6,218 research outputs found
Generation of eigenstates using the phase-estimation algorithm
The phase estimation algorithm is so named because it allows the estimation
of the eigenvalues associated with an operator. However it has been proposed
that the algorithm can also be used to generate eigenstates. Here we extend
this proposal for small quantum systems, identifying the conditions under which
the phase estimation algorithm can successfully generate eigenstates. We then
propose an implementation scheme based on an ion trap quantum computer. This
scheme allows us to illustrate two simple examples, one in which the algorithm
effectively generates eigenstates, and one in which it does not.Comment: 5 pages, 3 Figures, RevTeX4 Introduction expanded, typos correcte
Witnessing eigenstates for quantum simulation of Hamiltonian spectra
The efficient calculation of Hamiltonian spectra, a problem often intractable
on classical machines, can find application in many fields, from physics to
chemistry. Here, we introduce the concept of an "eigenstate witness" and
through it provide a new quantum approach which combines variational methods
and phase estimation to approximate eigenvalues for both ground and excited
states. This protocol is experimentally verified on a programmable silicon
quantum photonic chip, a mass-manufacturable platform, which embeds entangled
state generation, arbitrary controlled-unitary operations, and projective
measurements. Both ground and excited states are experimentally found with
fidelities >99%, and their eigenvalues are estimated with 32-bits of precision.
We also investigate and discuss the scalability of the approach and study its
performance through numerical simulations of more complex Hamiltonians. This
result shows promising progress towards quantum chemistry on quantum computers.Comment: 9 pages, 4 figures, plus Supplementary Material [New version with
minor typos corrected.
Postprocessing can speed up general quantum search algorithms
A general quantum search algorithm aims to evolve a quantum system from a
known source state to an unknown target state . It uses
a diffusion operator having source state as one of its eigenstates and
, where denotes the selective phase inversion of
state. It evolves to a particular state ,
call it w-state, in time steps where is and is a characteristic of the diffusion operator. Measuring
the w-state gives the target state with the success probability of
and applications of the algorithm can boost it from to
, making the total time complexity . In the special case
of Grover's algorithm, is and is very close to . A more
efficient way to boost the success probability is quantum amplitude
amplification provided we can efficiently implement . Such an efficient
implementation is not known so far. In this paper, we present an efficient
algorithm to approximate selective phase inversions of the unknown eigenstates
of an operator using phase estimation algorithm. This algorithm is used to
efficiently approximate which reduces the time complexity of general
algorithm to . Though algorithms are known to exist,
our algorithm offers physical implementation advantages.Comment: Accepted for publication in Physical Review A. arXiv admin note:
substantial text overlap with arXiv:1210.464
Quantum Metropolis Sampling
The original motivation to build a quantum computer came from Feynman who
envisaged a machine capable of simulating generic quantum mechanical systems, a
task that is believed to be intractable for classical computers. Such a machine
would have a wide range of applications in the simulation of many-body quantum
physics, including condensed matter physics, chemistry, and high energy
physics. Part of Feynman's challenge was met by Lloyd who showed how to
approximately decompose the time-evolution operator of interacting quantum
particles into a short sequence of elementary gates, suitable for operation on
a quantum computer. However, this left open the problem of how to simulate the
equilibrium and static properties of quantum systems. This requires the
preparation of ground and Gibbs states on a quantum computer. For classical
systems, this problem is solved by the ubiquitous Metropolis algorithm, a
method that basically acquired a monopoly for the simulation of interacting
particles. Here, we demonstrate how to implement a quantum version of the
Metropolis algorithm on a quantum computer. This algorithm permits to sample
directly from the eigenstates of the Hamiltonian and thus evades the sign
problem present in classical simulations. A small scale implementation of this
algorithm can already be achieved with today's technologyComment: revised versio
Encoding a Qubit into a Cavity Mode in Circuit-QED using Phase Estimation
Gottesman, Kitaev and Preskill have formulated a way of encoding a qubit into
an oscillator such that the qubit is protected against small shifts
(translations) in phase space. The idea underlying this encoding is that error
processes of low rate can be expanded into small shift errors. The qubit space
is defined as an eigenspace of two mutually commuting displacement operators
and which act as large shifts/translations in phase space. We
propose and analyze the approximate creation of these qubit states by coupling
the oscillator to a sequence of ancilla qubits. This preparation of the states
uses the idea of phase estimation where the phase of the displacement operator,
say , is approximately determined. We consider several possible forms of
phase estimation. We analyze the performance of repeated and adapative phase
estimation as the simplest and experimentally most viable schemes given a
realistic upper-limit on the number of photons in the oscillator. We propose a
detailed physical implementation of this protocol using the dispersive coupling
between a transmon ancilla qubit and a cavity mode in circuit-QED. We provide
an estimate that in a current experimental set-up one can prepare a good code
state from a squeezed vacuum state using rounds of adapative phase
estimation, lasting in total about sec., with (heralded) chance
of success.Comment: 24 pages, 15 figures. Some minor improvements to text and figures.
Some of the numerical data has been replaced by more accurate simulations.
The improved simulation shows that the code performs better than originally
anticipate
Analyzing many-body localization with a quantum computer
Many-body localization, the persistence against electron-electron
interactions of the localization of states with non-zero excitation energy
density, poses a challenge to current methods of theoretical and numerical
analysis. Numerical simulations have so far been limited to a small number of
sites, making it difficult to obtain reliable statements about the
thermodynamic limit. In this paper, we explore the ways in which a relatively
small quantum computer could be leveraged to study many-body localization. We
show that, in addition to studying time-evolution, a quantum computer can, in
polynomial time, obtain eigenstates at arbitrary energies to sufficient
accuracy that localization can be observed. The limitations of quantum
measurement, which preclude the possibility of directly obtaining the
entanglement entropy, make it difficult to apply some of the definitions of
many-body localization used in the recent literature. We discuss alternative
tests of localization that can be implemented on a quantum computer.Comment: 11 pages, 8 figures; slightly revised, published versio
Calculating energy derivatives for quantum chemistry on a quantum computer
Modeling chemical reactions and complicated molecular systems has been
proposed as the `killer application' of a future quantum computer. Accurate
calculations of derivatives of molecular eigenenergies are essential towards
this end, allowing for geometry optimization, transition state searches,
predictions of the response to an applied electric or magnetic field, and
molecular dynamics simulations. In this work, we survey methods to calculate
energy derivatives, and present two new methods: one based on quantum phase
estimation, the other on a low-order response approximation. We calculate
asymptotic error bounds and approximate computational scalings for the methods
presented. Implementing these methods, we perform the world's first geometry
optimization on an experimental quantum processor, estimating the equilibrium
bond length of the dihydrogen molecule to within 0.014 Angstrom of the full
configuration interaction value. Within the same experiment, we estimate the
polarizability of the H2 molecule, finding agreement at the equilibrium bond
length to within 0.06 a.u. (2% relative error).Comment: 19 pages, 1 page supplemental, 7 figures. v2 - tidied up and added
example to appendice
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