12 research outputs found
Unbounded Error Quantum Query Complexity
This work studies the quantum query complexity of Boolean functions in a
scenario where it is only required that the query algorithm succeeds with a
probability strictly greater than 1/2. We show that, just as in the
communication complexity model, the unbounded error quantum query complexity is
exactly half of its classical counterpart for any (partial or total) Boolean
function. Moreover, we show that the "black-box" approach to convert quantum
query algorithms into communication protocols by Buhrman-Cleve-Wigderson
[STOC'98] is optimal even in the unbounded error setting.
We also study a setting related to the unbounded error model, called the
weakly unbounded error setting, where the cost of a query algorithm is given by
q+log(1/2(p-1/2)), where q is the number of queries made and p>1/2 is the
success probability of the algorithm. In contrast to the case of communication
complexity, we show a tight Theta(log n) separation between quantum and
classical query complexity in the weakly unbounded error setting for a partial
Boolean function. We also show the asymptotic equivalence between them for some
well-studied total Boolean functions.Comment: 14 page
On the uselessness of quantum queries
Given a prior probability distribution over a set of possible oracle
functions, we define a number of queries to be useless for determining some
property of the function if the probability that the function has the property
is unchanged after the oracle responds to the queries. A familiar example is
the parity of a uniformly random Boolean-valued function over ,
for which classical queries are useless. We prove that if classical
queries are useless for some oracle problem, then quantum queries are also
useless. For such problems, which include classical threshold secret sharing
schemes, our result also gives a new way to obtain a lower bound on the quantum
query complexity, even in cases where neither the function nor the property to
be determined is Boolean
Optimal quantum query bounds for almost all Boolean functions
We show that almost all n-bit Boolean functions have bounded-error quantum
query complexity at least n/2, up to lower-order terms. This improves over an
earlier n/4 lower bound of Ambainis, and shows that van Dam's oracle
interrogation is essentially optimal for almost all functions. Our proof uses
the fact that the acceptance probability of a T-query algorithm can be written
as the sum of squares of degree-T polynomials.Comment: 8 pages LaTe
Weak Parity
We study the query complexity of Weak Parity: the problem of computing the
parity of an n-bit input string, where one only has to succeed on a 1/2+eps
fraction of input strings, but must do so with high probability on those inputs
where one does succeed. It is well-known that n randomized queries and n/2
quantum queries are needed to compute parity on all inputs. But surprisingly,
we give a randomized algorithm for Weak Parity that makes only
O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only
O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of
Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove
lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in
the quantum case for any eps>0. We show that improving our lower bounds is
intimately related to two longstanding open problems about Boolean functions:
the Sensitivity Conjecture, and the relationships between query complexity and
polynomial degree.Comment: 18 page
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of -query quantum algorithms in terms of the
unit ball of a space of degree- polynomials. Based on this, we obtain a
refined notion of approximate polynomial degree that equals the quantum query
complexity, answering a question of Aaronson et al. (CCC'16). Our proof is
based on a fundamental result of Christensen and Sinclair (J. Funct. Anal.,
1987) that generalizes the well-known Stinespring representation for quantum
channels to multilinear forms. Using our characterization, we show that many
polynomials of degree four are far from those coming from two-query quantum
algorithms. We also give a simple and short proof of one of the results of
Aaronson et al. showing an equivalence between one-query quantum algorithms and
bounded quadratic polynomials.Comment: 24 pages, 3 figures. v2: 27 pages, minor changes in response to
referee comment
Uselessness for an Oracle Model with Internal Randomness
We consider a generalization of the standard oracle model in which the oracle
acts on the target with a permutation selected according to internal random
coins. We describe several problems that are impossible to solve classically
but can be solved by a quantum algorithm using a single query; we show that
such infinity-vs-one separations between classical and quantum query
complexities can be constructed from much weaker separations.
We also give conditions to determine when oracle problems---either in the
standard model, or in any of the generalizations we consider---cannot be solved
with success probability better than random guessing would achieve. In the
oracle model with internal randomness where the goal is to gain any nonzero
advantage over guessing, we prove (roughly speaking) that quantum queries
are equivalent in power to classical queries, thus extending results of
Meyer and Pommersheim.Comment: 18 pages. v2. shortened, presentation improved, same result
Exponential improvement in precision for simulating sparse Hamiltonians
We provide a quantum algorithm for simulating the dynamics of sparse
Hamiltonians with complexity sublogarithmic in the inverse error, an
exponential improvement over previous methods. Specifically, we show that a
-sparse Hamiltonian acting on qubits can be simulated for time
with precision using queries and
additional 2-qubit gates, where . Unlike previous
approaches based on product formulas, the query complexity is independent of
the number of qubits acted on, and for time-varying Hamiltonians, the gate
complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our
algorithm is based on a significantly improved simulation of the continuous-
and fractional-query models using discrete quantum queries, showing that the
former models are not much more powerful than the discrete model even for very
small error. We also simplify the analysis of this conversion, avoiding the
need for a complex fault correction procedure. Our simplification relies on a
new form of "oblivious amplitude amplification" that can be applied even though
the reflection about the input state is unavailable. Finally, we prove new
lower bounds showing that our algorithms are optimal as a function of the
error.Comment: v1: 27 pages; Subsumes and improves upon results in arXiv:1308.5424.
v2: 28 pages, minor change
On the Fine-Grained Query Complexity of Symmetric Functions
This paper explores a fine-grained version of the Watrous conjecture,
including the randomized and quantum algorithms with success probabilities
arbitrarily close to . Our contributions include the following:
i) An analysis of the optimal success probability of quantum and randomized
query algorithms of two fundamental partial symmetric Boolean functions given a
fixed number of queries. We prove that for any quantum algorithm computing
these two functions using queries, there exist randomized algorithms using
queries that achieve the same success probability as the
quantum algorithm, even if the success probability is arbitrarily close to 1/2.
ii) We establish that for any total symmetric Boolean function , if a
quantum algorithm uses queries to compute with success probability
, then there exists a randomized algorithm using queries to
compute with success probability on a
fraction of inputs, where can be arbitrarily small
positive values. As a corollary, we prove a randomized version of
Aaronson-Ambainis Conjecture for total symmetric Boolean functions in the
regime where the success probability of algorithms can be arbitrarily close to
1/2.
iii) We present polynomial equivalences for several fundamental complexity
measures of partial symmetric Boolean functions. Specifically, we first prove
that for certain partial symmetric Boolean functions, quantum query complexity
is at most quadratic in approximate degree for any error arbitrarily close to
1/2. Next, we show exact quantum query complexity is at most quadratic in
degree. Additionally, we give the tight bounds of several complexity measures,
indicating their polynomial equivalence.Comment: accepted in ISAAC 202