arXiv subjects

Rolando D. Somma

Publications and source records attributed to Rolando D. Somma.

42 records · Page 3Linked to original sources

arXiv2012

Spectral Gap Amplification

A large number of problems in science can be solved by preparing a specific eigenstate of some Hamiltonian H. The generic cost of quantum algorithms for these problems is determined by the inverse spectral gap of H for that eigenstate and the cost of evolving with H for some fixed time. The goal of spectral gap amplification is to construct a Hamiltonian H' with the same eigenstate as H but a bigger spectral gap, requiring that constant-time evolutions with H' and H are implemented with nearly the same cost. We show that a quadratic spectral gap amplification is possible when H satisfies a frustration-free property and give H' for these cases. This results in quantum speedups for optimization problems. It also yields improved constructions for adiabatic simulations of quantum circuits and for the preparation of projected entangled pair states (PEPS), which play an important role in quantum many-body physics. Defining a suitable black-box model, we establish that the quadratic amplification is optimal for frustration-free Hamiltonians and that no spectral gap amplification is possible, in general, if the frustration-free property is removed. A corollary is that finding a similarity transformation between a stoquastic Hamiltonian and the corresponding stochastic matrix is hard in the black-box model, setting limits to the power of some classical methods that simulate quantum adiabatic evolutions.

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arXiv2012

Quantum Speedup by Quantum Annealing

We study the glued-trees problem of Childs et. al. in the adiabatic model of quantum computing and provide an annealing schedule to solve an oracular problem exponentially faster than classically possible. The Hamiltonians involved in the quantum annealing do not suffer from the so-called sign problem. Unlike the typical scenario, our schedule is efficient even though the minimum energy gap of the Hamiltonians is exponentially small in the problem size. We discuss generalizations based on initial-state randomization to avoid some slowdowns in adiabatic quantum computing due to small gaps.

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arXiv2008

Parameter Estimation with Mixed-State Quantum Computation

We present a quantum algorithm to estimate parameters at the quantum metrology limit using deterministic quantum computation with one bit. When the interactions occurring in a quantum system are described by a Hamiltonian $H= θH_0$, we estimate $θ$ by zooming in on previous estimations and by implementing an adaptive Bayesian procedure. The final result of the algorithm is an updated estimation of $θ$ whose variance has been decreased in proportion to the time of evolution under H. For the problem of estimating several parameters, we implement dynamical-decoupling techniques and use the results of single parameter estimation. The cases of discrete-time evolution and reference-frame alignment are also discussed within the adaptive approach.

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arXiv2006

A Quantum Approach to Classical Statistical Mechanics

We present a new approach to study the thermodynamic properties of $d$-dimensional classical systems by reducing the problem to the computation of ground state properties of a $d$-dimensional quantum model. This classical-to-quantum mapping allows us to deal with standard optimization methods, such as simulated and quantum annealing, on an equal basis. Consequently, we extend the quantum annealing method to simulate classical systems at finite temperatures. Using the adiabatic theorem of quantum mechanics, we derive the rates to assure convergence to the optimal thermodynamic state. For simulated and quantum annealing, we obtain the asymptotic rates of $T(t) \approx (p N) /(k_B \log t)$ and $γ(t) \approx (Nt)^{-\bar{c}/N}$, for the temperature and magnetic field, respectively. Other annealing strategies, as well as their potential speed-up, are also discussed.

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arXiv2006

Optimal Quantum Measurements of Expectation Values of Observables

Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state |psi> of the quantum system. Such expectation values can be measured by repeatedly preparing |psi> and coupling the system to an apparatus. For this method, the precision of the measured value scales as 1/sqrt(N) for N repetitions of the experiment. For the problem of estimating the parameter phi in an evolution exp(-i phi H), it is possible to achieve precision 1/N (the quantum metrology limit) provided that sufficient information about H and its spectrum is available. We consider the more general problem of estimating expectations of operators A with minimal prior knowledge of A. We give explicit algorithms that approach precision 1/N given a bound on the eigenvalues of A or on their tail distribution. These algorithms are particularly useful for simulating quantum systems on quantum computers because they enable efficient measurement of observables and correlation functions. Our algorithms are based on a method for efficiently measuring the complex overlap of |psi> and U|psi>, where U is an implementable unitary operator. We explicitly consider the issue of confidence levels in measuring observables and overlaps and show that, as expected, confidence levels can be improved exponentially with linear overhead. We further show that the algorithms given here can typically be parallelized with minimal increase in resource usage.

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arXiv2005

Quantum Computation, Complexity, and Many-Body Physics

Recently developed quantum algorithms suggest that quantum computers can solve certain problems and perform certain tasks more efficiently than conventional computers. Among other reasons, this is due to the possibility of creating non-classical correlations, or quantum entanglement, which is a phenomena hard or impossible to reproduce by classical-information methods. In this thesis I first investigate the simulation of quantum systems on a quantum computer constructed of two-level quantum elements or qubits. For this purpose, I present algebra mappings that allow one to obtain physical properties and compute correlation functions of fermionic, anyonic, and bosonic systems with such a computer. The results obtained show that the complexity of preparing a quantum state which contains the desired information for the computation is crucial. Second, I present a wide class of quantum computations, which could involve entangled states, that can be simulated with the same efficiency on both types of computers. The notion of generalized quantum entanglement then naturally emerges. This generalization of entanglement is based on the idea that entanglement is an observer-dependent concept, that is, relative to a set of preferred observables.

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