40 research outputs found
Holonomic quantum computation in subsystems
We introduce a generalized method of holonomic quantum computation (HQC)
based on encoding in subsystems. As an application, we propose a scheme for
applying holonomic gates to unencoded qubits by the use of a noisy ancillary
qubit. This scheme does not require initialization in a subspace since all
dynamical effects factor out as a transformation on the ancilla. We use this
approach to show how fault-tolerant HQC can be realized via 2-local
Hamiltonians with perturbative gadgets.Comment: Improved presentation, references adde
Robustness of operator quantum error correction with respect to initialization errors
In the theory of operator quantum error correction (OQEC), the notion of
correctability is defined under the assumption that states are perfectly
initialized inside a particular subspace, a factor of which (a subsystem)
contains the protected information. If the initial state of the system does not
belong entirely to the subspace in question, the restriction of the state to
the otherwise correctable subsystem may not remain invariant after the
application of noise and error correction. It is known that in the case of
decoherence-free subspaces and subsystems (DFSs) the condition for perfect
unitary evolution inside the code imposes more restrictive conditions on the
noise process if one allows imperfect initialization. It was believed that
these conditions are necessary if DFSs are to be able to protect imperfectly
encoded states from subsequent errors. By a similar argument, general OQEC
codes would also require more restrictive error-correction conditions for the
case of imperfect initialization. In this study, we examine this requirement by
looking at the errors on the encoded state. In order to quantitatively analyze
the errors in an OQEC code, we introduce a measure of the fidelity between the
encoded information in two states for the case of subsystem encoding. A major
part of the paper concerns the definition of the measure and the derivation of
its properties. In contrast to what was previously believed, we obtain that
more restrictive conditions are not necessary neither for DFSs nor for general
OQEC codes. This is because the effective noise that can arise inside the code
as a result of imperfect initialization is such that it can only increase the
fidelity of an imperfectly encoded state with a perfectly encoded one.Comment: 8 pages, no figure
Adiabatic Markovian Dynamics
We propose a theory of adiabaticity in quantum Markovian dynamics based on a
decomposition of the Hilbert space induced by the asymptotic behavior of the
Lindblad semigroup. A central idea of our approach is that the natural
generalization of the concept of eigenspace of the Hamiltonian in the case of
Markovian dynamics is a noiseless subsystem with a minimal noisy cofactor.
Unlike previous attempts to define adiabaticity for open systems, our approach
deals exclusively with physical entities and provides a simple, intuitive
picture at the underlying Hilbert-space level, linking the notion of
adiabaticity to the theory of noiseless subsystems. As an application of our
theory, we propose a framework for decoherence-assisted computation in
noiseless codes under general Markovian noise. We also formulate a
dissipation-driven approach to holonomic computation based on adiabatic
dragging of subsystems that is generally not achievable by non-dissipative
means.Comment: 4+3 page
General conditions for approximate quantum error correction and near-optimal recovery channels
We derive necessary and sufficient conditions for the approximate
correctability of a quantum code, generalizing the Knill-Laflamme conditions
for exact error correction. Our measure of success of the recovery operation is
the worst-case entanglement fidelity of the overall process. We show that the
optimal recovery fidelity can be predicted exactly from a dual optimization
problem on the environment causing the noise. We use this result to obtain an
easy-to-calculate estimate of the optimal recovery fidelity as well as a way of
constructing a class of near-optimal recovery channels that work within twice
the minimal error. In addition to standard subspace codes, our results hold for
subsystem codes and hybrid quantum-classical codes.Comment: minor clarifications, typos edited, references added
The Multi-round Process Matrix
We develop an extension of the process matrix (PM) framework for correlations
between quantum operations with no causal order that allows multiple rounds of
information exchange for each party compatibly with the assumption of
well-defined causal order of events locally. We characterise the higher-order
process describing such correlations, which we name the multi-round process
matrix (MPM), and formulate a notion of causal nonseparability for it that
extends the one for standard PMs. We show that in the multi-round case there
are novel manifestations of causal nonseparability that are not captured by a
naive application of the standard PM formalism: we exhibit an instance of an
operator that is both a valid PM and a valid MPM, but is causally separable in
the first case and can violate causal inequalities in the second case due to
the possibility of using a side channel.Comment: 24 pages with 6 figures, various improvements and corrections,
accepted in Quantu
Comment on "Nongeometric Conditional Phase Shift via Adiabatic Evolution of Dark Eigenstates: A New Approach to Quantum Computation"
In [Phys. Rev. Lett. 95, 080502 (2005)], Zheng proposed a scheme for
implementing a conditional phase shift via adiabatic passages. The author
claims that the gate is "neither of dynamical nor geometric origin" on the
grounds that the Hamiltonian does not follow a cyclic change. He further argues
that "in comparison with the adiabatic geometric gates, the nontrivial cyclic
loop is unnecessary, and thus the errors in obtaining the required solid angle
are avoided, which makes this new kind of phase gates superior to the geometric
gates." In this Comment, we point out that geometric operations, including
adiabatic holonomies, can be induced by noncyclic Hamiltonians, and show that
Zheng's gate is geometric. We also argue that the nontrivial loop responsible
for the phase shift is there, and it requires the same precision as in any
adiabatic geometric gate
Operational quantum theory without predefined time
The standard formulation of quantum theory assumes a predefined notion of
time. This is a major obstacle in the search for a quantum theory of gravity,
where the causal structure of space-time is expected to be dynamical and
fundamentally probabilistic in character. Here, we propose a generalized
formulation of quantum theory without predefined time or causal structure,
building upon a recently introduced operationally time-symmetric approach to
quantum theory. The key idea is a novel isomorphism between transformations and
states which depends on the symmetry transformation of time reversal. This
allows us to express the time-symmetric formulation in a time-neutral form with
a clear physical interpretation, and ultimately drop the assumption of time. In
the resultant generalized formulation, operations are associated with regions
that can be connected in networks with no directionality assumed for the
connections, generalizing the standard circuit framework and the process matrix
framework for operations without global causal order. The possible events in a
given region are described by positive semidefinite operators on a Hilbert
space at the boundary, while the connections between regions are described by
entangled states that encode a nontrivial symmetry and could be tested in
principle. We discuss how the causal structure of space-time could be
understood as emergent from properties of the operators on the boundaries of
compact space-time regions. The framework is compatible with indefinite causal
order, timelike loops, and other acausal structures.Comment: 15 pages, 7 figures, published version (this version covers the
second half of the original submission; the first half has been published
separately and is available at arXiv:1507.07745
Operator quantum error correction for continuous dynamics
We study the conditions under which a subsystem code is correctable in the
presence of noise that results from continuous dynamics. We consider the case
of Markovian dynamics as well as the general case of Hamiltonian dynamics of
the system and the environment, and derive necessary and sufficient conditions
on the Lindbladian and system-environment Hamiltonian, respectively. For the
case when the encoded information is correctable during an entire time
interval, the conditions we obtain can be thought of as generalizations of the
previously derived conditions for decoherence-free subsystems to the case where
the subsystem is time dependent. As a special case, we consider conditions for
unitary correctability. In the case of Hamiltonian evolution, the conditions
for unitary correctability concern only the effect of the Hamiltonian on the
system, whereas the conditions for general correctability concern the entire
system-environment Hamiltonian. We also derive conditions on the Hamiltonian
which depend on the initial state of the environment, as well as conditions for
correctability at only a particular moment of time. We discuss possible
implications of our results for approximate quantum error correction.Comment: 11 pages, no figures, essentially the published version, includes a
new section on correctability at only a particular moment of tim
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