29 research outputs found
Quantum Commuting Circuits and Complexity of Ising Partition Functions
Instantaneous quantum polynomial-time (IQP) computation is a class of quantum
computation consisting only of commuting two-qubit gates and is not universal
in the sense of standard quantum computation. Nevertheless, it has been shown
that if there is a classical algorithm that can simulate IQP efficiently, the
polynomial hierarchy (PH) collapses at the third level, which is highly
implausible. However, the origin of the classical intractability is still less
understood. Here we establish a relationship between IQP and computational
complexity of the partition functions of Ising models. We apply the established
relationship in two opposite directions. One direction is to find subclasses of
IQP that are classically efficiently simulatable in the strong sense, by using
exact solvability of certain types of Ising models. Another direction is
applying quantum computational complexity of IQP to investigate (im)possibility
of efficient classical approximations of Ising models with imaginary coupling
constants. Specifically, we show that there is no fully polynomial randomized
approximation scheme (FPRAS) for Ising models with almost all imaginary
coupling constants even on a planar graph of a bounded degree, unless the PH
collapses at the third level. Furthermore, we also show a multiplicative
approximation of such a class of Ising partition functions is at least as hard
as a multiplicative approximation for the output distribution of an arbitrary
quantum circuit.Comment: 36 pages, 5 figure
Stabilizer notation for Spekkens' toy theory
Spekkens has introduced a toy theory [Phys. Rev. A, 75, 032110 (2007)] in
order to argue for an epistemic view of quantum states. I describe a notation
for the theory (excluding certain joint measurements) which makes its
similarities and differences with the quantum mechanics of stabilizer states
clear. Given an application of the qubit stabilizer formalism, it is often
entirely straightforward to construct an analogous application of the notation
to the toy theory. This assists calculations within the toy theory, for example
of the number of possible states and transformations, and enables
superpositions to be defined for composite systems.Comment: 7+4 pages, 5 tables. v2: Clarifications added and typos fixed in
response to referee comment
Approximating Fractional Time Quantum Evolution
An algorithm is presented for approximating arbitrary powers of a black box
unitary operation, , where is a real number, and
is a black box implementing an unknown unitary. The complexity of
this algorithm is calculated in terms of the number of calls to the black box,
the errors in the approximation, and a certain `gap' parameter. For general
and large , one should apply a total of times followed by our procedure for approximating the fractional
power . An example is also given where for
large integers this method is more efficient than direct application of
copies of . Further applications and related algorithms are also
discussed.Comment: 13 pages, 2 figure
Accrediting outputs of noisy intermediate-scale quantum computing devices
We present an accreditation protocol for the outputs of noisy
intermediate-scale quantum devices. By testing entire circuits rather than
individual gates, our accreditation protocol can provide an upper-bound on the
variation distance between noisy and noiseless probability distribution of the
outputs of the target circuit of interest. Our accreditation protocol requires
implementation of quantum circuits no larger than the target circuit, therefore
it is practical in the near term and scalable in the long term. Inspired by
trap-based protocols for the verification of quantum computations, our
accreditation protocol assumes that noise in single-qubit gates is bounded (but
potentially gate-dependent) in diamond norm. We allow for arbitrary spatial and
temporal correlations in the noise affecting state preparation, measurements
and two-qubit gates. We describe how to implement our protocol on real-world
devices, and we also present a novel cryptographic protocol (which we call
`mesothetic' protocol) inspired by our accreditation protocol.Comment: Accepted versio
Mapping all classical spin models to a lattice gauge theory
In our recent work [Phys. Rev. Lett. 102, 230502 (2009)] we showed that the
partition function of all classical spin models, including all discrete
standard statistical models and all Abelian discrete lattice gauge theories
(LGTs), can be expressed as a special instance of the partition function of a
4-dimensional pure LGT with gauge group Z_2 (4D Z_2 LGT). This provides a
unification of models with apparently very different features into a single
complete model. The result uses an equality between the Hamilton function of
any classical spin model and the Hamilton function of a model with all possible
k-body Ising-type interactions, for all k, which we also prove. Here, we
elaborate on the proof of the result, and we illustrate it by computing
quantities of a specific model as a function of the partition function of the
4D Z_2 LGT. The result also allows one to establish a new method to compute the
mean-field theory of Z_2 LGTs with d > 3, and to show that computing the
partition function of the 4D Z_2 LGT is computationally hard (#P hard). The
proof uses techniques from quantum information.Comment: 21 pages, 21 figures; published versio
Photonic quantum information processing: a review
Photonic quantum technologies represent a promising platform for several
applications, ranging from long-distance communications to the simulation of
complex phenomena. Indeed, the advantages offered by single photons do make
them the candidate of choice for carrying quantum information in a broad
variety of areas with a versatile approach. Furthermore, recent technological
advances are now enabling first concrete applications of photonic quantum
information processing. The goal of this manuscript is to provide the reader
with a comprehensive review of the state of the art in this active field, with
a due balance between theoretical, experimental and technological results. When
more convenient, we will present significant achievements in tables or in
schematic figures, in order to convey a global perspective of the several
horizons that fall under the name of photonic quantum information.Comment: 36 pages, 6 figures, 634 references. Updated version with minor
changes and extended bibliograph