46,671 research outputs found
Average-Case Quantum Query Complexity
We compare classical and quantum query complexities of total Boolean
functions. It is known that for worst-case complexity, the gap between quantum
and classical can be at most polynomial. We show that for average-case
complexity under the uniform distribution, quantum algorithms can be
exponentially faster than classical algorithms. Under non-uniform distributions
the gap can even be super-exponential. We also prove some general bounds for
average-case complexity and show that the average-case quantum complexity of
MAJORITY under the uniform distribution is nearly quadratically better than the
classical complexity.Comment: 14 pages, LaTeX. Some parts rewritten. This version to appear in the
Journal of Physics
Oracles Are Subtle But Not Malicious
Theoretical computer scientists have been debating the role of oracles since
the 1970's. This paper illustrates both that oracles can give us nontrivial
insights about the barrier problems in circuit complexity, and that they need
not prevent us from trying to solve those problems.
First, we give an oracle relative to which PP has linear-sized circuits, by
proving a new lower bound for perceptrons and low- degree threshold
polynomials. This oracle settles a longstanding open question, and generalizes
earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More
importantly, it implies the first nonrelativizing separation of "traditional"
complexity classes, as opposed to interactive proof classes such as MIP and
MA-EXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does
not have circuits of size n^k for any fixed k. We present an alternative proof
of this fact, which shows that PP does not even have quantum circuits of size
n^k with quantum advice. To our knowledge, this is the first nontrivial lower
bound on quantum circuit size.
Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean
circuits in ZPP^NP. We show that the NP queries in this algorithm cannot be
parallelized by any relativizing technique, by giving an oracle relative to
which ZPP^||NP and even BPP^||NP have linear-size circuits. On the other hand,
we also show that the NP queries could be parallelized if P=NP. Thus, classes
such as ZPP^||NP inhabit a "twilight zone," where we need to distinguish
between relativizing and black-box techniques. Our results on this subject have
implications for computational learning theory as well as for the circuit
minimization problem.Comment: 20 pages, 1 figur
Approximating Fractional Time Quantum Evolution
An algorithm is presented for approximating arbitrary powers of a black box
unitary operation, , where is a real number, and
is a black box implementing an unknown unitary. The complexity of
this algorithm is calculated in terms of the number of calls to the black box,
the errors in the approximation, and a certain `gap' parameter. For general
and large , one should apply a total of times followed by our procedure for approximating the fractional
power . An example is also given where for
large integers this method is more efficient than direct application of
copies of . Further applications and related algorithms are also
discussed.Comment: 13 pages, 2 figure
Quantum Coins
One of the earliest cryptographic applications of quantum information was to
create quantum digital cash that could not be counterfeited. In this paper, we
describe a new type of quantum money: quantum coins, where all coins of the
same denomination are represented by identical quantum states. We state
desirable security properties such as anonymity and unforgeability and propose
two candidate quantum coin schemes: one using black box operations, and another
using blind quantum computation.Comment: 12 pages, 4 figure
- β¦