3,628 research outputs found
Lower Bounds on Quantum Query and Learning Graph Complexities
In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models.
First, we study lower bounds on learning graph complexity. We consider two types of learning graphs: adaptive and, more restricted, non-adaptive learning graphs. We express both adaptive and non-adaptive learning graph complexities of Boolean-valued functions (i.e., decision problems) as semidefinite minimization problems, and derive their dual problems. For various functions, we construct feasible solutions to these dual problems, thereby obtaining lower bounds on the learning graph complexity of the functions. Most notably, we prove an almost optimal Omega(n^(9/7)/sqrt(log n)) lower bound on the non-adaptive learning graph complexity of the Triangle problem. We also prove an Omega(n^(1-2^(k-2)/(2^k-1))) lower bound on the adaptive learning graph complexity of the k-Distinctness problem, which matches the complexity of the best known quantum query algorithm for this problem.
Second, we construct optimal adversary lower bounds for various decision problems. Our main procedure for constructing them is to embed the adversary matrix into a larger matrix whose properties are easier to analyze. This embedding procedure imposes certain requirements on the size of the input alphabet. We prove optimal Omega(n^(1/3)) adversary lower bounds for the Collision and Set Equality problems, provided that the alphabet size is at least Omega(n^2). An optimal lower bound for Collision was previously proven using the polynomial method, while our lower bound for Set Equality is new. (An optimal lower bound for Set Equality was also independently and at about the same time proven by Zhandry using the polynomial method [arXiv, 2013].)
We compare the power of non-adaptive learning graphs and quantum query algorithms that only utilize the knowledge on the possible positions of certificates in the input string. To do that, we introduce a notion of a certificate structure of a decision problem. Using the adversary method and the dual formulation of the learning graph complexity, we show that, for every certificate structure, there exists a decision problem possessing this certificate structure such that its non-adaptive learning graph and quantum query complexities differ by at most a constant multiplicative factor. For a special case of certificate structures, we construct a relatively general class of problems having this property. This construction generalizes the adversary lower bound for the k-Sum problem derived recently by Belovs and Spalek [ACM ITCS, 2013].
We also construct an optimal Omega(n^(2/3)) adversary lower bound for the Element Distinctness problem with minimal non-trivial alphabet size, which equals the length of the input. Due to the strict requirement on the alphabet size, here we cannot use the embedding procedure, and the construction of the adversary matrix heavily relies on the representation theory of the symmetric group. While an optimal lower bound for Element Distinctness using the polynomial method had been proven for any input alphabet, an optimal adversary construction was previously only known for alphabets of size at least Omega(n^2).
Finally, we introduce the Enhanced Find-Two problem and we study its query complexity. The Enhanced Find-Two problem is, given n elements such that exactly k of them are marked, find two distinct marked elements using the following resources:
(1) one initial copy of the uniform superposition over all marked elements,
(2) an oracle that reflects across this superposition, and
(3) an oracle that tests if an element is marked.
This relational problem arises in the study of quantum proofs of knowledge. We prove that its query complexity is Theta(min{sqrt(n/k),sqrt(k)})
Adversary Lower Bound for Element Distinctness with Small Range
The Element Distinctness problem is to decide whether each character of an
input string is unique. The quantum query complexity of Element Distinctness is
known to be ; the polynomial method gives a tight lower bound
for any input alphabet, while a tight adversary construction was only known for
alphabets of size .
We construct a tight adversary lower bound for Element
Distinctness with minimal non-trivial alphabet size, which equals the length of
the input. This result may help to improve lower bounds for other related query
problems.Comment: 22 pages. v2: one figure added, updated references, and minor typos
fixed. v3: minor typos fixe
A New Quantum Lower Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs
We give a new version of the adversary method for proving lower bounds on
quantum query algorithms. The new method is based on analyzing the eigenspace
structure of the problem at hand. We use it to prove a new and optimal strong
direct product theorem for 2-sided error quantum algorithms computing k
independent instances of a symmetric Boolean function: if the algorithm uses
significantly less than k times the number of queries needed for one instance
of the function, then its success probability is exponentially small in k. We
also use the polynomial method to prove a direct product theorem for 1-sided
error algorithms for k threshold functions with a stronger bound on the success
probability. Finally, we present a quantum algorithm for evaluating solutions
to systems of linear inequalities, and use our direct product theorems to show
that the time-space tradeoff of this algorithm is close to optimal.Comment: 16 pages LaTeX. Version 2: title changed, proofs significantly
cleaned up and made selfcontained. This version to appear in the proceedings
of the STOC 06 conferenc
Quantum Zero-Error Algorithms Cannot be Composed
We exhibit two black-box problems, both of which have an efficient quantum
algorithm with zero-error, yet whose composition does not have an efficient
quantum algorithm with zero-error. This shows that quantum zero-error
algorithms cannot be composed. In oracle terms, we give a relativized world
where ZQP^{ZQP}\=ZQP, while classically we always have ZPP^{ZPP}=ZPP.Comment: 7 pages LaTeX. 2nd version slightly rewritte
Optimal quantum algorithm for polynomial interpolation
We consider the number of quantum queries required to determine the
coefficients of a degree-d polynomial over GF(q). A lower bound shown
independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2+1/2
quantum queries are needed to solve this problem with bounded error, whereas an
algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We
show that the lower bound is achievable: d/2+1/2 quantum queries suffice to
determine the polynomial with bounded error. Furthermore, we show that d/2+1
queries suffice to achieve probability approaching 1 for large q. These upper
bounds improve results of Boneh and Zhandry on the insecurity of cryptographic
protocols against quantum attacks. We also show that our algorithm's success
probability as a function of the number of queries is precisely optimal.
Furthermore, the algorithm can be implemented with gate complexity poly(log q)
with negligible decrease in the success probability. We end with a conjecture
about the quantum query complexity of multivariate polynomial interpolation.Comment: 17 pages, minor improvements, added conjecture about multivariate
interpolatio
Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries
Generalizing earlier work characterizing the quantum query complexity of
computing a function of an unknown classical ``black box'' function drawn from
some set of such black box functions, we investigate a more general quantum
query model in which the goal is to compute functions of N by N ``black box''
unitary matrices drawn from a set of such matrices, a problem with applications
to determining properties of quantum physical systems. We characterize the
existence of an algorithm for such a query problem, with given error and number
of queries, as equivalent to the feasibility of a certain set of semidefinite
programming constraints, or equivalently the infeasibility of a dual of these
constraints, which we construct. Relaxing the primal constraints to correspond
to mere pairwise near-orthogonality of the final states of a quantum computer,
conditional on black-box inputs having distinct function values, rather than
bounded-error determinability of the function value via a single measurement on
the output states, we obtain a relaxed primal program the feasibility of whose
dual still implies the nonexistence of a quantum algorithm. We use this to
obtain a generalization, to our not-necessarily-commutative setting, of the
``spectral adversary method'' for quantum query lower bounds.Comment: Dagstuhl Seminar Proceedings 06391, "Algorithms and Complexity for
Continuous Problems," ed. S. Dahlke, K. Ritter, I. H. Sloan, J. F. Traub
(2006), available electronically at
http://drops.dagstuhl.de/portals/index.php?semnr=0639
Quantum Weakly Nondeterministic Communication Complexity
We study the weakest model of quantum nondeterminism in which a classical
proof has to be checked with probability one by a quantum protocol. We show the
first separation between classical nondeterministic communication complexity
and this model of quantum nondeterministic communication complexity for a total
function. This separation is quadratic.Comment: 12 pages. v3: minor correction
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