19,349 research outputs found
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
Adiabatic quantum computation: Noise in the adiabatic theorem and using the Jordan-Wigner transform to find effective Hamiltonians
This thesis explores two mathematical aspects of adiabatic quantum computation. Adiabatic quantum computation depends on the adiabatic theorem of quantum mechanics, and (a) we provide a rigorous formulation of the adiabatic theorem with explicit definitions of constants, and (b) we bound error in the adiabatic approximation under conditions of noise and experimental error. We apply the new results to a standard example of violation of the adiabatic approximation, and to a superconducting flux qubit.
Further, adiabatic quantum computation requires large ground-state energy gaps throughout a Hamiltonian evolution if it is to solve problems in polynomial time. We identify a class of random Hamiltonians with non-nearest-neighbor interactions and a ground-state energy gap of , where is the number of qubits. We also identify two classes of Hamiltonians with non-nearest-neighbor interactions whose ground state can be found in polynomial time with adiabatic quantum computing. We then use the Jordan-Wigner transformation to derive equivalent results for Hamiltonians defined using Pauli operators
Adiabatic quantum computation along quasienergies
The parametric deformations of quasienergies and eigenvectors of unitary
operators are applied to the design of quantum adiabatic algorithms. The
conventional, standard adiabatic quantum computation proceeds along
eigenenergies of parameter-dependent Hamiltonians. By contrast, discrete
adiabatic computation utilizes adiabatic passage along the quasienergies of
parameter-dependent unitary operators. For example, such computation can be
realized by a concatenation of parameterized quantum circuits, with an
adiabatic though inevitably discrete change of the parameter. A design
principle of adiabatic passage along quasienergy is recently proposed: Cheon's
quasienergy and eigenspace anholonomies on unitary operators is available to
realize anholonomic adiabatic algorithms [Tanaka and Miyamoto, Phys. Rev. Lett.
98, 160407 (2007)], which compose a nontrivial family of discrete adiabatic
algorithms. It is straightforward to port a standard adiabatic algorithm to an
anholonomic adiabatic one, except an introduction of a parameter |v>, which is
available to adjust the gaps of the quasienergies to control the running time
steps. In Grover's database search problem, the costs to prepare |v> for the
qualitatively different, i.e., power or exponential, running time steps are
shown to be qualitatively different. Curiously, in establishing the equivalence
between the standard quantum computation based on the circuit model and the
anholonomic adiabatic quantum computation model, it is shown that the cost for
|v> to enlarge the gaps of the eigenvalue is qualitatively negligible.Comment: 11 pages, 2 figure
Effective Physical Processes and Active Information in Quantum Computing
The recent debate on hypercomputation has arisen new questions both on the
computational abilities of quantum systems and the Church-Turing Thesis role in
Physics. We propose here the idea of "effective physical process" as the
essentially physical notion of computation. By using the Bohm and Hiley active
information concept we analyze the differences between the standard form
(quantum gates) and the non-standard one (adiabatic and morphogenetic) of
Quantum Computing, and we point out how its Super-Turing potentialities derive
from an incomputable information source in accordance with Bell's constraints.
On condition that we give up the formal concept of "universality", the
possibility to realize quantum oracles is reachable. In this way computation is
led back to the logic of physical world.Comment: 10 pages; Added references for sections 2 and
Fault tolerance for holonomic quantum computation
We review an approach to fault-tolerant holonomic quantum computation on
stabilizer codes. We explain its workings as based on adiabatic dragging of the
subsystem containing the logical information around suitable loops along which
the information remains protected.Comment: 16 pages, this is a chapter in the book "Quantum Error Correction",
edited by Daniel A. Lidar and Todd A. Brun, (Cambridge University Press,
2013), at
http://www.cambridge.org/us/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/quantum-error-correctio
Adiabatic graph-state quantum computation
Measurement-based quantum computation (MBQC) and holonomic quantum
computation (HQC) are two very different computational methods. The computation
in MBQC is driven by adaptive measurements executed in a particular order on a
large entangled state. In contrast in HQC the system starts in the ground
subspace of a Hamiltonian which is slowly changed such that a transformation
occurs within the subspace. Following the approach of Bacon and Flammia, we
show that any measurement-based quantum computation on a graph state with
\emph{gflow} can be converted into an adiabatically driven holonomic
computation, which we call \emph{adiabatic graph-state quantum computation}
(AGQC). We then investigate how properties of AGQC relate to the properties of
MBQC, such as computational depth. We identify a trade-off that can be made
between the number of adiabatic steps in AGQC and the norm of as well
as the degree of , in analogy to the trade-off between the number of
measurements and classical post-processing seen in MBQC. Finally the effects of
performing AGQC with orderings that differ from standard MBQC are investigated.Comment: 25 pages, 3 figure
Minor-Embedding in Adiabatic Quantum Computation: I. The Parameter Setting Problem
We show that the NP-hard quadratic unconstrained binary optimization (QUBO)
problem on a graph can be solved using an adiabatic quantum computer that
implements an Ising spin-1/2 Hamiltonian, by reduction through minor-embedding
of in the quantum hardware graph . There are two components to this
reduction: embedding and parameter setting. The embedding problem is to find a
minor-embedding of a graph in , which is a subgraph of
such that can be obtained from by contracting edges. The
parameter setting problem is to determine the corresponding parameters, qubit
biases and coupler strengths, of the embedded Ising Hamiltonian. In this paper,
we focus on the parameter setting problem. As an example, we demonstrate the
embedded Ising Hamiltonian for solving the maximum independent set (MIS)
problem via adiabatic quantum computation (AQC) using an Ising spin-1/2 system.
We close by discussing several related algorithmic problems that need to be
investigated in order to facilitate the design of adiabatic algorithms and AQC
architectures.Comment: 17 pages, 5 figures, submitte
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