3,257 research outputs found
Quantum vs Classical Proofs and Subset Verification
We study the ability of efficient quantum verifiers to decide properties of
exponentially large subsets given either a classical or quantum witness. We
develop a general framework that can be used to prove that QCMA machines, with
only classical witnesses, cannot verify certain properties of subsets given
implicitly via an oracle. We use this framework to prove an oracle separation
between QCMA and QMA using an "in-place" permutation oracle, making the first
progress on this question since Aaronson and Kuperberg in 2007. We also use the
framework to prove a particularly simple standard oracle separation between
QCMA and AM.Comment: 23 pages, presentation and notation clarified, small errors fixe
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
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
Oracle Complexity Classes and Local Measurements on Physical Hamiltonians
The canonical problem for the class Quantum Merlin-Arthur (QMA) is that of
estimating ground state energies of local Hamiltonians. Perhaps surprisingly,
[Ambainis, CCC 2014] showed that the related, but arguably more natural,
problem of simulating local measurements on ground states of local Hamiltonians
(APX-SIM) is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that
APX-SIM is P^QMA[log]-complete, for P^QMA[log] the class of languages decidable
by a P machine making a logarithmic number of adaptive queries to a QMA oracle.
In this work, we show that APX-SIM is P^QMA[log]-complete even when restricted
to more physical Hamiltonians, obtaining as intermediate steps a variety of
related complexity-theoretic results.
We first give a sequence of results which together yield P^QMA[log]-hardness
for APX-SIM on well-motivated Hamiltonians: (1) We show that for NP, StoqMA,
and QMA oracles, a logarithmic number of adaptive queries is equivalent to
polynomially many parallel queries. These equalities simplify the proofs of our
subsequent results. (2) Next, we show that the hardness of APX-SIM is preserved
under Hamiltonian simulations (a la [Cubitt, Montanaro, Piddock, 2017]). As a
byproduct, we obtain a full complexity classification of APX-SIM, showing it is
complete for P, P^||NP, P^||StoqMA, or P^||QMA depending on the Hamiltonians
employed. (3) Leveraging the above, we show that APX-SIM is P^QMA[log]-complete
for any family of Hamiltonians which can efficiently simulate spatially sparse
Hamiltonians, including physically motivated models such as the 2D Heisenberg
model.
Our second focus considers 1D systems: We show that APX-SIM remains
P^QMA[log]-complete even for local Hamiltonians on a 1D line of 8-dimensional
qudits. This uses a number of ideas from above, along with replacing the "query
Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.Comment: 38 pages, 3 figure
Energy efficient mining on a quantum-enabled blockchain using light
We outline a quantum-enabled blockchain architecture based on a consortium of
quantum servers. The network is hybridised, utilising digital systems for
sharing and processing classical information combined with a fibre--optic
infrastructure and quantum devices for transmitting and processing quantum
information. We deliver an energy efficient interactive mining protocol enacted
between clients and servers which uses quantum information encoded in light and
removes the need for trust in network infrastructure. Instead, clients on the
network need only trust the transparent network code, and that their devices
adhere to the rules of quantum physics. To demonstrate the energy efficiency of
the mining protocol, we elaborate upon the results of two previous experiments
(one performed over 1km of optical fibre) as applied to this work. Finally, we
address some key vulnerabilities, explore open questions, and observe
forward--compatibility with the quantum internet and quantum computing
technologies.Comment: 25 pages, 5 figure
Parallel repetition for entangled k-player games via fast quantum search
We present two parallel repetition theorems for the entangled value of
multi-player, one-round free games (games where the inputs come from a product
distribution). Our first theorem shows that for a -player free game with
entangled value , the -fold repetition of
has entangled value at most , where is the answer length of any
player. In contrast, the best known parallel repetition theorem for the
classical value of two-player free games is , due to Barak, et al. (RANDOM 2009). This
suggests the possibility of a separation between the behavior of entangled and
classical free games under parallel repetition.
Our second theorem handles the broader class of free games where the
players can output (possibly entangled) quantum states. For such games, the
repeated entangled value is upper bounded by . We also show that the dependence of the exponent
on is necessary: we exhibit a -player free game and such
that .
Our analysis exploits the novel connection between communication protocols
and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP
2014). We demonstrate that better communication protocols yield better parallel
repetition theorems: our first theorem crucially uses a quantum search protocol
by Aaronson and Ambainis, which gives a quadratic speed-up for distributed
search problems. Finally, our results apply to a broader class of games than
were previously considered before; in particular, we obtain the first parallel
repetition theorem for entangled games involving more than two players, and for
games involving quantum outputs.Comment: This paper is a significantly revised version of arXiv:1411.1397,
which erroneously claimed strong parallel repetition for free entangled
games. Fixed author order to alphabetica
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