2,256 research outputs found
Quantum speedup of Monte Carlo methods
Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomised or quantum subrou-tine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algo-rithm can also be used to estimate the total variation distance between probability distributions efficiently.
Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem
We study the approximation of the smallest eigenvalue of a Sturm-Liouville
problem in the classical and quantum settings. We consider a univariate
Sturm-Liouville eigenvalue problem with a nonnegative function from the
class and study the minimal number n(\e) of function evaluations
or queries that are necessary to compute an \e-approximation of the smallest
eigenvalue. We prove that n(\e)=\Theta(\e^{-1/2}) in the (deterministic)
worst case setting, and n(\e)=\Theta(\e^{-2/5}) in the randomized setting.
The quantum setting offers a polynomial speedup with {\it bit} queries and an
exponential speedup with {\it power} queries. Bit queries are similar to the
oracle calls used in Grover's algorithm appropriately extended to real valued
functions. Power queries are used for a number of problems including phase
estimation. They are obtained by considering the propagator of the discretized
system at a number of different time moments. They allow us to use powers of
the unitary matrix , where is an
matrix obtained from the standard discretization of the Sturm-Liouville
differential operator. The quantum implementation of power queries by a number
of elementary quantum gates that is polylog in is an open issue.Comment: 33 page
Quantum SDP-Solvers: Better upper and lower bounds
Brand\~ao and Svore very recently gave quantum algorithms for approximately
solving semidefinite programs, which in some regimes are faster than the
best-possible classical algorithms in terms of the dimension of the problem
and the number of constraints, but worse in terms of various other
parameters. In this paper we improve their algorithms in several ways, getting
better dependence on those other parameters. To this end we develop new
techniques for quantum algorithms, for instance a general way to efficiently
implement smooth functions of sparse Hamiltonians, and a generalized
minimum-finding procedure.
We also show limits on this approach to quantum SDP-solvers, for instance for
combinatorial optimizations problems that have a lot of symmetry. Finally, we
prove some general lower bounds showing that in the worst case, the complexity
of every quantum LP-solver (and hence also SDP-solver) has to scale linearly
with when , which is the same as classical.Comment: v4: 69 pages, small corrections and clarifications. This version will
appear in Quantu
Quantum Chebyshev's Inequality and Applications
In this paper we provide new quantum algorithms with polynomial speed-up for
a range of problems for which no such results were known, or we improve
previous algorithms. First, we consider the approximation of the frequency
moments of order in the multi-pass streaming model with
updates (turnstile model). We design a -pass quantum streaming algorithm
with memory satisfying a tradeoff of ,
whereas the best classical algorithm requires . Then,
we study the problem of estimating the number of edges and the number
of triangles given query access to an -vertex graph. We describe optimal
quantum algorithms that perform and
queries respectively. This is
a quadratic speed-up compared to the classical complexity of these problems.
For this purpose we develop a new quantum paradigm that we call Quantum
Chebyshev's inequality. Namely we demonstrate that, in a certain model of
quantum sampling, one can approximate with relative error the mean of any
random variable with a number of quantum samples that is linear in the ratio of
the square root of the variance to the mean. Classically the dependency is
quadratic. Our algorithm subsumes a previous result of Montanaro [Mon15]. This
new paradigm is based on a refinement of the Amplitude Estimation algorithm of
Brassard et al. [BHMT02] and of previous quantum algorithms for the mean
estimation problem. We show that this speed-up is optimal, and we identify
another common model of quantum sampling where it cannot be obtained. For our
applications, we also adapt the variable-time amplitude amplification technique
of Ambainis [Amb10] into a variable-time amplitude estimation algorithm.Comment: 27 pages; v3: better presentation, lower bound in Theorem 4.3 is ne
Assessing, testing, and challenging the computational power of quantum devices
Randomness is an intrinsic feature of quantum theory. The outcome of any measurement will be random, sampled from a probability distribution that is defined by the measured quantum state. The task of sampling from a prescribed probability distribution therefore seems to be a natural technological application of quantum devices. And indeed, certain random sampling tasks have been proposed to experimentally demonstrate the speedup of quantum over classical computation, so-called “quantum computational supremacy”.
In the research presented in this thesis, I investigate the complexity-theoretic and physical foundations of quantum sampling algorithms. Using the theory of computational complexity, I assess the computational power of natural quantum simulators and close loopholes in the complexity-theoretic argument for the classical intractability of quantum samplers (Part I). In particular, I prove anticoncentration for quantum circuit families that give rise to a 2-design and review methods for proving average-case hardness. I present quantum random sampling schemes that are tailored to large-scale quantum simulation hardware but at the same time rise up to the highest standard in terms of their complexity-theoretic underpinning. Using methods from property testing and quantum system identification, I shed light on the question, how and under which conditions quantum sampling devices can be tested or verified in regimes that are not simulable on classical computers (Part II). I present a no-go result that prevents efficient verification of quantum random sampling schemes as well as approaches using which this no-go result can be circumvented. In particular, I develop fully efficient verification protocols in what I call the measurement-device-dependent scenario in which single-qubit measurements are assumed to function with high accuracy. Finally, I try to understand the physical mechanisms governing the computational boundary between classical and quantum computing devices by challenging their computational power using tools from computational physics and the theory of computational complexity (Part III). I develop efficiently computable measures of the infamous Monte Carlo sign problem and assess those measures both in terms of their practicability as a tool for alleviating or easing the sign problem and the computational complexity of this task.
An overarching theme of the thesis is the quantum sign problem which arises due to destructive interference between paths – an intrinsically quantum effect. The (non-)existence of a sign problem takes on the role as a criterion which delineates the boundary between classical and quantum computing devices. I begin the thesis by identifying the quantum sign problem as a root of the computational intractability of quantum output probabilities. It turns out that the intricate structure of the probability distributions the sign problem gives rise to, prohibits their verification from few samples. In an ironic twist, I show that assessing the intrinsic sign problem of a quantum system is again an intractable problem
Quantum Amplitude Amplification and Estimation
Consider a Boolean function that partitions set
between its good and bad elements, where is good if and bad
otherwise. Consider also a quantum algorithm such that is a quantum superposition of the
elements of , and let denote the probability that a good element is
produced if is measured. If we repeat the process of running ,
measuring the output, and using to check the validity of the result, we
shall expect to repeat times on the average before a solution is found.
*Amplitude amplification* is a process that allows to find a good after an
expected number of applications of and its inverse which is proportional to
, assuming algorithm makes no measurements. This is a
generalization of Grover's searching algorithm in which was restricted to
producing an equal superposition of all members of and we had a promise
that a single existed such that . Our algorithm works whether or
not the value of is known ahead of time. In case the value of is known,
we can find a good after a number of applications of and its inverse
which is proportional to even in the worst case. We show that this
quadratic speedup can also be obtained for a large family of search problems
for which good classical heuristics exist. Finally, as our main result, we
combine ideas from Grover's and Shor's quantum algorithms to perform amplitude
estimation, a process that allows to estimate the value of . We apply
amplitude estimation to the problem of *approximate counting*, in which we wish
to estimate the number of such that . We obtain optimal
quantum algorithms in a variety of settings.Comment: 32 pages, no figure
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