Quantum simulations of one dimensional quantum systems

We present quantum algorithms for the simulation of quantum systems in one spatial dimension, which result in quantum speedups that range from superpolynomial to polynomial. We first describe a method to simulate the evolution of the quantum harmonic oscillator (QHO) based on a refined analysis of the Trotter-Suzuki formula that exploits the Lie algebra structure. For total evolution time $t$ and precision $\epsilon>0$, the complexity of our method is $O(\exp(\gamma \sqrt{\log(N/\epsilon)}))$, where $\gamma>0$ is a constant and $N$ is the quantum number associated with an "energy cutoff" of the initial state. Remarkably, this complexity is subpolynomial in $N/\epsilon$. We also provide a method to prepare discrete versions of the eigenstates of the QHO of complexity polynomial in $\log(N)/\epsilon$, where $N$ is the dimension or number of points in the discretization. This method may be of independent interest as it provides a way to prepare, e.g., quantum states with Gaussian-like amplitudes. Next, we consider a system with a quartic potential. Our numerical simulations suggest a method for simulating the evolution of sublinear complexity $\tilde O(N^{1/3+o(1)})$, for constant $t$ and $\epsilon$. We also analyze complex one-dimensional systems and prove a complexity bound $\tilde O(N)$, under fairly general assumptions. Our quantum algorithms may find applications in other problems. As an example, we discuss the fractional Fourier transform, a generalization of the Fourier transform that is useful for signal analysis and can be formulated in terms of the evolution of the QHO.Comment: 25 pages, 9 fig

A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation

We present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by an energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.Comment: 5 page

Quantum algorithms for Gibbs sampling and hitting-time estimation

We present quantum algorithms for solving two problems regarding stochastic processes. The first algorithm prepares the thermal Gibbs state of a quantum system and runs in time almost linear in $\sqrt{N \beta/{\cal Z}}$ and polynomial in $\log(1/\epsilon)$, where $N$ is the Hilbert space dimension, $\beta$ is the inverse temperature, ${\cal Z}$ is the partition function, and $\epsilon$ is the desired precision of the output state. Our quantum algorithm exponentially improves the dependence on $1/\epsilon$ and quadratically improves the dependence on $\beta$ of known quantum algorithms for this problem. The second algorithm estimates the hitting time of a Markov chain. For a sparse stochastic matrix $P$, it runs in time almost linear in $1/(\epsilon \Delta^{3/2})$, where $\epsilon$ is the absolute precision in the estimation and $\Delta$ is a parameter determined by $P$, and whose inverse is an upper bound of the hitting time. Our quantum algorithm quadratically improves the dependence on $1/\epsilon$ and $1/\Delta$ of the analog classical algorithm for hitting-time estimation. Both algorithms use tools recently developed in the context of Hamiltonian simulation, spectral gap amplification, and solving linear systems of equations.Comment: 13 page

Security of Decoy-State Protocols for General Photon-Number-Splitting Attacks

Decoy-state protocols provide a way to defeat photon-number splitting attacks in quantum cryptography implemented with weak coherent pulses. We point out that previous security analyses of such protocols relied on assumptions about eavesdropping attacks that considered treating each pulse equally and independently. We give an example to demonstrate that, without such assumptions, the security parameters of previous decoy-state implementations could be worse than the ones claimed. Next we consider more general photon-number splitting attacks, which correlate different pulses, and give an estimation procedure for the number of single photon signals with rigorous security statements. The impact of our result is that previous analyses of the number of times a decoy-state quantum cryptographic system can be reused before it makes a weak key must be revised.Comment: 9 pages and 4 figure

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.Comment: PhD Thesis; Figures compressed; If needed, contact the author for a version in spanish. (Full abstract is in the thesis file

Quantum eigenvalue estimation via time series analysis

We present an efficient method for estimating the eigenvalues of a Hamiltonian $H$ from the expectation values of the evolution operator for various times. For a given quantum state $\rho$, our method outputs a list of eigenvalue estimates and approximate probabilities. Each probability depends on the support of $\rho$ in those eigenstates of $H$ associated with eigenvalues within an arbitrarily small range. The complexity of our method is polynomial in the inverse of a given precision parameter $\epsilon$, which is the gap between eigenvalue estimates. Unlike the well-known quantum phase estimation algorithm that uses the quantum Fourier transform, our method does not require large ancillary systems, large sequences of controlled operations, or preserving coherence between experiments, and is therefore more attractive for near-term applications. The output of our method can be used to compute spectral properties of $H$ and other expectation values efficiently, within additive error proportional to $\epsilon$.Comment: 10 pages, 6 figs. New section with numerical result

Quantum circuit synthesis for generalized coherent states

We present a method that outputs a sequence of simple unitary operations to prepare a given quantum state that is a generalized coherent state. Our method takes as inputs the expectation values of some relevant observables on the state to be prepared. Such expectation values can be estimated by performing projective measurements on $O(M^3 \log(M/\delta)/\epsilon^2)$ copies of the state, where $M$ is the dimension of an associated Lie algebra, $\epsilon$ is a precision parameter, and $1-\delta$ is the required confidence level. The method can be implemented on a classical computer and runs in time $O(M^4 \log(M/\epsilon))$. It provides $O(M \log(M/\epsilon))$ simple unitaries that form the sequence. The number of all computational resources is then polynomial in $M$, making the whole procedure very efficient in those cases where $M$ is significantly smaller than the Hilbert space dimension. When the algebra of relevant observables is determined by some Pauli matrices, each simple unitary may be easily decomposed into two-qubit gates. We discuss applications to quantum state tomography and classical simulations of quantum circuits.Comment: 7+2 pages, 2 figure

An exact real-space renormalization method and applications

We present a numerical method based on real-space renormalization that outputs the exact ground space of "frustration-free" Hamiltonians. The complexity of our method is polynomial in the degeneracy of the ground spaces of the Hamiltonians involved in the renormalization steps. We apply the method to obtain the full ground spaces of two spin systems. The first system is a spin-1/2 Heisenberg model with four-spin cyclic-exchange interactions defined on a square lattice. In this case, we study finite lattices of up to 160 spins and find a triplet ground state that differs from the singlet ground states obtained in C.D. Batista and S. Trugman, Phys. Rev. Lett. 93, 217202 (2004). We characterize such a triplet state as consisting of a triplon that propagates in a background of fluctuating singlet dimers. The second system is a family of spin-1/2 Heisenberg chains with uniaxial exchange anisotropy and next-nearest neighbor interactions. In this case, the method finds a ground-space degeneracy that scales quadratically with the system size and outputs the full ground space efficiently. Our method can substantially outperform methods based on exact diagonalization and is more efficient than other renormalization methods when the ground-space degeneracy is large.Comment: 10 pages, 8 Figs. Typos correcte

Improved Bounds for Eigenpath Traversal

We present a bound on the length of the path defined by the ground states of a continuous family of Hamiltonians in terms of the spectral gap G. We use this bound to obtain a significant improvement over the cost of recently proposed methods for quantum adiabatic state transformations and eigenpath traversal. In particular, we prove that a method based on evolution randomization, which is a simple extension of adiabatic quantum computation, has an average cost of order 1/G^2, and a method based on fixed-point search, has a maximum cost of order 1/G^(3/2). Additionally, if the Hamiltonians satisfy a frustration-free property, such costs can be further improved to order 1/G^(3/2) and 1/G, respectively. Our methods offer an important advantage over adiabatic quantum computation when the gap is small, where the cost is of order 1/G^3.Comment: 10 pages, 1 figur

Exponential improvement in precision for Hamiltonian-evolution simulation

We provide a quantum method for simulating Hamiltonian evolution with complexity polynomial in the logarithm of the inverse error. This is an exponential improvement over existing methods for Hamiltonian simulation. In addition, its scaling with respect to time is close to linear, and its scaling with respect to the time derivative of the Hamiltonian is logarithmic. These scalings improve upon most existing methods. Our method is to use a compressed Lie-Trotter formula, based on recent ideas for efficient discrete-time simulations of continuous-time quantum query algorithms.Comment: 8 pages, 1 figure, updated result in appendi