30 research outputs found
Universal quantum computation with little entanglement
We show that universal quantum computation can be achieved in the standard
pure-state circuit model while, at any time, the entanglement entropy of all
bipartitions is small---even tending to zero with growing system size. The
result is obtained by showing that a quantum computer operating within a small
region around the set of unentangled states still has universal computational
power, and by using continuity of entanglement entropy. In fact an analogous
conclusion applies to every entanglement measure which is continuous in a
certain natural sense, which amounts to a large class. Other examples include
the geometric measure, localizable entanglement, smooth epsilon-measures,
multipartite concurrence, squashed entanglement, and several others. We discuss
implications of these results for the believed role of entanglement as a key
necessary resource for quantum speed-ups
Classical simulation complexity of extended Clifford circuits
Clifford gates are a winsome class of quantum operations combining
mathematical elegance with physical significance. The Gottesman-Knill theorem
asserts that Clifford computations can be classically efficiently simulated but
this is true only in a suitably restricted setting. Here we consider Clifford
computations with a variety of additional ingredients: (a) strong vs. weak
simulation, (b) inputs being computational basis states vs. general product
states, (c) adaptive vs. non-adaptive choices of gates for circuits involving
intermediate measurements, (d) single line outputs vs. multi-line outputs. We
consider the classical simulation complexity of all combinations of these
ingredients and show that many are not classically efficiently simulatable
(subject to common complexity assumptions such as P not equal to NP). Our
results reveal a surprising proximity of classical to quantum computing power
viz. a class of classically simulatable quantum circuits which yields universal
quantum computation if extended by a purely classical additional ingredient
that does not extend the class of quantum processes occurring.Comment: 17 pages, 1 figur
Classical simulations of Abelian-group normalizer circuits with intermediate measurements
Quantum normalizer circuits were recently introduced as generalizations of
Clifford circuits [arXiv:1201.4867]: a normalizer circuit over a finite Abelian
group is composed of the quantum Fourier transform (QFT) over G, together
with gates which compute quadratic functions and automorphisms. In
[arXiv:1201.4867] it was shown that every normalizer circuit can be simulated
efficiently classically. This result provides a nontrivial example of a family
of quantum circuits that cannot yield exponential speed-ups in spite of usage
of the QFT, the latter being a central quantum algorithmic primitive. Here we
extend the aforementioned result in several ways. Most importantly, we show
that normalizer circuits supplemented with intermediate measurements can also
be simulated efficiently classically, even when the computation proceeds
adaptively. This yields a generalization of the Gottesman-Knill theorem (valid
for n-qubit Clifford operations [quant-ph/9705052, quant-ph/9807006] to quantum
circuits described by arbitrary finite Abelian groups. Moreover, our
simulations are twofold: we present efficient classical algorithms to sample
the measurement probability distribution of any adaptive-normalizer
computation, as well as to compute the amplitudes of the state vector in every
step of it. Finally we develop a generalization of the stabilizer formalism
[quant-ph/9705052, quant-ph/9807006] relative to arbitrary finite Abelian
groups: for example we characterize how to update stabilizers under generalized
Pauli measurements and provide a normal form of the amplitudes of generalized
stabilizer states using quadratic functions and subgroup cosets.Comment: 26 pages+appendices. Title has changed in this second version. To
appear in Quantum Information and Computation, Vol.14 No.3&4, 201
Certifiability criterion for large-scale quantum systems
Can one certify the preparation of a coherent, many-body quantum state by
measurements with bounded accuracy in the presence of noise and decoherence?
Here, we introduce a criterion to assess the fragility of large-scale quantum
states which is based on the distinguishability of orthogonal states after the
action of very small amounts of noise. States which do not pass this criterion
are called asymptotically incertifiable. We show that, if a coherent quantum
state is asymptotically incertifiable, there exists an incoherent mixture (with
entropy at least log 2) which is experimentally indistinguishable from the
initial state. The Greenberger-Horne-Zeilinger states are examples of such
asymptotically incertifiable states. More generally, we prove that any
so-called macroscopic superposition state is asymptotically incertifiable. We
also provide examples of quantum states that are experimentally
indistinguishable from highly incoherent mixtures, i.e., with an almost-linear
entropy in the number of qubits. Finally we show that all unique ground states
of local gapped Hamiltonians (in any dimension) are certifiable.Comment: 21 pages, 1 figure; V2: title changed plus minor changes in the text
and some additional reference
A Non-Commuting Stabilizer Formalism
We propose a non-commutative extension of the Pauli stabilizer formalism. The
aim is to describe a class of many-body quantum states which is richer than the
standard Pauli stabilizer states. In our framework, stabilizer operators are
tensor products of single-qubit operators drawn from the group , where and . We
provide techniques to efficiently compute various properties related to
bipartite entanglement, expectation values of local observables, preparation by
means of quantum circuits, parent Hamiltonians etc. We also highlight
significant differences compared to the Pauli stabilizer formalism. In
particular, we give examples of states in our formalism which cannot arise in
the Pauli stabilizer formalism, such as topological models that support
non-Abelian anyons.Comment: 52 page
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
Commuting quantum circuits: efficient classical simulations versus hardness results
The study of quantum circuits composed of commuting gates is particularly
useful to understand the delicate boundary between quantum and classical
computation. Indeed, while being a restricted class, commuting circuits exhibit
genuine quantum effects such as entanglement. In this paper we show that the
computational power of commuting circuits exhibits a surprisingly rich
structure. First we show that every 2-local commuting circuit acting on d-level
systems and followed by single-qudit measurements can be efficiently simulated
classically with high accuracy. In contrast, we prove that such strong
simulations are hard for 3-local circuits. Using sampling methods we further
show that all commuting circuits composed of exponentiated Pauli operators
e^{i\theta P} can be simulated efficiently classically when followed by
single-qubit measurements. Finally, we show that commuting circuits can
efficiently simulate certain non-commutative processes, related in particular
to constant-depth quantum circuits. This gives evidence that the power of
commuting circuits goes beyond classical computation.Comment: 19 page