6 research outputs found
Perfect zero knowledge for quantum multiprover interactive proofs
In this work we consider the interplay between multiprover interactive
proofs, quantum entanglement, and zero knowledge proofs - notions that are
central pillars of complexity theory, quantum information and cryptography. In
particular, we study the relationship between the complexity class MIP, the
set of languages decidable by multiprover interactive proofs with quantumly
entangled provers, and the class PZKMIP, which is the set of languages
decidable by MIP protocols that furthermore possess the perfect zero
knowledge property.
Our main result is that the two classes are equal, i.e., MIP
PZKMIP. This result provides a quantum analogue of the celebrated result of
Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP
PZKMIP (in other words, all classical multiprover interactive protocols can be
made zero knowledge). We prove our result by showing that every MIP
protocol can be efficiently transformed into an equivalent zero knowledge
MIP protocol in a manner that preserves the completeness-soundness gap.
Combining our transformation with previous results by Slofstra (Forum of
Mathematics, Pi 2019) and Fitzsimons, Ji, Vidick and Yuen (STOC 2019), we
obtain the corollary that all co-recursively enumerable languages (which
include undecidable problems as well as all decidable problems) have zero
knowledge MIP protocols with vanishing promise gap
Composably secure device-independent encryption with certified deletion
We study the task of encryption with certified deletion (ECD) introduced by
Broadbent and Islam (2019), but in a device-independent setting: we show that
it is possible to achieve this task even when the honest parties do not trust
their quantum devices. Moreover, we define security for the ECD task in a
composable manner and show that our ECD protocol satisfies conditions that lead
to composable security. Our protocol is based on device-independent quantum key
distribution (DIQKD), and in particular the parallel DIQKD protocol based on
the magic square non-local game, given by Jain, Miller and Shi (2020). To
achieve certified deletion, we use a property of the magic square game observed
by Fu and Miller (2017), namely that a two-round variant of the game can be
used to certify deletion of a single random bit. In order to achieve certified
deletion security for arbitrarily long messages from this property, we prove a
parallel repetition theorem for two-round non-local games, which may be of
independent interest.Comment: 46 pages, 2 figure
Low-degree testing for quantum states, and a quantum entangled games PCP for QMA
We show that given an explicit description of a multiplayer game, with a
classical verifier and a constant number of players, it is QMA-hard, under
randomized reductions, to distinguish between the cases when the players have a
strategy using entanglement that succeeds with probability 1 in the game, or
when no such strategy succeeds with probability larger than 1/2. This proves
the "games quantum PCP conjecture" of Fitzsimons and the second author
(ITCS'15), albeit under randomized reductions. The core component in our
reduction is a construction of a family of two-player games for testing
-qubit maximally entangled states. For any integer , we give a test
in which questions from the verifier are bits long, and answers are
bits long. We show that for any constant
, any strategy that succeeds with probability at least
in the test must use a state that is within distance
from a state that is locally equivalent to a maximally
entangled state on qubits, for some universal constant . The
construction is based on the classical plane-vs-point test for multivariate
low-degree polynomials of Raz and Safra (STOC'97). We extend the classical test
to the quantum regime by executing independent copies of the test in the
generalized Pauli and bases over , where is a
sufficiently large prime power, and combine the two through a test for the
Pauli twisted commutation relations. Our main complexity-theoretic result is
obtained by combining this family of games with constructions of PCPs of
proximity introduced by Ben-Sasson et al. (CCC'05), and crucially relies on a
linear property of such PCPs. Another consequence of our results is a
deterministic reduction from the games quantum PCP conjecture to a suitable
formulation of the Hamiltonian quantum PCP conjecture.Comment: 59 pages. Game sized reduced from quasipolynomial to polynomial,
yielding improved complexity-theoretic result
Hardness amplification for entangled games via anchoring
We study the parallel repetition of one-round games involving players that can use quantum entanglement. A major open question in this area is whether parallel repetition reduces the entangled value of a game at an exponential rate - in other words, does an analogue of Raz's parallel repetition theorem hold for games with players sharing quantum entanglement? Previous results only apply to special classes of games.
We introduce a class of games we call anchored. We then introduce a simple transformation on games called anchoring, inspired in part by the Feige-Kilian transformation, that turns any (multiplayer) game into an anchored game. Unlike the Feige-Kilian transformation, our anchoring transformation is completeness preserving.
We prove an exponential-decay parallel repetition theorem for anchored games that involve any number of entangled players. We also prove a threshold version of our parallel repetition theorem for anchored games.
Together, our parallel repetition theorems and anchoring transformation provide the first hardness amplification techniques for general entangled games. We give an application to the games version of the Quantum PCP Conjecture
MIP*=RE
We show that the class MIP* of languages that can be decided by a classical
verifier interacting with multiple all-powerful quantum provers sharing
entanglement is equal to the class RE of recursively enumerable languages. Our
proof builds upon the quantum low-degree test of (Natarajan and Vidick, FOCS
2018) and the classical low-individual degree test of (Ji, et al., 2020) by
integrating recent developments from (Natarajan and Wright, FOCS 2019) and
combining them with the recursive compression framework of (Fitzsimons et al.,
STOC 2019).
An immediate byproduct of our result is that there is an efficient reduction
from the Halting Problem to the problem of deciding whether a two-player
nonlocal game has entangled value or at most . Using a known
connection, undecidability of the entangled value implies a negative answer to
Tsirelson's problem: we show, by providing an explicit example, that the
closure of the set of quantum tensor product correlations is strictly
included in the set of quantum commuting correlations. Following work
of (Fritz, Rev. Math. Phys. 2012) and (Junge et al., J. Math. Phys. 2011) our
results provide a refutation of Connes' embedding conjecture from the theory of
von Neumann algebras.Comment: 206 pages. v2: Updated to use arXiv:2009.12982. New appendi