unitary) into a product of different time-step unitaries. The algorithm product-decomposes $U$ in a chosen operator basis by identifying a certain symmetry of $U$ that is intimately related to the number of gates in the decomposition. We illustrate the algorithm by first obtaining a polynomial decomposition in the Pauli basis of the $n$-qubit Quantum State Transfer unitary by Di Franco et. al. (Phys. Rev. Lett. 101, 23050…
Recently developed quantum algorithms suggest that in principle, quantum computers can solve problems such as simulation of physical systems more efficiently than classical computers. Much remains to be done to implement these conceptual ideas into actual quantum computers. As a small-scale demonstration of their capability, we simulate a simple many-fermion problem, the Fano-Anderson model, using liquid state Nuclear Magnetic Resonance (NMR). We carefully designed our experiment so that the resource requirement would scale up polynomially with the size of the quantum system to be simulated. T…
In this paper, we briefly discuss the methodology for simulating a quantum computer which performs Shor's algorithm on a 7-qubit system to factorise 15. Using this simulation and the overlooked quantum brachistochrone method, we devised a Monte Carlo algorithm to calculate the expected time a theoretical quantum computer could perform this calculation under the same energy conditions as current working quantum computers. We found that, experimentally, a nuclear magnetic resonance quantum computer would take $1.59 \pm 0.04$ s to perform our simulated computation, whereas the expected optimal ti…
The phase resolution of interferometers is limited by the so-called Heisenberg limit, which states that the optimum phase sensitivity is inversely proportional to the number of interfering particles N, a 1/sqrt{N} improvement over the standard quantum limit. We have used simulated annealing, a global optimization strategy, to systematically search for quantum interferometer input states that approach the Heisenberg limited uncertainty in estimates of the interferometer phase shift. We compare the performance of these states to that of other non-classical states already known to yield Heisenber…
The Quantum Fourier Transformation ($QFT$) is a key building block for a whole wealth of quantum algorithms. Despite its proven efficiency, only a few proof-of-principle demonstrations have been reported. Here we utilize $QFT$ to enhance the performance of a quantum sensor. We implement the $QFT$ algorithm in a hybrid quantum register consisting of a nitrogen-vacancy (NV) center electron spin and three nuclear spins. The $QFT$ runs on the nuclear spins and serves to process the sensor - NV electron spin signal. We demonstrate $QFT$ for quantum (spins) and classical signals (radio frequency (RF…
We investigate evolution of quantum correlations in ensembles of two-qubit nuclear spin systems via nuclear magnetic resonance techniques. We use discord as a measure of quantum correlations and the Werner state as an explicit example. We first introduce different ways of measuring discord and geometric discord in two-qubit systems and then describe the following experimental studies: (a) We quantitatively measure discord for Werner-like states prepared using an entangling pulse sequence. An initial thermal state with zero discord is gradually and periodically transformed into a mixed state wi…
If the interaction between qubits in a quantum computer has a non-diagonal form (e.g. the Heisenberg interaction), then one must be able to "switch it off" in order to prevent uncontrolled propagation of states. Therefore, such QC schemes typically demand local control of the interaction strength between each pair of neighboring qubits. Here we demonstrate that this degree of control is not necessary: it suffices to switch the interaction collectively - something that can in principle be achieved by global fields rather than with local manipulations. This observation may offer a significant si…
Large quantum simulators, with sufficiently many qubits to be impossible to simulate classically, become hard to experimentally validate. We propose two tests of a quantum simulator with Heisenberg interaction in a linear chain of spins. In the first, we propagate half of a singlet state through a chain of spin with a ferromagnetic interaction and subsequently recover the state with an antiferromagnetic interaction. The antiferromagnetic interaction is intrinsic to the system while the ferromagnetic one can be simulated by a sequence of time-dependent controls of the antiferromagnetic interact…
Two qubit density matrices, which are of X-shape, are a natural generalization of Bell Diagonal States (BDSs) recently simulated on the IBM quantum device. We generalize the previous results and propose a quantum circuit for simulation of a general two qubit X-state, implement it on the same quantum device, and study its entanglement for several values of the extended parameter space. We also show that their X-shape is approximately robust against noisy quantum gates. To further physically motivate this study, we invoke the two-spin Heisenberg XYZ system and show that for a wide class of initi…
We show how quantum metrology protocols that seek to estimate the parameters of a Hamiltonian that exhibits a quantum phase transition can be efficiently simulated on an exponentially smaller quantum computer. Specifically, by exploiting the fact that the ground state of such a Hamiltonian changes drastically around its phase transition point, we construct a suitable observable from which one can estimate the relevant parameters of the Hamiltonian with Heisenberg scaling precision. We then show how, for the one-dimensional Ising Hamiltonian with transverse magnetic field acting on N spins, suc…
High-Field NMR Characterization and Indirect $J$-Spectroscopy of a Nuclear Spin Chain [U-$^{13}$C,$^{15}$N]-butyronitrile
Novosibirsk State University（新西伯利亚国立大学）；International Tomography Center SB RAS（俄罗斯科学院西伯利亚分院国际 tomography 中心）；Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学）；Helmholtz Institute Mainz（美因茨亥姆霍兹研究所）；Department of Chemistry, University of Miami（迈阿密大学化学系）；Frost Institute for Chemistry and Molecular Science（弗斯特化学与分子科学研究所）
We demonstrate that the optimal states in lossy quantum interferometry may be efficiently simulated using low rank matrix product states. We argue that this should be expected in all realistic quantum metrological protocols with uncorrelated noise and is related to the elusive nature of the Heisenberg precision scaling in presence of decoherence.
Quantum systems are the future candidates for computers and information processing devices. Information about quantum states and processes may be incomplete and scattered in these systems. We use a quantum version of Belief Propagation(BP) Algorithm to integrate the distributed information. In this algorithm the distributed information, which is in the form of density matrix, can be approximated to local structures. The validity of this algorithm is measured in comparison with Suzuki-Trotter(ST) method, using simulated information. ST in 3-body Heisenberg example gives a more accurate answer, …
The common spin Hamiltonians such as the Ising, XY, or Heisenberg model do not have ground states that are the graph states needed in measurement-based quantum computation. Various highly-entangled many-body states have been suggested as a universal resource for this type of computation, however, it is not easy to preserve these states in solid-state systems due to their short coherence times. Here we propose a scheme for generating a Hamiltonian that has a cluster state as ground state. Our approach employs a series of pulse sequences inspired by established NMR techniques and holds promise f…
Detailed experimental investigations of thermal and magnetic properties are presented for Cu(NH3)4SO4.H2O, an ideal uniform Heisenberg spin half chain compound. A comparison of these properties with relevant spin models is also presented. The temperature dependent magnetic susceptibility and specific heat data has been compared with the exact solution for uniform Heisenberg chain model derived by means of Bethe ansatz technique. Field dependent isothermal magnetization curves are simulated by Quantum Monte Carlo technique and compared with the corresponding experimental ones. Specific heat as …
