1,519 research outputs found
Direct Application of the Phase Estimation Algorithm to Find the Eigenvalues of the Hamiltonians
The eigenvalue of a Hamiltonian, , can be estimated through the
phase estimation algorithm given the matrix exponential of the Hamiltonian,
. The difficulty of this exponentiation impedes the
applications of the phase estimation algorithm particularly when
is composed of non-commuting terms. In this paper, we present a method to use
the Hamiltonian matrix directly in the phase estimation algorithm by using an
ancilla based framework: In this framework, we also show how to find the power
of the Hamiltonian matrix-which is necessary in the phase estimation
algorithm-through the successive applications. This may eliminate the necessity
of matrix exponential for the phase estimation algorithm and therefore provide
an efficient way to estimate the eigenvalues of particular Hamiltonians. The
classical and quantum algorithmic complexities of the framework are analyzed
for the Hamiltonians which can be written as a sum of simple unitary matrices
and shown that a Hamiltonian of order written as a sum of number of
simple terms can be used in the phase estimation algorithm with
number of qubits and number of quantum operations, where is the
number of iterations in the phase estimation. In addition, we use the
Hamiltonian of the hydrogen molecule as an example system and present the
simulation results for finding its ground state energy.Comment: 10 pages, 3 figure
A Universal Quantum Circuit Scheme For Finding Complex Eigenvalues
We present a general quantum circuit design for finding eigenvalues of
non-unitary matrices on quantum computers using the iterative phase estimation
algorithm. In particular, we show how the method can be used for the simulation
of resonance states for quantum systems
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
Quantum computing is powerful because unitary operators describing the
time-evolution of a quantum system have exponential size in terms of the number
of qubits present in the system. We develop a new "Singular value
transformation" algorithm capable of harnessing this exponential advantage,
that can apply polynomial transformations to the singular values of a block of
a unitary, generalizing the optimal Hamiltonian simulation results of Low and
Chuang. The proposed quantum circuits have a very simple structure, often give
rise to optimal algorithms and have appealing constant factors, while usually
only use a constant number of ancilla qubits. We show that singular value
transformation leads to novel algorithms. We give an efficient solution to a
certain "non-commutative" measurement problem and propose a new method for
singular value estimation. We also show how to exponentially improve the
complexity of implementing fractional queries to unitaries with a gapped
spectrum. Finally, as a quantum machine learning application we show how to
efficiently implement principal component regression. "Singular value
transformation" is conceptually simple and efficient, and leads to a unified
framework of quantum algorithms incorporating a variety of quantum speed-ups.
We illustrate this by showing how it generalizes a number of prominent quantum
algorithms, including: optimal Hamiltonian simulation, implementing the
Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude
amplification, robust oblivious amplitude amplification, fast QMA
amplification, fast quantum OR lemma, certain quantum walk results and several
quantum machine learning algorithms. In order to exploit the strengths of the
presented method it is useful to know its limitations too, therefore we also
prove a lower bound on the efficiency of singular value transformation, which
often gives optimal bounds.Comment: 67 pages, 1 figur
Hamiltonian Simulation by Qubitization
We present the problem of approximating the time-evolution operator
to error , where the Hamiltonian is the
projection of a unitary oracle onto the state created by
another unitary oracle. Our algorithm solves this with a query complexity
to both oracles that is optimal
with respect to all parameters in both the asymptotic and non-asymptotic
regime, and also with low overhead, using at most two additional ancilla
qubits. This approach to Hamiltonian simulation subsumes important prior art
considering Hamiltonians which are -sparse or a linear combination of
unitaries, leading to significant improvements in space and gate complexity,
such as a quadratic speed-up for precision simulations. It also motivates
useful new instances, such as where is a density matrix. A key
technical result is `qubitization', which uses the controlled version of these
oracles to embed any in an invariant subspace. A large
class of operator functions of can then be computed with optimal
query complexity, of which is a special case.Comment: 23 pages, 1 figure; v2: updated notation; v3: accepted versio
- …