298 research outputs found
On the Complexity of Random Quantum Computations and the Jones Polynomial
There is a natural relationship between Jones polynomials and quantum
computation. We use this relationship to show that the complexity of evaluating
relative-error approximations of Jones polynomials can be used to bound the
classical complexity of approximately simulating random quantum computations.
We prove that random quantum computations cannot be classically simulated up to
a constant total variation distance, under the assumption that (1) the
Polynomial Hierarchy does not collapse and (2) the average-case complexity of
relative-error approximations of the Jones polynomial matches the worst-case
complexity over a constant fraction of random links. Our results provide a
straightforward relationship between the approximation of Jones polynomials and
the complexity of random quantum computations.Comment: 8 pages, 4 figure
Simulating Hamiltonian dynamics using many-qudit Hamiltonians and local unitary control
When can a quantum system of finite dimension be used to simulate another
quantum system of finite dimension? What restricts the capacity of one system
to simulate another? In this paper we complete the program of studying what
simulations can be done with entangling many-qudit Hamiltonians and local
unitary control. By entangling we mean that every qudit is coupled to every
other qudit, at least indirectly. We demonstrate that the only class of
finite-dimensional entangling Hamiltonians that aren't universal for simulation
is the class of entangling Hamiltonians on qubits whose Pauli operator
expansion contains only terms coupling an odd number of systems, as identified
by Bremner et. al. [Phys. Rev. A, 69, 012313 (2004)]. We show that in all other
cases entangling many-qudit Hamiltonians are universal for simulation
Quantum Sampling Problems, BosonSampling and Quantum Supremacy
There is a large body of evidence for the potential of greater computational
power using information carriers that are quantum mechanical over those
governed by the laws of classical mechanics. But the question of the exact
nature of the power contributed by quantum mechanics remains only partially
answered. Furthermore, there exists doubt over the practicality of achieving a
large enough quantum computation that definitively demonstrates quantum
supremacy. Recently the study of computational problems that produce samples
from probability distributions has added to both our understanding of the power
of quantum algorithms and lowered the requirements for demonstration of fast
quantum algorithms. The proposed quantum sampling problems do not require a
quantum computer capable of universal operations and also permit physically
realistic errors in their operation. This is an encouraging step towards an
experimental demonstration of quantum algorithmic supremacy. In this paper, we
will review sampling problems and the arguments that have been used to deduce
when sampling problems are hard for classical computers to simulate. Two
classes of quantum sampling problems that demonstrate the supremacy of quantum
algorithms are BosonSampling and IQP Sampling. We will present the details of
these classes and recent experimental progress towards demonstrating quantum
supremacy in BosonSampling.Comment: Survey paper first submitted for publication in October 2016. 10
pages, 4 figures, 1 tabl
Instantaneous Quantum Computation
We examine theoretic architectures and an abstract model for a restricted
class of quantum computation, called here instantaneous quantum computation
because it allows for essentially no temporal structure within the quantum
dynamics. Using the theory of binary matroids, we argue that the paradigm is
rich enough to enable sampling from probability distributions that cannot,
classically, be sampled from efficiently and accurately. This paradigm also
admits simple interactive proof games that may convince a skeptic of the
existence of truly quantum effects. Furthermore, these effects can be created
using significantly fewer qubits than are required for running Shor's
Algorithm.Comment: Significantly rewritten for clarity, more explanation adde
Approximation Algorithms for Complex-Valued Ising Models on Bounded Degree Graphs
We study the problem of approximating the Ising model partition function with
complex parameters on bounded degree graphs. We establish a deterministic
polynomial-time approximation scheme for the partition function when the
interactions and external fields are absolutely bounded close to zero.
Furthermore, we prove that for this class of Ising models the partition
function does not vanish. Our algorithm is based on an approach due to Barvinok
for approximating evaluations of a polynomial based on the location of the
complex zeros and a technique due to Patel and Regts for efficiently computing
the leading coefficients of graph polynomials on bounded degree graphs.
Finally, we show how our algorithm can be extended to approximate certain
output probability amplitudes of quantum circuits.Comment: 12 pages, 0 figures, published versio
Achieving quantum supremacy with sparse and noisy commuting quantum computations
The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O(sqrt(n) log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP circuit whose output probability distribution is sufficiently anticoncentrated, there is a polynomial-time classical algorithm that simulates sampling from the resulting distribution, up to constant accuracy in total variation distance. However, we show that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form. These results demonstrate the challenges faced by experiments designed to demonstrate quantum supremacy over classical computation, and how these challenges can be overcome
IQP Sampling and Verifiable Quantum Advantage: Stabilizer Scheme and Classical Security
Sampling problems demonstrating beyond classical computing power with noisy
intermediate-scale quantum (NISQ) devices have been experimentally realized. In
those realizations, however, our trust that the quantum devices faithfully
solve the claimed sampling problems is usually limited to simulations of
smaller-scale instances and is, therefore, indirect. The problem of verifiable
quantum advantage aims to resolve this critical issue and provides us with
greater confidence in a claimed advantage. Instantaneous quantum
polynomial-time (IQP) sampling has been proposed to achieve beyond classical
capabilities with a verifiable scheme based on quadratic-residue codes (QRC).
Unfortunately, this verification scheme was recently broken by an attack
proposed by Kahanamoku-Meyer. In this work, we revive IQP-based verifiable
quantum advantage by making two major contributions. Firstly, we introduce a
family of IQP sampling protocols called the \emph{stabilizer scheme}, which
builds on results linking IQP circuits, the stabilizer formalism, coding
theory, and an efficient characterization of IQP circuit correlation functions.
This construction extends the scope of existing IQP-based schemes while
maintaining their simplicity and verifiability. Secondly, we introduce the
\emph{Hidden Structured Code} (HSC) problem as a well-defined mathematical
challenge that underlies the stabilizer scheme. To assess classical security,
we explore a class of attacks based on secret extraction, including the
Kahanamoku-Meyer's attack as a special case. We provide evidence of the
security of the stabilizer scheme, assuming the hardness of the HSC problem. We
also point out that the vulnerability observed in the original QRC scheme is
primarily attributed to inappropriate parameter choices, which can be naturally
rectified with proper parameter settings.Comment: 22 pages, 3 figure
Interpersonal representations of touch in somatosensory cortex are modulated by perspective
Observing others being touched activates similar brain areas as those activated when one experiences a touch oneself. Event-related potential (ERP) studies have revealed that modulation of somatosensory components by observed touch occurs within 100 ms after stimulus onset, and such vicarious effects have been taken as evidence for empathy for others' tactile experiences. In previous studies body parts have been presented from a first person perspective. This raises the question of the extent to which somatosensory activation by observed touch to body parts depends on the perspective from which the body part is observed. In this study (N = 18), we examined the modulation of somatosensory ERPs by observed touch delivered to another person's hand when viewed as if from a first person versus a third person perspective. We found that vicarious touch effects primarily consist of two separable components in the early stages of somatosensory processing: an anatomical mapping for touch in first person perspective at P45, and a specular (mirror like) mapping for touch in third person perspective at P100. This is consistent with suggestions that vicarious representations exist to support predictions for one's own bodily events, but also to enable predictions of a social or interpersonal kind, at distinct temporal stages
A practical scheme for quantum computation with any two-qubit entangling gate
Which gates are universal for quantum computation? Although it is well known
that certain gates on two-level quantum systems (qubits), such as the
controlled-not (CNOT), are universal when assisted by arbitrary one-qubit
gates, it has only recently become clear precisely what class of two-qubit
gates is universal in this sense. Here we present an elementary proof that any
entangling two-qubit gate is universal for quantum computation, when assisted
by one-qubit gates. A proof of this important result for systems of arbitrary
finite dimension has been provided by J. L. and R. Brylinski
[arXiv:quant-ph/0108062, 2001]; however, their proof relies upon a long
argument using advanced mathematics. In contrast, our proof provides a simple
constructive procedure which is close to optimal and experimentally practical
[C. M. Dawson and A. Gilchrist, online implementation of the procedure
described herein (2002), http://www.physics.uq.edu.au/gqc/].Comment: 3 pages, online implementation of procedure described can be found at
http://www.physics.uq.edu.au/gqc
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