41 research outputs found
On pole-swapping algorithms for the eigenvalue problem
Pole-swapping algorithms, which are generalizations of the QZ algorithm for
the generalized eigenvalue problem, are studied. A new modular (and therefore
more flexible) convergence theory that applies to all pole-swapping algorithms
is developed. A key component of all such algorithms is a procedure that swaps
two adjacent eigenvalues in a triangular pencil. An improved swapping routine
is developed, and its superiority over existing methods is demonstrated by a
backward error analysis and numerical tests. The modularity of the new
convergence theory and the generality of the pole-swapping approach shed new
light on bi-directional chasing algorithms, optimally packed shifts, and bulge
pencils, and allow the design of novel algorithms
Approximate Quantum Circuit Synthesis using Block-Encodings
One of the challenges in quantum computing is the synthesis of unitary
operators into quantum circuits with polylogarithmic gate complexity. Exact
synthesis of generic unitaries requires an exponential number of gates in
general. We propose a novel approximate quantum circuit synthesis technique by
relaxing the unitary constraints and interchanging them for ancilla qubits via
block-encodings. This approach combines smaller block-encodings, which are
easier to synthesize, into quantum circuits for larger operators. Due to the
use of block-encodings, our technique is not limited to unitary operators and
can also be applied for the synthesis of arbitrary operators. We show that
operators which can be approximated by a canonical polyadic expression with a
polylogarithmic number of terms can be synthesized with polylogarithmic gate
complexity with respect to the matrix dimension
Chemistry on quantum computers with virtual quantum subspace expansion
Several novel methods for performing calculations relevant to quantum
chemistry on quantum computers have been proposed but not yet explored
experimentally. Virtual quantum subspace expansion [T. Takeshita et al., Phys.
Rev. X 10, 011004 (2020)] is one such algorithm developed for modeling complex
molecules using their full orbital space and without the need for additional
quantum resources. We implement this method on the IBM Q platform and calculate
the potential energy curves of the hydrogen and lithium dimers using only two
qubits and simple classical post-processing. A comparable level of accuracy
would require twenty qubits with previous approaches. We also develop an
approach to minimize the impact of experimental noise on the stability of a
generalized eigenvalue problem that is a crucial component of the algorithm.
Our results demonstrate that virtual quantum subspace expansion works well in
practice
A rational QZ method
We propose a rational QZ method for the solution of the dense, unsymmetric
generalized eigenvalue problem. This generalization of the classical QZ method
operates implicitly on a Hessenberg, Hessenberg pencil instead of on a
Hessenberg, triangular pencil. Whereas the QZ method performs nested subspace
iteration driven by a polynomial, the rational QZ method allows for nested
subspace iteration driven by a rational function, this creates the additional
freedom of selecting poles. In this article we study Hessenberg, Hessenberg
pencils, link them to rational Krylov subspaces, propose a direct reduction
method to such a pencil, and introduce the implicit rational QZ step. The link
with rational Krylov subspaces allows us to prove essential uniqueness
(implicit Q theorem) of the rational QZ iterates as well as convergence of the
proposed method. In the proofs, we operate directly on the pencil instead of
rephrasing it all in terms of a single matrix. Numerical experiments are
included to illustrate competitiveness in terms of speed and accuracy with the
classical approach. Two other types of experiments exemplify new possibilities.
First we illustrate that good pole selection can be used to deflate the
original problem during the reduction phase, and second we use the rational QZ
method to implicitly filter a rational Krylov subspace in an iterative method
A multishift, multipole rational QZ method with aggressive early deflation
The rational QZ method generalizes the QZ method by implicitly supporting
rational subspace iteration. In this paper we extend the rational QZ method by
introducing shifts and poles of higher multiplicity in the Hessenberg pencil,
which is a pencil consisting of two Hessenberg matrices. The result is a
multishift, multipole iteration on block Hessenberg pencils which allows one to
stick to real arithmetic for a real input pencil. In combination with optimally
packed shifts and aggressive early deflation as an advanced deflation technique
we obtain an efficient method for the dense generalized eigenvalue problem. In
the numerical experiments we compare the results with state-of-the-art routines
for the generalized eigenvalue problem and show that we are competitive in
terms of speed and accuracy
Estimating Eigenenergies from Quantum Dynamics: A Unified Noise-Resilient Measurement-Driven Approach
Ground state energy estimation in physics and chemistry is one of the most
promising applications of quantum computing. In this paper, we introduce a
novel measurement-driven approach that finds eigenenergies by collecting
real-time measurements and post-processing them using the machinery of dynamic
mode decomposition (DMD). We provide theoretical and numerical evidence that
our method converges rapidly even in the presence of noise and show that our
method is isomorphic to matrix pencil methods developed independently across
various scientific communities. Our DMD-based strategy can systematically
mitigate perturbative noise and stands out as a promising hybrid
quantum-classical eigensolver
HamLib: A library of Hamiltonians for benchmarking quantum algorithms and hardware
In order to characterize and benchmark computational hardware, software, and
algorithms, it is essential to have many problem instances on-hand. This is no
less true for quantum computation, where a large collection of real-world
problem instances would allow for benchmarking studies that in turn help to
improve both algorithms and hardware designs. To this end, here we present a
large dataset of qubit-based quantum Hamiltonians. The dataset, called HamLib
(for Hamiltonian Library), is freely available online and contains problem
sizes ranging from 2 to 1000 qubits. HamLib includes problem instances of the
Heisenberg model, Fermi-Hubbard model, Bose-Hubbard model, molecular electronic
structure, molecular vibrational structure, MaxCut, Max-k-SAT, Max-k-Cut,
QMaxCut, and the traveling salesperson problem. The goals of this effort are
(a) to save researchers time by eliminating the need to prepare problem
instances and map them to qubit representations, (b) to allow for more thorough
tests of new algorithms and hardware, and (c) to allow for reproducibility and
standardization across research studies