2,435 research outputs found
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
On Perfect Completeness for QMA
Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with
one-sided error, has been an open problem for years. This note helps to explain
why the problem is difficult, by using ideas from real analysis to give a
"quantum oracle" relative to which they are different. As a byproduct, we find
that there are facts about quantum complexity classes that are classically
relativizing but not quantumly relativizing, among them such "trivial"
containments as BQP in ZQEXP.Comment: 9 pages. To appear in Quantum Information & Computatio
Quantum Information and the PCP Theorem
We show how to encode (classical) bits by a single
quantum state of size O(n) qubits, such that: for any constant and
any , the values of the bits
can be retrieved from by a one-round
Arthur-Merlin interactive protocol of size polynomial in . This shows how to
go around Holevo-Nayak's Theorem, using Arthur-Merlin proofs.
We use the new representation to prove the following results:
1) Interactive proofs with quantum advice: We show that the class
contains ALL languages. That is, for any language (even non-recursive), the
membership (for of length ) can be proved by a polynomial-size
quantum interactive proof, where the verifier is a polynomial-size quantum
circuit with working space initiated with some quantum state
(depending only on and ). Moreover, the interactive proof that we give
is of only one round, and the messages communicated are classical.
2) PCP with only one query: We show that the membership (for
of length ) can be proved by a logarithmic-size quantum state ,
together with a polynomial-size classical proof consisting of blocks of length
bits each, such that after measuring the state the
verifier only needs to read {\bf one} block of the classical proof.
While the first result is a straight forward consequence of the new
representation, the second requires an additional machinery of quantum
low-degree-test that may be interesting in its own right.Comment: 30 page
Generalized Quantum Arthur-Merlin Games
This paper investigates the role of interaction and coins in public-coin
quantum interactive proof systems (also called quantum Arthur-Merlin games).
While prior works focused on classical public coins even in the quantum
setting, the present work introduces a generalized version of quantum
Arthur-Merlin games where the public coins can be quantum as well: the verifier
can send not only random bits, but also halves of EPR pairs. First, it is
proved that the class of two-turn quantum Arthur-Merlin games with quantum
public coins, denoted qq-QAM in this paper, does not change by adding a
constant number of turns of classical interactions prior to the communications
of the qq-QAM proof systems. This can be viewed as a quantum analogue of the
celebrated collapse theorem for AM due to Babai. To prove this collapse
theorem, this paper provides a natural complete problem for qq-QAM: deciding
whether the output of a given quantum circuit is close to a totally mixed
state. This complete problem is on the very line of the previous studies
investigating the hardness of checking the properties related to quantum
circuits, and is of independent interest. It is further proved that the class
qq-QAM_1 of two-turn quantum-public-coin quantum Arthur-Merlin proof systems
with perfect completeness gives new bounds for standard well-studied classes of
two-turn interactive proof systems. Finally, the collapse theorem above is
extended to comprehensively classify the role of interaction and public coins
in quantum Arthur-Merlin games: it is proved that, for any constant m>1, the
class of problems having an m-turn quantum Arthur-Merlin proof system is either
equal to PSPACE or equal to the class of problems having a two-turn quantum
Arthur-Merlin game of a specific type, which provides a complete set of quantum
analogues of Babai's collapse theorem.Comment: 31 pages + cover page, the proof of Lemma 27 (Lemma 24 in v1) is
corrected, and a new completeness result is adde
Direct certification of a class of quantum simulations
One of the main challenges in the field of quantum simulation and computation
is to identify ways to certify the correct functioning of a device when a
classical efficient simulation is not available. Important cases are situations
in which one cannot classically calculate local expectation values of state
preparations efficiently. In this work, we develop weak-membership formulations
of the certification of ground state preparations. We provide a non-interactive
protocol for certifying ground states of frustration-free Hamiltonians based on
simple energy measurements of local Hamiltonian terms. This certification
protocol can be applied to classically intractable analog quantum simulations:
For example, using Feynman-Kitaev Hamiltonians, one can encode universal
quantum computation in such ground states. Moreover, our certification protocol
is applicable to ground states encodings of IQP circuits demonstration of
quantum supremacy. These can be certified efficiently when the error is
polynomially bounded.Comment: 10 pages, corrected a small error in Eqs. (2) and (5
Why Philosophers Should Care About Computational Complexity
One might think that, once we know something is computable, how efficiently
it can be computed is a practical question with little further philosophical
importance. In this essay, I offer a detailed case that one would be wrong. In
particular, I argue that computational complexity theory---the field that
studies the resources (such as time, space, and randomness) needed to solve
computational problems---leads to new perspectives on the nature of
mathematical knowledge, the strong AI debate, computationalism, the problem of
logical omniscience, Hume's problem of induction, Goodman's grue riddle, the
foundations of quantum mechanics, economic rationality, closed timelike curves,
and several other topics of philosophical interest. I end by discussing aspects
of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and
beyond," MIT Press, 2012. Some minor clarifications and corrections; new
references adde
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