15 research outputs found
Scalable quantum search using trapped ions
We propose a scalable implementation of Grover's quantum search algorithm in
a trapped-ion quantum information processor. The system is initialized in an
entangled Dicke state by using simple adiabatic techniques. The
inversion-about-average and the oracle operators take the form of single
off-resonant laser pulses, addressing, respectively, all and half of the ions
in the trap. This is made possible by utilizing the physical symmetrie of the
trapped-ion linear crystal. The physical realization of the algorithm
represents a dramatic simplification: each logical iteration (oracle and
inversion about average) requires only two physical interaction steps, in
contrast to the large number of concatenated gates required by previous
approaches. This does not only facilitate the implementation, but also
increases the overall fidelity of the algorithm.Comment: 6 pages, 2 figure
Physical realization of coupled Hilbert-space mirrors for quantum-state engineering
Manipulation of superpositions of discrete quantum states has a mathematical
counterpart in the motion of a unit-length statevector in an N-dimensional
Hilbert space. Any such statevector motion can be regarded as a succession of
two-dimensional rotations. But the desired statevector change can also be
treated as a succession of reflections, the generalization of Householder
transformations. In multidimensional Hilbert space such reflection sequences
offer more efficient procedures for statevector manipulation than do sequences
of rotations. We here show how such reflections can be designed for a system
with two degenerate levels - a generalization of the traditional two-state atom
- that allows the construction of propagators for angular momentum states. We
use the Morris-Shore transformation to express the propagator in terms of
Morris-Shore basis states and Cayley-Klein parameters, which allows us to
connect properties of laser pulses to Hilbert-space motion. Under suitable
conditions on the couplings and the common detuning, the propagators within
each set of degenerate states represent products of generalized Householder
reflections, with orthogonal vectors. We propose physical realizations of this
novel geometrical object with resonant, near-resonant and far-off-resonant
laser pulses. We give several examples of implementations in real atoms or
molecules.Comment: 15 pages, 6 figure
Population trapping in three-state quantum loops revealed by Householder reflections
Population trapping occurs when a particular quantum-state superposition is
immune to action by a specific interaction, such as the well-known dark state
in a three-state lambda system. We here show that in a three-state loop
linkage, a Hilbert-space Householder reflection breaks the loop and presents
the linkage as a single chain. With certain conditions on the interaction
parameters, this chain can break into a simple two-state system and an
additional spectator state. Alternatively, a two-photon resonance condition in
this Householder-basis chain can be enforced, which heralds the existence of
another spectator state. These spectator states generalize the usual dark state
to include contributions from all three bare basis states and disclose hidden
population trapping effects, and hence hidden constants of motion. Insofar as a
spectator state simplifies the overall dynamics, its existence facilitates the
derivation of analytic solutions and the design of recipes for quantum state
engineering in the loop system. Moreover, it is shown that a suitable sequence
of Householder transformations can cast an arbitrary N-dimensional hermitian
Hamiltonian into a tridiagonal form. The implication is that a general N-state
system, with arbitrary linkage patterns where each state connects to any other
state, can be reduced to an equivalent chainwise-connected system, with
nearest-neighbor interactions only, with ensuing possibilities for discovering
hidden multidimensional spectator states and constants of motion
Efficient quantum computation of molecular forces and other energy gradients
While most work on the quantum simulation of chemistry has focused on
computing energy surfaces, a similarly important application requiring subtly
different algorithms is the computation of energy derivatives. Almost all
molecular properties can be expressed an energy derivative, including molecular
forces, which are essential for applications such as molecular dynamics
simulations. Here, we introduce new quantum algorithms for computing molecular
energy derivatives with significantly lower complexity than prior methods.
Under cost models appropriate for noisy-intermediate scale quantum devices we
demonstrate how low rank factorizations and other tomography schemes can be
optimized for energy derivative calculations. We perform numerics revealing
that our techniques reduce the number of circuit repetitions required by many
orders of magnitude for even modest systems. In the context of fault-tolerant
algorithms, we develop new methods of estimating energy derivatives with
Heisenberg limited scaling incorporating state-of-the-art techniques for block
encoding fermionic operators. Our results suggest that the calculation of
forces on a single nucleus may be of similar cost to estimating energies of
chemical systems, but that further developments are needed for quantum
computers to meaningfully assist with molecular dynamics simulations.Comment: 48 pages, 14 page appendix, 10 figures. v2 contains updated lambdas
(rescaling factors) for sparse FT encodings and some NISQ methods, obtained
by localizing orbital