3 research outputs found
Fermionic Linear Optics Revisited
We provide an alternative view of the efficient classical simulatibility of
fermionic linear optics in terms of Slater determinants. We investigate the
generic effects of two-mode measurements on the Slater number of fermionic
states. We argue that most such measurements are not capable (in conjunction
with fermion linear optics) of an efficient exact implementation of universal
quantum computation. Our arguments do not apply to the two-mode parity
measurement, for which exact quantum computation becomes possible, see
quant-ph/0401066.Comment: 16 pages, submitted to the special issue of Foundation of Physics in
honor of Asher Peres' 70th birthda
Spectral estimation for Hamiltonians: A comparison between classical imaginary-time evolution and quantum real-time evolution
We consider the task of spectral estimation of local quantum Hamiltonians. The spectral estimation is performed by estimating the oscillation frequencies or decay rates of signals representing the time evolution of states. We present a classical Monte Carlo (MC) scheme which efficiently estimates an imaginary-time, decaying signal for stoquastic (i.e. sign-problem-free) local Hamiltonians. The decay rates in this signal correspond to Hamiltonian eigenvalues (with associated eigenstates present in an input state) and can be extracted using a classical signal processing method like ESPRIT. We compare the efficiency of this MC scheme to its quantum counterpart in which one extracts eigenvalues of a general local Hamiltonian from a real-time, oscillatory signal obtained through quantum phase estimation circuits, again using the ESPRIT method. We prove that the ESPRIT method can resolve S = poly(n) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) quantum and classical effort through the quantum phase estimation circuits, assuming efficient preparation of the input state. We prove that our Monte Carlo scheme plus the ESPRIT method can resolve S = O(1) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) purely classical effort for stoquastic Hamiltonians, requiring some access structure to the input state. However, we also show that under these assumptions, i.e. S = O(1) eigenvalues, assuming a 1/poly(n) gap between them and some access structure to the input state, one can achieve this with poly(n) purely classical effort for general local Hamiltonians. These results thus quantify some opportunities and limitations of Monte Carlo methods for spectral estimation of Hamiltonians. We numerically compare the MC eigenvalue estimation scheme (for stoquastic Hamiltonians) and the quantum-phase-estimation-based eigenvalue estimation scheme by implementing them for an archetypal stoquastic Hamiltonian system: the transverse field Ising chain
The problem of equilibration and the computation of correlation functions on a quantum computer
We address the question of how a quantum computer can be used to simulate
experiments on quantum systems in thermal equilibrium. We present two
approaches for the preparation of the equilibrium state on a quantum computer.
For both approaches, we show that the output state of the algorithm, after long
enough time, is the desired equilibrium. We present a numerical analysis of one
of these approaches for small systems. We show how equilibrium
(time)-correlation functions can be efficiently estimated on a quantum
computer, given a preparation of the equilibrium state. The quantum algorithms
that we present are hard to simulate on a classical computer. This indicates
that they could provide an exponential speedup over what can be achieved with a
classical device.Comment: 25 pages LaTex + 8 figures; various additional comments, results and
correction