32,233 research outputs found
An efficient quantum algorithm for spectral estimation
We develop an efficient quantum implementation of an important signal
processing algorithm for line spectral estimation: the matrix pencil method,
which determines the frequencies and damping factors of signals consisting of
finite sums of exponentially damped sinusoids. Our algorithm provides a
quantum speedup in a natural regime where the sampling rate is much higher
than the number of sinusoid components. Along the way, we develop techniques
that are expected to be useful for other quantum algorithms as
well—consecutive phase estimations to efficiently make products of asymmetric
low rank matrices classically accessible and an alternative method to
efficiently exponentiate non-Hermitian matrices. Our algorithm features an
efficient quantum–classical division of labor: the time-critical steps are
implemented in quantum superposition, while an interjacent step, requiring
much fewer parameters, can operate classically. We show that frequencies and
damping factors can be obtained in time logarithmic in the number of sampling
points, exponentially faster than known classical algorithms
A Quasi-Random Approach to Matrix Spectral Analysis
Inspired by the quantum computing algorithms for Linear Algebra problems
[HHL,TaShma] we study how the simulation on a classical computer of this type
of "Phase Estimation algorithms" performs when we apply it to solve the
Eigen-Problem of Hermitian matrices. The result is a completely new, efficient
and stable, parallel algorithm to compute an approximate spectral decomposition
of any Hermitian matrix. The algorithm can be implemented by Boolean circuits
in parallel time with a total cost of Boolean
operations. This Boolean complexity matches the best known rigorous parallel time algorithms, but unlike those algorithms our algorithm is
(logarithmically) stable, so further improvements may lead to practical
implementations.
All previous efficient and rigorous approaches to solve the Eigen-Problem use
randomization to avoid bad condition as we do too. Our algorithm makes further
use of randomization in a completely new way, taking random powers of a unitary
matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian
perturbation and a random polynomial power are sufficient to ensure almost
pairwise independence of the phases is the main technical
contribution of this work. This randomization enables us, given a Hermitian
matrix with well separated eigenvalues, to sample a random eigenvalue and
produce an approximate eigenvector in parallel time and
Boolean complexity. We conjecture that further improvements of
our method can provide a stable solution to the full approximate spectral
decomposition problem with complexity similar to the complexity (up to a
logarithmic factor) of sampling a single eigenvector.Comment: Replacing previous version: parallel algorithm runs in total
complexity and not . However, the depth of the
implementing circuit is : hence comparable to fastest
eigen-decomposition algorithms know
Real-Time Krylov Theory for Quantum Computing Algorithms
Quantum computers provide new avenues to access ground and excited state
properties of systems otherwise difficult to simulate on classical hardware.
New approaches using subspaces generated by real-time evolution have shown
efficiency in extracting eigenstate information, but the full capabilities of
such approaches are still not understood. In recent work, we developed the
variational quantum phase estimation (VQPE) method, a compact and efficient
real-time algorithm to extract eigenvalues on quantum hardware. Here we build
on that work by theoretically and numerically exploring a generalized Krylov
scheme where the Krylov subspace is constructed through a parametrized
real-time evolution, which applies to the VQPE algorithm as well as others. We
establish an error bound that justifies the fast convergence of our spectral
approximation. We also derive how the overlap with high energy eigenstates
becomes suppressed from real-time subspace diagonalization and we visualize the
process that shows the signature phase cancellations at specific eigenenergies.
We investigate various algorithm implementations and consider performance when
stochasticity is added to the target Hamiltonian in the form of spectral
statistics. To demonstrate the practicality of such real-time evolution, we
discuss its application to fundamental problems in quantum computation such as
electronic structure predictions for strongly correlated systems
Approximating Fractional Time Quantum Evolution
An algorithm is presented for approximating arbitrary powers of a black box
unitary operation, , where is a real number, and
is a black box implementing an unknown unitary. The complexity of
this algorithm is calculated in terms of the number of calls to the black box,
the errors in the approximation, and a certain `gap' parameter. For general
and large , one should apply a total of times followed by our procedure for approximating the fractional
power . An example is also given where for
large integers this method is more efficient than direct application of
copies of . Further applications and related algorithms are also
discussed.Comment: 13 pages, 2 figure
Adiabatic Quantum State Generation and Statistical Zero Knowledge
The design of new quantum algorithms has proven to be an extremely difficult
task. This paper considers a different approach to the problem, by studying the
problem of 'quantum state generation'. This approach provides intriguing links
between many different areas: quantum computation, adiabatic evolution,
analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing
Markov chains, the complexity class statistical zero knowledge, quantum random
walks, and more.
We first show that many natural candidates for quantum algorithms can be cast
as a state generation problem. We define a paradigm for state generation,
called 'adiabatic state generation' and develop tools for adiabatic state
generation which include methods for implementing very general Hamiltonians and
ways to guarantee non negligible spectral gaps. We use our tools to prove that
adiabatic state generation is equivalent to state generation in the standard
quantum computing model, and finally we show how to apply our techniques to
generate interesting superpositions related to Markov chains.Comment: 35 pages, two figure
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