172 research outputs found
On Cauchy problem for first order nonlinear functional differential equations of non-Volterra’s type
summary:On the segment consider the problem where is a continuous, in general nonlinear operator satisfying Carathéodory condition, and . The effective sufficient conditions guaranteeing the solvability and unique solvability of the considered problem are established. Examples verifying the optimality of obtained results are given, as well
The Fragility of Quantum Information?
We address the question whether there is a fundamental reason why quantum
information is more fragile than classical information. We show that some
answers can be found by considering the existence of quantum memories and their
dimensional dependence.Comment: Essay on quantum information: no new results. Ten pages, published in
Lec. Notes in Comp. Science, Vol. 7505, pp. 47-56 (2012. One reference adde
No-go Theorem for One-way Quantum Computing on Naturally Occurring Two-level Systems
One-way quantum computing achieves the full power of quantum computation by
performing single particle measurements on some many-body entangled state,
known as the resource state. As single particle measurements are relatively
easy to implement, the preparation of the resource state becomes a crucial
task. An appealing approach is simply to cool a strongly correlated quantum
many-body system to its ground state. In addition to requiring the ground state
of the system to be universal for one-way quantum computing, we also want the
Hamiltonian to have non-degenerate ground state protected by a fixed energy
gap, to involve only two-body interactions, and to be frustration-free so that
measurements in the course of the computation leave the remaining particles in
the ground space. Recently, significant efforts have been made to the search of
resource states that appear naturally as ground states in spin lattice systems.
The approach is proved to be successful in spin-5/2 and spin-3/2 systems. Yet,
it remains an open question whether there could be such a natural resource
state in a spin-1/2, i.e., qubit system. Here, we give a negative answer to
this question by proving that it is impossible for a genuinely entangled qubit
states to be a non-degenerate ground state of any two-body frustration-free
Hamiltonian. What is more, we prove that every spin-1/2 frustration-free
Hamiltonian with two-body interaction always has a ground state that is a
product of single- or two-qubit states, a stronger result that is interesting
independent of the context of one-way quantum computing.Comment: 5 pages, 1 figur
Quantum simulations under translational symmetry
We investigate the power of quantum systems for the simulation of Hamiltonian
time evolutions on a cubic lattice under the constraint of translational
invariance. Given a set of translationally invariant local Hamiltonians and
short range interactions we determine time evolutions which can and those that
can not be simulated. Whereas for general spin systems no finite universal set
of generating interactions is shown to exist, universality turns out to be
generic for quadratic bosonic and fermionic nearest-neighbor interactions when
supplemented by all translationally invariant on-site Hamiltonians.Comment: 9 pages, 2 figures, references added, minor change
Simulation of quantum circuits by ow-rank sotabilizer decompositions
Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of stabilizer rank, which for a pure state ψ is defined to be the smallest integer χ such that ψ is a superposition of χ stabilizer states.
Here we develop a comprehensive mathematical theory of the stabilizer rank and the
related approximate stabilizer rank. We also present a suite of classical simulation
algorithms with broader applicability and significantly improved performance over the
previous state-of-the-art. A new feature is the capability to simulate circuits composed
of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation
methods and used them to simulate quantum algorithms with 40-50 qubits and over
60 non-Clifford gates, without resorting to high-performance computers. We report a
simulation of the Quantum Approximate Optimization Algorithm in which we process
superpositions of χ ∼ 106
stabilizer states and sample from the full n-bit output distribution, improving on previous simulations which used ∼ 103
stabilizer states and
sampled only from single-qubit marginals. We also simulated instances of the Hidden
Shift algorithm with circuits including up to 64 T gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a
circuit’s non-Clifford components
Quantum algorithms for spin models and simulable gate sets for quantum computation
We present elementary mappings between classical lattice models and quantum
circuits. These mappings provide a general framework to obtain efficiently
simulable quantum gate sets from exactly solvable classical models. For
example, we recover and generalize the simulability of Valiant's match-gates by
invoking the solvability of the free-fermion eight-vertex model. Our mappings
furthermore provide a systematic formalism to obtain simple quantum algorithms
to approximate partition functions of lattice models in certain
complex-parameter regimes. For example, we present an efficient quantum
algorithm for the six-vertex model as well as a 2D Ising-type model. We finally
show that simulating our quantum algorithms on a classical computer is as hard
as simulating universal quantum computation (i.e. BQP-complete).Comment: 6 pages, 2 figure
Experimental magic state distillation for fault-tolerant quantum computing
Any physical quantum device for quantum information processing is subject to
errors in implementation. In order to be reliable and efficient, quantum
computers will need error correcting or error avoiding methods. Fault-tolerance
achieved through quantum error correction will be an integral part of quantum
computers. Of the many methods that have been discovered to implement it, a
highly successful approach has been to use transversal gates and specific
initial states. A critical element for its implementation is the availability
of high-fidelity initial states such as |0> and the Magic State. Here we report
an experiment, performed in a nuclear magnetic resonance (NMR) quantum
processor, showing sufficient quantum control to improve the fidelity of
imperfect initial magic states by distilling five of them into one with higher
fidelity
Quantum Speedup by Quantum Annealing
We study the glued-trees problem of Childs et. al. in the adiabatic model of
quantum computing and provide an annealing schedule to solve an oracular
problem exponentially faster than classically possible. The Hamiltonians
involved in the quantum annealing do not suffer from the so-called sign
problem. Unlike the typical scenario, our schedule is efficient even though the
minimum energy gap of the Hamiltonians is exponentially small in the problem
size. We discuss generalizations based on initial-state randomization to avoid
some slowdowns in adiabatic quantum computing due to small gaps.Comment: 7 page
Structure of 2D Topological Stabilizer Codes
We provide a detailed study of the general structure of two-dimensional
topological stabilizer quantum error correcting codes, including subsystem
codes. Under the sole assumption of translational invariance, we show that all
such codes can be understood in terms of the homology of string operators that
carry a certain topological charge. In the case of subspace codes, we prove
that two codes are equivalent under a suitable set of local transformations if
and only they have equivalent topological charges. Our approach emphasizes
local properties of the codes over global ones.Comment: 54 pages, 11 figures, version accepted in journal, improved
presentation and result
Thermodyamic bounds on Drude weights in terms of almost-conserved quantities
We consider one-dimensional translationally invariant quantum spin (or
fermionic) lattices and prove a Mazur-type inequality bounding the
time-averaged thermodynamic limit of a finite-temperature expectation of a
spatio-temporal autocorrelation function of a local observable in terms of
quasi-local conservation laws with open boundary conditions. Namely, the
commutator between the Hamiltonian and the conservation law of a finite chain
may result in boundary terms only. No reference to techniques used in Suzuki's
proof of Mazur bound is made (which strictly applies only to finite-size
systems with exact conservation laws), but Lieb-Robinson bounds and exponential
clustering theorems of quasi-local C^* quantum spin algebras are invoked
instead. Our result has an important application in the transport theory of
quantum spin chains, in particular it provides rigorous non-trivial examples of
positive finite-temperature spin Drude weight in the anisotropic Heisenberg XXZ
spin 1/2 chain [Phys. Rev. Lett. 106, 217206 (2011)].Comment: version as accepted by Communications in Mathematical Physics (22
pages with 2 pdf-figures
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