621 research outputs found
Classical Algorithms from Quantum and Arthur-Merlin Communication Protocols
In recent years, the polynomial method from circuit complexity has been applied to several fundamental problems and obtains the state-of-the-art running times (e.g., R. Williams\u27s n^3 / 2^{Omega(sqrt{log n})} time algorithm for APSP). As observed in [Alman and Williams, STOC 2017], almost all applications of the polynomial method in algorithm design ultimately rely on certain (probabilistic) low-rank decompositions of the computation matrices corresponding to key subroutines. They suggest that making use of low-rank decompositions directly could lead to more powerful algorithms, as the polynomial method is just one way to derive such a decomposition.
Inspired by their observation, in this paper, we study another way of systematically constructing low-rank decompositions of matrices which could be used by algorithms - communication protocols. Since their introduction, it is known that various types of communication protocols lead to certain low-rank decompositions (e.g., P protocols/rank, BQP protocols/approximate rank). These are usually interpreted as approaches for proving communication lower bounds, while in this work we explore the other direction.
We have the following two generic algorithmic applications of communication protocols:
- Quantum Communication Protocols and Deterministic Approximate Counting. Our first connection is that a fast BQP communication protocol for a function f implies a fast deterministic additive approximate counting algorithm for a related pair counting problem. Applying known BQP communication protocols, we get fast deterministic additive approximate counting algorithms for Count-OV (#OV), Sparse Count-OV and Formula of SYM circuits. In particular, our approximate counting algorithm for #OV runs in near-linear time for all dimensions d = o(log^2 n). Previously, even no truly-subquadratic time algorithm was known for d = omega(log n).
- Arthur-Merlin Communication Protocols and Faster Satisfying-Pair Algorithms. Our second connection is that a fast AM^{cc} protocol for a function f implies a faster-than-bruteforce algorithm for f-Satisfying-Pair. Using the classical Goldwasser-Sisper AM protocols for approximating set size, we obtain a new algorithm for approximate Max-IP_{n,c log n} in time n^{2 - 1/O(log c)}, matching the state-of-the-art algorithms in [Chen, CCC 2018].
We also apply our second connection to shed some light on long-standing open problems in communication complexity. We show that if the Longest Common Subsequence (LCS) problem admits a fast (computationally efficient) AM^{cc} protocol (polylog(n) complexity), then polynomial-size Formula-SAT admits a 2^{n - n^{1-delta}} time algorithm for any constant delta > 0, which is conjectured to be unlikely by a recent work [Abboud and Bringmann, ICALP 2018]. The same holds even for a fast (computationally efficient) PH^{cc} protocol
AM with Multiple Merlins
We introduce and study a new model of interactive proofs: AM(k), or
Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known
MIP, here the assumption is that each Merlin receives an independent random
challenge from Arthur. One motivation for this model (which we explore in
detail) comes from the close analogies between it and the quantum complexity
class QMA(k), but the AM(k) model is also natural in its own right.
We illustrate the power of multiple Merlins by giving an AM(2) protocol for
3SAT, in which the Merlins' challenges and responses consist of only
n^{1/2+o(1)} bits each. Our protocol has the consequence that, assuming the
Exponential Time Hypothesis (ETH), any algorithm for approximating a dense CSP
with a polynomial-size alphabet must take n^{(log n)^{1-o(1)}} time. Algorithms
nearly matching this lower bound are known, but their running times had never
been previously explained. Brandao and Harrow have also recently used our 3SAT
protocol to show quasipolynomial hardness for approximating the values of
certain entangled games.
In the other direction, we give a simple quasipolynomial-time approximation
algorithm for free games, and use it to prove that, assuming the ETH, our 3SAT
protocol is essentially optimal. More generally, we show that multiple Merlins
never provide more than a polynomial advantage over one: that is, AM(k)=AM for
all k=poly(n). The key to this result is a subsampling theorem for free games,
which follows from powerful results by Alon et al. and Barak et al. on
subsampling dense CSPs, and which says that the value of any free game can be
closely approximated by the value of a logarithmic-sized random subgame.Comment: 48 page
Two Results about Quantum Messages
We show two results about the relationship between quantum and classical
messages. Our first contribution is to show how to replace a quantum message in
a one-way communication protocol by a deterministic message, establishing that
for all partial Boolean functions we
have . This bound was previously
known for total functions, while for partial functions this improves on results
by Aaronson, in which either a log-factor on the right hand is present, or the
left hand side is , and in which also no entanglement is
allowed.
In our second contribution we investigate the power of quantum proofs over
classical proofs. We give the first example of a scenario, where quantum proofs
lead to exponential savings in computing a Boolean function. The previously
only known separation between the power of quantum and classical proofs is in a
setting where the input is also quantum.
We exhibit a partial Boolean function , such that there is a one-way
quantum communication protocol receiving a quantum proof (i.e., a protocol of
type QMA) that has cost for , whereas every one-way quantum
protocol for receiving a classical proof (protocol of type QCMA) requires
communication
Quantum superiority for verifying NP-complete problems with linear optics
Demonstrating quantum superiority for some computational task will be a
milestone for quantum technologies and would show that computational advantages
are possible not only with a universal quantum computer but with simpler
physical devices. Linear optics is such a simpler but powerful platform where
classically-hard information processing tasks, such as Boson Sampling, can be
in principle implemented. In this work, we study a fundamentally different type
of computational task to achieve quantum superiority using linear optics,
namely the task of verifying NP-complete problems. We focus on a protocol by
Aaronson et al. (2008) that uses quantum proofs for verification. We show that
the proof states can be implemented in terms of a single photon in an equal
superposition over many optical modes. Similarly, the tests can be performed
using linear-optical transformations consisting of a few operations: a global
permutation of all modes, simple interferometers acting on at most four modes,
and measurement using single-photon detectors. We also show that the protocol
can tolerate experimental imperfections.Comment: 10 pages, 6 figures, minor corrections, results unchange
The Power of Unentanglement
The class QMA(k). introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error?
In this paper, we make progress on all of the above questions.
* We give a protocol by which a verifier can be convinced that a 3SAT formula of size m is satisfiable, with constant soundness, given Õ (√m) unentangled quantum witnesses with O(log m) qubits each. Our protocol relies on the existence of very short PCPs.
* We show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2.
* We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one
QIP = PSPACE
We prove that the complexity class QIP, which consists of all problems having
quantum interactive proof systems, is contained in PSPACE. This containment is
proved by applying a parallelized form of the matrix multiplicative weights
update method to a class of semidefinite programs that captures the
computational power of quantum interactive proofs. As the containment of PSPACE
in QIP follows immediately from the well-known equality IP = PSPACE, the
equality QIP = PSPACE follows.Comment: 21 pages; v2 includes corrections and minor revision
Testing product states, quantum Merlin-Arthur games and tensor optimisation
We give a test that can distinguish efficiently between product states of n
quantum systems and states which are far from product. If applied to a state
psi whose maximum overlap with a product state is 1-epsilon, the test passes
with probability 1-Theta(epsilon), regardless of n or the local dimensions of
the individual systems. The test uses two copies of psi. We prove correctness
of this test as a special case of a more general result regarding stability of
maximum output purity of the depolarising channel. A key application of the
test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain
several structural results that had been previously conjectured, including the
fact that efficient soundness amplification is possible and that two Merlins
can simulate many Merlins: QMA(k)=QMA(2) for k>=2. Building on a previous
result of Aaronson et al, this implies that there is an efficient quantum
algorithm to verify 3-SAT with constant soundness, given two unentangled proofs
of O(sqrt(n) polylog(n)) qubits. We also show how QMA(2) with log-sized proofs
is equivalent to a large number of problems, some related to quantum
information (such as testing separability of mixed states) as well as problems
without any apparent connection to quantum mechanics (such as computing
injective tensor norms of 3-index tensors). As a consequence, we obtain many
hardness-of-approximation results, as well as potential algorithmic
applications of methods for approximating QMA(2) acceptance probabilities.
Finally, our test can also be used to construct an efficient test for
determining whether a unitary operator is a tensor product, which is a
generalisation of classical linearity testing.Comment: 44 pages, 1 figure, 7 appendices; v6: added references, rearranged
sections, added discussion of connections to classical CS. Final version to
appear in J of the AC
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