93,247 research outputs found
Quantum algorithm for non-homogeneous linear partial differential equations
We describe a quantum algorithm for preparing states that encode solutions of
non-homogeneous linear partial differential equations. The algorithm is a
continuous-variable version of matrix inversion: it efficiently inverts
differential operators that are polynomials in the variables and their partial
derivatives. The output is a quantum state whose wavefunction is proportional
to a specific solution of the non-homogeneous differential equation, which can
be measured to reveal features of the solution. The algorithm consists of three
stages: preparing fixed resource states in ancillary systems, performing
Hamiltonian simulation, and measuring the ancilla systems. The algorithm can be
carried out using standard methods for gate decompositions, but we improve this
in two ways. First, we show that for a wide class of differential operators, it
is possible to derive exact decompositions for the gates employed in
Hamiltonian simulation. This avoids the need for costly commutator
approximations, reducing gate counts by orders of magnitude. Additionally, we
employ methods from machine learning to find explicit circuits that prepare the
required resource states. We conclude by studying an example application of the
algorithm: solving Poisson's equation in electrostatics.Comment: 9 pages, 6 figure
Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity
We show that there exists a universal quantum Turing machine (UQTM) that can
simulate every other QTM until the other QTM has halted and then halt itself
with probability one. This extends work by Bernstein and Vazirani who have
shown that there is a UQTM that can simulate every other QTM for an arbitrary,
but preassigned number of time steps. As a corollary to this result, we give a
rigorous proof that quantum Kolmogorov complexity as defined by Berthiaume et
al. is invariant, i.e. depends on the choice of the UQTM only up to an additive
constant. Our proof is based on a new mathematical framework for QTMs,
including a thorough analysis of their halting behaviour. We introduce the
notion of mutually orthogonal halting spaces and show that the information
encoded in an input qubit string can always be effectively decomposed into a
classical and a quantum part.Comment: 18 pages, 1 figure. The operation R is now really a quantum operation
(it was not before); corrected some typos, III.B more readable, Conjecture
3.15 is now a theore
Universal Quantum Computation with ideal Clifford gates and noisy ancillas
We consider a model of quantum computation in which the set of elementary
operations is limited to Clifford unitaries, the creation of the state ,
and qubit measurement in the computational basis. In addition, we allow the
creation of a one-qubit ancilla in a mixed state , which should be
regarded as a parameter of the model. Our goal is to determine for which
universal quantum computation (UQC) can be efficiently simulated. To answer
this question, we construct purification protocols that consume several copies
of and produce a single output qubit with higher polarization. The
protocols allow one to increase the polarization only along certain ``magic''
directions. If the polarization of along a magic direction exceeds a
threshold value (about 65%), the purification asymptotically yields a pure
state, which we call a magic state. We show that the Clifford group operations
combined with magic states preparation are sufficient for UQC. The connection
of our results with the Gottesman-Knill theorem is discussed.Comment: 15 pages, 4 figures, revtex
Anomalies in the Entanglement Properties of the Square Lattice Heisenberg Model
We compute the bipartite entanglement properties of the spin-half
square-lattice Heisenberg model by a variety of numerical techniques that
include valence bond quantum Monte Carlo (QMC), stochastic series expansion
QMC, high temperature series expansions and zero temperature coupling constant
expansions around the Ising limit. We find that the area law is always
satisfied, but in addition to the entanglement entropy per unit boundary
length, there are other terms that depend logarithmically on the subregion
size, arising from broken symmetry in the bulk and from the existence of
corners at the boundary. We find that the numerical results are anomalous in
several ways. First, the bulk term arising from broken symmetry deviates from
an exact calculation that can be done for a mean-field Neel state. Second, the
corner logs do not agree with the known results for non-interacting Boson
modes. And, third, even the finite temperature mutual information shows an
anomalous behavior as T goes to zero, suggesting that T->0 and L->infinity
limits do not commute. These calculations show that entanglement entropy
demonstrates a very rich behavior in d>1, which deserves further attention.Comment: 12 pages, 7 figures, 2 tables. Numerical values in Table I correcte
Quantum computing and the entanglement frontier - Rapporteur talk at the 25th Solvay Conference
Quantum information science explores the frontier of highly complex quantum states,
the "entanglement frontier". This study is motivated by the observation (widely believed
but unproven) that classical systems cannot simulate highly entangled quantum systems
efficiently, and we hope to hasten the day when well controlled quantum systems can
perform tasks surpassing what can be done in the classical world. One way to achieve
such "quantum supremacy" would be to run an algorithm on a quantum computer which
solves a problem with a super-polynomial speedup relative to classical computers, but
there may be other ways that can be achieved sooner, such as simulating exotic quantum
states of strongly correlated matter. To operate a large scale quantum computer reliably
we will need to overcome the debilitating effects of decoherence, which might be done
using "standard" quantum hardware protected by quantum error-correcting codes, or by
exploiting the nonabelian quantum statistics of anyons realized in solid state systems,
or by combining both methods. Only by challenging the entanglement frontier will we
learn whether Nature provides extravagant resources far beyond what the classical world
would allow
Programmable coherent linear quantum operations with high-dimensional optical spatial modes
A simple and flexible scheme for high-dimensional linear quantum operations
on optical transverse spatial modes is demonstrated. The quantum Fourier
transformation (QFT) and quantum state tomography (QST) via symmetric
informationally complete positive operator-valued measures (SIC POVMs) are
implemented with dimensionality of 15. The matrix fidelity of QFT is 0.85,
while the statistical fidelity of SIC POVMs and fidelity of QST are ~0.97 and
up to 0.853, respectively. We believe that our device has the potential for
further exploration of high-dimensional spatial entanglement provided by
spontaneous parametric down conversion in nonlinear crystals
- …