132 research outputs found
Span Programs and Quantum Space Complexity
While quantum computers hold the promise of significant computational speedups, the limited size of early quantum machines motivates the study of space-bounded quantum computation. We relate the quantum space complexity of computing a function f with one-sided error to the logarithm of its span program size, a classical quantity that is well-studied in attempts to prove formula size lower bounds.
In the more natural bounded error model, we show that the amount of space needed for a unitary quantum algorithm to compute f with bounded (two-sided) error is lower bounded by the logarithm of its approximate span program size. Approximate span programs were introduced in the field of quantum algorithms but not studied classically. However, the approximate span program size of a function is a natural generalization of its span program size.
While no non-trivial lower bound is known on the span program size (or approximate span program size) of any concrete function, a number of lower bounds are known on the monotone span program size. We show that the approximate monotone span program size of f is a lower bound on the space needed by quantum algorithms of a particular form, called monotone phase estimation algorithms, to compute f. We then give the first non-trivial lower bound on the approximate span program size of an explicit function
Approximate Span Programs
Span programs are a model of computation that have been used to design
quantum algorithms, mainly in the query model. For any decision problem, there
exists a span program that leads to an algorithm with optimal quantum query
complexity, but finding such an algorithm is generally challenging.
We consider new ways of designing quantum algorithms using span programs. We
show how any span program that decides a problem can also be used to decide
"property testing" versions of , or more generally, approximate the span
program witness size, a property of the input related to . For example,
using our techniques, the span program for OR, which can be used to design an
optimal algorithm for the OR function, can also be used to design optimal
algorithms for: threshold functions, in which we want to decide if the Hamming
weight of a string is above a threshold or far below, given the promise that
one of these is true; and approximate counting, in which we want to estimate
the Hamming weight of the input. We achieve these results by relaxing the
requirement that 1-inputs hit some target exactly in the span program, which
could make design of span programs easier.
We also give an exposition of span program structure, which increases the
understanding of this important model. One implication is alternative
algorithms for estimating the witness size when the phase gap of a certain
unitary can be lower bounded. We show how to lower bound this phase gap in some
cases.
As applications, we give the first upper bounds in the adjacency query model
on the quantum time complexity of estimating the effective resistance between
and , , of , and, when is a lower
bound on , by our phase gap lower bound, we can obtain , both using space
Quantum Communication Complexity of Distributed Set Joins
Computing set joins of two inputs is a common task in database theory. Recently, Van Gucht, Williams, Woodruff and Zhang [PODS 2015] considered the complexity of such problems in the natural model of (classical) two-party communication complexity and obtained tight bounds for the complexity of several important distributed set joins.
In this paper we initiate the study of the quantum communication complexity of distributed set joins. We design a quantum protocol for distributed Boolean matrix multiplication, which corresponds to computing the composition join of two databases, showing that the product of two n times n Boolean matrices, each owned by one of two respective parties, can be computed with widetilde-O(sqrt{n} ell^{3/4}) qubits of communication, where ell denotes the number of non-zero entries of the product. Since Van Gucht et al. showed that the classical communication complexity of this problem is widetilde-Theta(n sqrt{ell}), our quantum algorithm outperforms classical protocols whenever the output matrix is sparse. We also show a quantum lower bound and a matching classical upper bound on the communication complexity of distributed matrix multiplication over F_2.
Besides their applications to database theory, the communication complexity of set joins is interesting due to its connections to direct product theorems in communication complexity. In this work we also introduce a notion of all-pairs product theorem, and relate this notion to standard direct product theorems in communication complexity
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 (*)
Partial-indistinguishability obfuscation using braids
An obfuscator is an algorithm that translates circuits into
functionally-equivalent similarly-sized circuits that are hard to understand.
Efficient obfuscators would have many applications in cryptography. Until
recently, theoretical progress has mainly been limited to no-go results. Recent
works have proposed the first efficient obfuscation algorithms for classical
logic circuits, based on a notion of indistinguishability against
polynomial-time adversaries. In this work, we propose a new notion of
obfuscation, which we call partial-indistinguishability. This notion is based
on computationally universal groups with efficiently computable normal forms,
and appears to be incomparable with existing definitions. We describe universal
gate sets for both classical and quantum computation, in which our definition
of obfuscation can be met by polynomial-time algorithms. We also discuss some
potential applications to testing quantum computers. We stress that the
cryptographic security of these obfuscators, especially when composed with
translation from other gate sets, remains an open question.Comment: 21 pages,Proceedings of TQC 201
Span Programs and Quantum Space Complexity
While quantum computers hold the promise of significant computational speedups, the limited size of early quantum machines motivates the study of space-bounded quantum computation. We relate the quantum space complexity of computing a function f with one-sided error to the logarithm of its span program size, a classical quantity that is well-studied in attempts to prove formula size lower bounds.
In the more natural bounded error model, we show that the amount of space needed for a unitary quantum algorithm to compute f with bounded (two-sided) error is lower bounded by the logarithm of its approximate span program size. Approximate span programs were introduced in the field of quantum algorithms but not studied classically. However, the approximate span program size of a function is a natural generalization of its span program size.
While no non-trivial lower bound is known on the span program size (or approximate span program size) of any concrete function, a number of lower bounds are known on the monotone span program size. We show that the approximate monotone span program size of f is a lower bound on the space needed by quantum algorithms of a particular form, called monotone phase estimation algorithms, to compute f. We then give the first non-trivial lower bound on the app
Collision Finding with Many Classical or Quantum Processors
In this thesis, we investigate the cost of finding collisions in a black-box function, a problem that is of fundamental importance in cryptanalysis. Inspired by the excellent performance of the heuristic rho method of collision finding, we define several new models of complexity that take into account the cost of moving information across a large space, and lay the groundwork for studying the performance of classical and quantum algorithms in these models
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
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