850 research outputs found
A monomial matrix formalism to describe quantum many-body states
We propose a framework to describe and simulate a class of many-body quantum
states. We do so by considering joint eigenspaces of sets of monomial unitary
matrices, called here "M-spaces"; a unitary matrix is monomial if precisely one
entry per row and column is nonzero. We show that M-spaces encompass various
important state families, such as all Pauli stabilizer states and codes, the
AKLT model, Kitaev's (abelian and non-abelian) anyon models, group coset
states, W states and the locally maximally entanglable states. We furthermore
show how basic properties of M-spaces can transparently be understood by
manipulating their monomial stabilizer groups. In particular we derive a
unified procedure to construct an eigenbasis of any M-space, yielding an
explicit formula for each of the eigenstates. We also discuss the computational
complexity of M-spaces and show that basic problems, such as estimating local
expectation values, are NP-hard. Finally we prove that a large subclass of
M-spaces---containing in particular most of the aforementioned examples---can
be simulated efficiently classically with a unified method.Comment: 11 pages + appendice
Entanglement growth in quench dynamics with variable range interactions
Studying entanglement growth in quantum dynamics provides both insight into
the underlying microscopic processes and information about the complexity of
the quantum states, which is related to the efficiency of simulations on
classical computers. Recently, experiments with trapped ions, polar molecules,
and Rydberg excitations have provided new opportunities to observe dynamics
with long-range interactions. We explore nonequilibrium coherent dynamics after
a quantum quench in such systems, identifying qualitatively different behavior
as the exponent of algebraically decaying spin-spin interactions in a
transverse Ising chain is varied. Computing the build-up of bipartite
entanglement as well as mutual information between distant spins, we identify
linear growth of entanglement entropy corresponding to propagation of
quasiparticles for shorter range interactions, with the maximum rate of growth
occurring when the Hamiltonian parameters match those for the quantum phase
transition. Counter-intuitively, the growth of bipartite entanglement for
long-range interactions is only logarithmic for most regimes, i.e.,
substantially slower than for shorter range interactions. Experiments with
trapped ions allow for the realization of this system with a tunable
interaction range, and we show that the different phenomena are robust for
finite system sizes and in the presence of noise. These results can act as a
direct guide for the generation of large-scale entanglement in such
experiments, towards a regime where the entanglement growth can render existing
classical simulations inefficient.Comment: 17 pages, 7 figure
General fixed points of quasi-local frustration-free quantum semigroups: from invariance to stabilization
We investigate under which conditions a mixed state on a finite-dimensional
multipartite quantum system may be the unique, globally stable fixed point of
frustration-free semigroup dynamics subject to specified quasi-locality
constraints. Our central result is a linear-algebraic necessary and sufficient
condition for a generic (full-rank) target state to be frustration-free
quasi-locally stabilizable, along with an explicit procedure for constructing
Markovian dynamics that achieve stabilization. If the target state is not
full-rank, we establish sufficiency under an additional condition, which is
naturally motivated by consistency with pure-state stabilization results yet
provably not necessary in general. Several applications are discussed, of
relevance to both dissipative quantum engineering and information processing,
and non-equilibrium quantum statistical mechanics. In particular, we show that
a large class of graph product states (including arbitrary thermal graph
states) as well as Gibbs states of commuting Hamiltonians are frustration-free
stabilizable relative to natural quasi-locality constraints. Likewise, we
provide explicit examples of non-commuting Gibbs states and non-trivially
entangled mixed states that are stabilizable despite the lack of an underlying
commuting structure, albeit scalability to arbitrary system size remains in
this case an open question.Comment: 44 pages, main results are improved, several proofs are more
streamlined, application section is refine
Generalized parity measurements
Measurements play an important role in quantum computing (QC), by either
providing the nonlinearity required for two-qubit gates (linear optics QC), or
by implementing a quantum algorithm using single-qubit measurements on a highly
entangled initial state (cluster state QC). Parity measurements can be used as
building blocks for preparing arbitrary stabilizer states, and, together with
1-qubit gates are universal for quantum computing. Here we generalize parity
gates by using a higher dimensional (qudit) ancilla. This enables us to go
beyond the stabilizer/graph state formalism and prepare other types of
multi-particle entangled states. The generalized parity module introduced here
can prepare in one-shot, heralded by the outcome of the ancilla, a large class
of entangled states, including GHZ_n, W_n, Dicke states D_{n,k}, and, more
generally, certain sums of Dicke states, like G_n states used in secret
sharing. For W_n states it provides an exponential gain compared to linear
optics based methods.Comment: 7 pages, 1 fig; updated to the published versio
A proposal for implementing an n-qubit controlled-rotation gate with three-level superconducting qubit systems in cavity QED
We present a way for implementing an n-qubit controlled-rotation gate with
three-level superconducting qubit systems in cavity QED. The two logical states
of a qubit are represented by the two lowest levels of each system while a
higher-energy level is used for the gate implementation. The method operates
essentially by preparing a state conditioned on the states of the control
qubits, creating a single photon in the cavity mode, and then performing an
arbitrary rotation on the states of the target qubit with assistance of the
cavity photon. It is interesting to note that the basic operational steps for
implementing the proposed gate do not increase with the number of qubits,
and the gate operation time decreases as the number of qubits increases. This
proposal is quite general, which can be applied to various types of
superconducting devices in a cavity or coupled to a resonator.Comment: Six figures, accepted by Journal of Physics: Condensed Matte
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