3,505 research outputs found
Sculpting Quantum Speedups
Given a problem which is intractable for both quantum and classical
algorithms, can we find a sub-problem for which quantum algorithms provide an
exponential advantage? We refer to this problem as the "sculpting problem." In
this work, we give a full characterization of sculptable functions in the query
complexity setting. We show that a total function f can be restricted to a
promise P such that Q(f|_P)=O(polylog(N)) and R(f|_P)=N^{Omega(1)}, if and only
if f has a large number of inputs with large certificate complexity. The proof
uses some interesting techniques: for one direction, we introduce new
relationships between randomized and quantum query complexity in various
settings, and for the other direction, we use a recent result from
communication complexity due to Klartag and Regev. We also characterize
sculpting for other query complexity measures, such as R(f) vs. R_0(f) and
R_0(f) vs. D(f).
Along the way, we prove some new relationships for quantum query complexity:
for example, a nearly quadratic relationship between Q(f) and D(f) whenever the
promise of f is small. This contrasts with the recent super-quadratic query
complexity separations, showing that the maximum gap between classical and
quantum query complexities is indeed quadratic in various settings - just not
for total functions!
Lastly, we investigate sculpting in the Turing machine model. We show that if
there is any BPP-bi-immune language in BQP, then every language outside BPP can
be restricted to a promise which places it in PromiseBQP but not in PromiseBPP.
Under a weaker assumption, that some problem in BQP is hard on average for
P/poly, we show that every paddable language outside BPP is sculptable in this
way.Comment: 30 page
Improved Quantum Communication Complexity Bounds for Disjointness and Equality
We prove new bounds on the quantum communication complexity of the
disjointness and equality problems. For the case of exact and non-deterministic
protocols we show that these complexities are all equal to n+1, the previous
best lower bound being n/2. We show this by improving a general bound for
non-deterministic protocols of de Wolf. We also give an O(sqrt{n}c^{log^*
n})-qubit bounded-error protocol for disjointness, modifying and improving the
earlier O(sqrt{n}log n) protocol of Buhrman, Cleve, and Wigderson, and prove an
Omega(sqrt{n}) lower bound for a large class of protocols that includes the
BCW-protocol as well as our new protocol.Comment: 11 pages LaTe
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
An Application of Quantum Finite Automata to Interactive Proof Systems
Quantum finite automata have been studied intensively since their
introduction in late 1990s as a natural model of a quantum computer with
finite-dimensional quantum memory space. This paper seeks their direct
application to interactive proof systems in which a mighty quantum prover
communicates with a quantum-automaton verifier through a common communication
cell. Our quantum interactive proof systems are juxtaposed to
Dwork-Stockmeyer's classical interactive proof systems whose verifiers are
two-way probabilistic automata. We demonstrate strengths and weaknesses of our
systems and further study how various restrictions on the behaviors of
quantum-automaton verifiers affect the power of quantum interactive proof
systems.Comment: This is an extended version of the conference paper in the
Proceedings of the 9th International Conference on Implementation and
Application of Automata, Lecture Notes in Computer Science, Springer-Verlag,
Kingston, Canada, July 22-24, 200
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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