41 research outputs found
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
Analyzing Prospects for Quantum Advantage in Topological Data Analysis
Lloyd et al. were first to demonstrate the promise of quantum algorithms for
computing Betti numbers, a way to characterize topological features of data
sets. Here, we propose, analyze, and optimize an improved quantum algorithm for
topological data analysis (TDA) with reduced scaling, including a method for
preparing Dicke states based on inequality testing, a more efficient amplitude
estimation algorithm using Kaiser windows, and an optimal implementation of
eigenvalue projectors based on Chebyshev polynomials. We compile our approach
to a fault-tolerant gate set and estimate constant factors in the Toffoli
complexity. Our analysis reveals that super-quadratic quantum speedups are only
possible for this problem when targeting a multiplicative error approximation
and the Betti number grows asymptotically. Further, we propose a dequantization
of the quantum TDA algorithm that shows that having exponentially large
dimension and Betti number are necessary, but insufficient conditions, for
super-polynomial advantage. We then introduce and analyze specific problem
examples which have parameters in the regime where super-polynomial advantages
may be achieved, and argue that quantum circuits with tens of billions of
Toffoli gates can solve seemingly classically intractable instances.Comment: 54 pages, 7 figures. Added a number of theorems and lemmas to clarify
findings and also a discussion in the main text and new appendix about
variants of our problems with high Betti numbers that are challenging for
recent classical algorithm
Quantum computation beyond the circuit model
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2008.Includes bibliographical references (p. 133-144).The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of quantum computers. However, several other models of quantum computation exist which provide useful alternative frameworks for both discovering new quantum algorithms and devising new physical implementations of quantum computers. In this thesis, I first present necessary background material for a general physics audience and discuss existing models of quantum computation. Then, I present three new results relating to various models of quantum computation: a scheme for improving the intrinsic fault tolerance of adiabatic quantum computers using quantum error detecting codes, a proof that a certain problem of estimating Jones polynomials is complete for the one clean qubit complexity class, and a generalization of perturbative gadgets which allows k-body interactions to be directly simulated using 2-body interactions. Lastly, I discuss general principles regarding quantum computation that I learned in the course of my research, and using these principles I propose directions for future research.by Stephen Paul Jordan.Ph.D
Many-body quantum magic
Magic (non-stabilizerness) is a necessary but "expensive" kind of "fuel" to
drive universal fault-tolerant quantum computation. To properly study and
characterize the origin of quantum "complexity" in computation as well as
physics, it is crucial to develop a rigorous understanding of the
quantification of magic. Previous studies of magic mostly focused on small
systems and largely relied on the discrete Wigner formalism (which is only well
behaved in odd prime power dimensions). Here we present an initiatory study of
the magic of genuinely many-body quantum states (with focus on the important
case of many qubits) at a quantitative level. We first address the basic
question of how magical a many-body state can be, and show that the maximum and
typical magic of an -qubit state is essentially , simultaneously for a
range of natural resource measures. As a corollary, we show that the resource
theory of magic with stabilizer-preserving free operations is asymptotically
reversible. In the quest for explicit, scalable cases of highly entangled
states whose magic can be understood, we connect the magic of hypergraph states
with the second-order nonlinearity of their underlying Boolean functions. Next,
we go on and investigate many-body magic in practical and physical contexts. We
first consider Pauli measurement-based quantum computation, in which magic is a
necessary feature of the initial resource state. We show that -qubit states
with nearly magic, or indeed almost all states, cannot supply nontrivial
speedups over classical computers. We then present an example of analyzing the
magic of "natural" condensed matter systems. We apply the Boolean function
techniques to derive explicit bounds on the magic of the ground states of
certain 2D symmetry-protected topological (SPT) phases, and comment on possible
further connections between magic and the quantum complexity of matter.Comment: 15 pages, 3 figure