57 research outputs found
Adversary Lower Bound for the Orthogonal Array Problem
We prove a quantum query lower bound \Omega(n^{(d+1)/(d+2)}) for the problem
of deciding whether an input string of size n contains a k-tuple which belongs
to a fixed orthogonal array on k factors of strength d<=k-1 and index 1,
provided that the alphabet size is sufficiently large. Our lower bound is tight
when d=k-1.
The orthogonal array problem includes the following problems as special
cases: k-sum problem with d=k-1, k-distinctness problem with d=1, k-pattern
problem with d=0, (d-1)-degree problem with 1<=d<=k-1, unordered search with
d=0 and k=1, and graph collision with d=0 and k=2.Comment: 13 page
Improved Classical and Quantum Algorithms for Subset-Sum
We present new classical and quantum algorithms for solving random subset-sum
instances. First, we improve over the Becker-Coron-Joux algorithm (EUROCRYPT
2011) from downto
, using more general representations with
values in .
Next, we improve the state of the art of quantum algorithms for this problem
in several directions. By combining the Howgrave-Graham-Joux algorithm
(EUROCRYPT 2010) and quantum search, we devise an algorithm with asymptotic
cost , lower than the cost of the quantum
walk based on the same classical algorithm proposed by Bernstein, Jeffery,
Lange and Meurer (PQCRYPTO 2013). This algorithm has the advantage of using
\emph{classical} memory with quantum random access, while the previously known
algorithms used the quantum walk framework, and required \emph{quantum} memory
with quantum random access.
We also propose new quantum walks for subset-sum, performing better than the
previous best time complexity of given by
Helm and May (TQC 2018). We combine our new techniques to reach a time
. This time is dependent on a heuristic on
quantum walk updates, formalized by Helm and May, that is also required by the
previous algorithms. We show how to partially overcome this heuristic, and we
obtain an algorithm with quantum time
requiring only the standard classical subset-sum heuristics
A Unified Framework of Quantum Walk Search
Many quantum algorithms critically rely on quantum walk search, or the use of quantum walks to speed up search problems on graphs. However, the main results on quantum walk search are scattered over different, incomparable frameworks, such as the hitting time framework, the MNRS framework, and the electric network framework. As a consequence, a number of pieces are currently missing. For example, recent work by Ambainis et al. (STOC\u2720) shows how quantum walks starting from the stationary distribution can always find elements quadratically faster. In contrast, the electric network framework allows quantum walks to start from an arbitrary initial state, but it only detects marked elements.
We present a new quantum walk search framework that unifies and strengthens these frameworks, leading to a number of new results. For example, the new framework effectively finds marked elements in the electric network setting. The new framework also allows to interpolate between the hitting time framework, minimizing the number of walk steps, and the MNRS framework, minimizing the number of times elements are checked for being marked. This allows for a more natural tradeoff between resources. In addition to quantum walks and phase estimation, our new algorithm makes use of quantum fast-forwarding, similar to the recent results by Ambainis et al. This perspective also enables us to derive more general complexity bounds on the quantum walk algorithms, e.g., based on Monte Carlo type bounds of the corresponding classical walk. As a final result, we show how in certain cases we can avoid the use of phase estimation and quantum fast-forwarding, answering an open question of Ambainis et al
Quantum Subroutine Composition
An important tool in algorithm design is the ability to build algorithms from
other algorithms that run as subroutines. In the case of quantum algorithms, a
subroutine may be called on a superposition of different inputs, which
complicates things. For example, a classical algorithm that calls a subroutine
times, where the average probability of querying the subroutine on input
is , and the cost of the subroutine on input is , incurs
expected cost from all subroutine queries. While this
statement is obvious for classical algorithms, for quantum algorithms, it is
much less so, since naively, if we run a quantum subroutine on a superposition
of inputs, we need to wait for all branches of the superposition to terminate
before we can apply the next operation. We nonetheless show an analogous
quantum statement (*): If is the average query weight on over all
queries, the cost from all quantum subroutine queries is .
Here the query weight on for a particular query is the probability of
measuring in the input register if we were to measure right before the
query.
We prove this result using the technique of multidimensional quantum walks,
recently introduced in arXiv:2208.13492. We present a more general version of
their quantum walk edge composition result, which yields variable-time quantum
walks, generalizing variable-time quantum search, by, for example, replacing
the update cost with , where
is the cost to move from vertex to vertex . The same technique
that allows us to compose quantum subroutines in quantum walks can also be used
to compose in any quantum algorithm, which is how we prove (*)
Applications of the Adversary Method in Quantum Query Algorithms
In the thesis, we use a recently developed tight characterisation of quantum
query complexity, the adversary bound, to develop new quantum algorithms and
lower bounds. Our results are as follows:
* We develop a new technique for the construction of quantum algorithms:
learning graphs.
* We use learning graphs to improve quantum query complexity of the triangle
detection and the -distinctness problems.
* We prove tight lower bounds for the -sum and the triangle sum problems.
* We construct quantum algorithms for some subgraph-finding problems that are
optimal in terms of query, time and space complexities.
* We develop a generalisation of quantum walks that connects electrical
properties of a graph and its quantum hitting time. We use it to construct a
time-efficient quantum algorithm for 3-distinctness.Comment: PhD Thesis, 169 page
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