arXiv subjects

R. D. Somma

Publications and source records attributed to R. D. Somma.

8 recordsLinked to original sources

arXiv2011

Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space

We consider the manifold of all quantum many-body states that can be generated by arbitrary time-dependent local Hamiltonians in a time that scales polynomially in the system size, and show that it occupies an exponentially small volume in Hilbert space. This implies that the overwhelming majority of states in Hilbert space are not physical as they can only be produced after an exponentially long time. We establish this fact by making use of a time-dependent generalization of the Suzuki-Trotter expansion, followed by a counting argument. This also demonstrates that a computational model based on arbitrarily rapidly changing Hamiltonians is no more powerful than the standard quantum circuit model.

quant-ph↗

arXiv2010

Fast quantum algorithms for traversing paths of eigenstates

Consider a path of non-degenerate eigenstates of unitary operators or Hamiltonians with minimum eigenvalue gap G. The eigenpath traversal problem is to transform one or more copies of the initial to the final eigenstate. Solutions to this problem have applications ranging from quantum physics simulation to optimization. For Hamiltonians, the conventional way of doing this is by applying the adiabatic theorem. We give ``digital'' methods for performing the transformation that require no assumption on path continuity or differentiability other than the absence of large jumps. Given sufficient information about eigenvalues and overlaps between states on the path, the transformation can be accomplished with complexity O(L/G log(L/e)), where L is the angular length of the path and e is a specified bound on the error of the output state. We show that the required information can be obtained in a first set of transformations, whose complexity per state transformed has an additional factor that depends logarithmically on a maximum angular velocity along the path. This velocity is averaged over constant angular distances and does not require continuity. Our methods have substantially better behavior than conventional adiabatic algorithms, with fewer conditions on the path. They also improve on the previously best digital methods and demonstrate that path length and the gap are the primary parameters that determine the complexity of state transformation along a path.

quant-ph↗

arXiv2009

Necessary Condition for the Quantum Adiabatic Approximation

A gapped quantum system that is adiabatically perturbed remains approximately in its eigenstate after the evolution. We prove that, for constant gap, general quantum processes that approximately prepare the final eigenstate require a minimum time proportional to the ratio of the length of the eigenstate path to the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We also give a necessary condition for the adiabatic approximation that depends on local properties of the path, which is appropriate when the gap varies.

quant-ph↗

arXiv2009

Quantum state preparation by phase randomization

A computation in adiabatic quantum computing is implemented by traversing a path of nondegenerate eigenstates of a continuous family of Hamiltonians. We introduce a method that traverses a discretized form of the path: At each step we apply the instantaneous Hamiltonian for a random time. The resulting decoherence approximates a projective measurement onto the desired eigenstate, achieving a version of the quantum Zeno effect. If negative evolution times can be implemented with constant overhead, then the average absolute evolution time required by our method is O(L^2/Delta) for constant error probability, where L is the length of the path of eigenstates and Delta is the minimum spectral gap of the Hamiltonian. Making explicit the dependence on the path length is useful for cases where L does not depend on Delta. The complexity of our method has a logarithmic improvement over previous algorithms of this type. The same cost applies to the discrete-time case, where a family of unitary operators is given and each unitary and its inverse can be used. Restriction to positive evolution times incurs an error that decreases exponentially with the cost. Applications of this method to unstructured search and quantum sampling are considered. In particular, we discuss the quantum simulated annealing algorithm for solving combinatorial optimization problems. This algorithm provides a quadratic speed-up in the gap of the stochastic matrix over its classical counterpart implemented via Markov chain Monte Carlo.

quant-ph↗

arXiv2008

Efficient discrete-time simulations of continuous-time quantum query algorithms

The continuous-time query model is a variant of the discrete query model in which queries can be interleaved with known operations (called "driving operations") continuously in time. Interesting algorithms have been discovered in this model, such as an algorithm for evaluating nand trees more efficiently than any classical algorithm. Subsequent work has shown that there also exists an efficient algorithm for nand trees in the discrete query model; however, there is no efficient conversion known for continuous-time query algorithms for arbitrary problems. We show that any quantum algorithm in the continuous-time query model whose total query time is T can be simulated by a quantum algorithm in the discrete query model that makes O[T log(T) / log(log(T))] queries. This is the first upper bound that is independent of the driving operations (i.e., it holds even if the norm of the driving Hamiltonian is very large). A corollary is that any lower bound of T queries for a problem in the discrete-time query model immediately carries over to a lower bound of Ω[T log(log(T))/log (T)] in the continuous-time query model.

quant-ph↗

arXiv2008

Quantum Simulations of Classical Annealing Processes

We describe a quantum algorithm that solves combinatorial optimization problems by quantum simulation of a classical simulated annealing process. Our algorithm exploits quantum walks and the quantum Zeno effect induced by evolution randomization. It requires order $1/\sqrtδ$ steps to find an optimal solution with bounded error probability, where $δ$ is the minimum spectral gap of the stochastic matrices used in the classical annealing process. This is a quadratic improvement over the order $1/δ$ steps required by the latter.

quant-ph↗

arXiv2006

Lower bounds for the fidelity of entangled state preparation

Estimating the fidelity of state preparation in multi-qubit systems is generally a time-consuming task. Nevertheless, this complexity can be reduced if the desired state can be characterized by certain symmetries measurable with the corresponding experimental setup. In this paper we give simple expressions to estimate the fidelity of multi-qubit state preparation for rotational-invariant, stabilizer, and generalized coherent states. We specifically discuss the cat, W-type, and generalized coherent states, and obtain efficiently measurable lower bounds for the fidelity. We use these techniques to estimate the fidelity of a quantum simulation of an Ising-like interacting model using two trapped ions. These results are directly applicable to experiments using fidelity-based entanglement witnesses, such as quantum simulations and quantum computation.

quant-ph↗

arXiv2001

Phase diagram of the XXZ chain with next-nearest-neighbor interactions

We calculate the quantum phase diagram of the {\it XXZ} chain with nearest-neighbor (NN) $J_{1}$ and next-NN exchange $J_{2}$ with anisotropies $Δ_{1}$ and $Δ_{2}$ respectively. In particular we consider the case $Δ_{1}=-Δ_{2}$ to interpolate between the {\it XX} chain ($% Δ_{i}=0$) and the isotropic model with ferromagnetic $J_{2}$. For $% Δ_{1}<-1$, a ferromagnetic and two antiferromagnetic phases exist. For $| Δ_{i}| <1$, the boundary between the dimer and spin fluid phases is determined by the method of crossing of excitation spectra. For large $J_{2}/J_{1}$, this method seems to indicate the existence of a second spin fluid critical phase. However, an analysis of the spin stiffness and magnetic susceptibility for $Δ_{1}=Δ_{2}=1$ suggest that a small gap is present.

cond-mat↗