302 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
On the Complexity of Zero Gap MIP*
The class is the set of languages decidable by multiprover
interactive proofs with quantum entangled provers. It was recently shown by Ji,
Natarajan, Vidick, Wright and Yuen that is equal to
, the set of recursively enumerable languages. In particular this
shows that the complexity of approximating the quantum value of a non-local
game is equivalent to the complexity of the Halting problem.
In this paper we investigate the complexity of deciding whether the quantum
value of a non-local game is exactly . This problem corresponds to a
complexity class that we call zero gap , denoted by
, where there is no promise gap between the verifier's
acceptance probabilities in the YES and NO cases. We prove that
extends beyond the first level of the arithmetical hierarchy
(which includes and its complement ), and in fact
is equal to , the class of languages that can be decided by quantified
formulas of the form .
Combined with the previously known result that (the
commuting operator variant of ) is equal to ,
our result further highlights the fascinating connection between various models
of quantum multiprover interactive proofs and different classes in
computability theory.Comment: Fixed typos and edited protocol to more smoothly follow from
reference
NEEXP is Contained in MIP*
We study multiprover interactive proof systems. The power of classical multiprover interactive proof systems, in which the provers do not share entanglement, was characterized in a famous work by Babai, Fortnow, and Lund (Computational Complexity 1991), whose main result was the equality MIP = NEXP. The power of quantum multiprover interactive proof systems, in which the provers are allowed to share entanglement, has proven to be much more difficult to characterize. The best known lower-bound on MIP* is NEXP ⊆ MIP*, due to Ito and Vidick (FOCS 2012). As for upper bounds, MIP* could be as large as RE, the class of recursively enumerable languages.
The main result of this work is the inclusion of NEEXP = NTIME[2^(2poly(n))] ⊆ MIP*. This is an exponential improvement over the prior lower bound and shows that proof systems with entangled provers are at least exponentially more powerful than classical provers. In our protocol the verifier delegates a classical, exponentially large MIP protocol for NEEXP to two entangled provers: the provers obtain their exponentially large questions by measuring their shared state, and use a classical PCP to certify the correctness of their exponentially-long answers. For the soundness of our protocol, it is crucial that each player should not only sample its own question correctly but also avoid performing measurements that would reveal the other player's sampled question. We ensure this by commanding the players to perform a complementary measurement, relying on the Heisenberg uncertainty principle to prevent the forbidden measurements from being performed
Robust self-testing of many-qubit states
We introduce a simple two-player test which certifies that the players apply
tensor products of Pauli and observables on the tensor
product of EPR pairs. The test has constant robustness: any strategy
achieving success probability within an additive of the optimal
must be -close, in the appropriate distance
measure, to the honest -qubit strategy. The test involves -bit questions
and -bit answers. The key technical ingredient is a quantum version of the
classical linearity test of Blum, Luby, and Rubinfeld.
As applications of our result we give (i) the first robust self-test for
EPR pairs; (ii) a quantum multiprover interactive proof system for the local
Hamiltonian problem with a constant number of provers and classical questions
and answers, and a constant completeness-soundness gap independent of system
size; (iii) a robust protocol for delegated quantum computation.Comment: 36 pages. Improves upon and supersedes our earlier submission
arXiv:1512.0209
Complexity Lower Bounds for Computing the Approximately-Commuting Operator Value of Non-Local Games to High Precision
We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is NP-complete to decide whether the classical value of a non-local game is 1 or 1- epsilon, promised that one of the two is the case. Furthermore, as long as epsilon is small enough, this result does not depend on the gap epsilon. In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap epsilon decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of MIP^{co}(2,1,1,s) (proofs with two provers, one round, completeness probability 1, soundness probability s, and commuting-operator strategies) can increase without bound as the gap 1-s gets arbitrarily small.
Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class PZK-MIP^{co}_{delta}(2,1,1,s) of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap 1-s gets arbitrarily small. While we do not know any computable time upper bound on the class MIP^{co}, a result of the first author and Vidick shows that for s = 1-1/poly(f(n)) and delta = 1/poly(f(n)), the class MIP^{co}_delta(2,1,1,s), with constant communication from the provers, is contained in TIME(exp(poly(f(n)))). We give a lower bound of coNTIME(f(n)) (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis
Unitary Complexity and the Uhlmann Transformation Problem
State transformation problems such as compressing quantum information or
breaking quantum commitments are fundamental quantum tasks. However, their
computational difficulty cannot easily be characterized using traditional
complexity theory, which focuses on tasks with classical inputs and outputs.
To study the complexity of such state transformation tasks, we introduce a
framework for unitary synthesis problems, including notions of reductions and
unitary complexity classes. We use this framework to study the complexity of
transforming one entangled state into another via local operations. We
formalize this as the Uhlmann Transformation Problem, an algorithmic version of
Uhlmann's theorem. Then, we prove structural results relating the complexity of
the Uhlmann Transformation Problem, polynomial space quantum computation, and
zero knowledge protocols.
The Uhlmann Transformation Problem allows us to characterize the complexity
of a variety of tasks in quantum information processing, including decoding
noisy quantum channels, breaking falsifiable quantum cryptographic assumptions,
implementing optimal prover strategies in quantum interactive proofs, and
decoding the Hawking radiation of black holes. Our framework for unitary
complexity thus provides new avenues for studying the computational complexity
of many natural quantum information processing tasks.Comment: 126 pages, comments welcom
Quantum proof systems for iterated exponential time, and beyond
© 2019 Copyright held by the owner/author(s). Publication rights licensed to ACM. We show that any language solvable in nondeterministic time exp(exp(· · · exp(n))), where the number of iterated exponentials is an arbitrary function R(n), can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness 1 and soundness 1 − exp(−C exp(· · · exp(n))), where the number of iterated exponentials is R(n) − 1 and C > 0 is a universal constant. The result was previously known for R = 1 and R = 2; we obtain it for any time-constructible function R. The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC’17). As a separate consequence of this technique we obtain a different proof of Slofstra’s recent result on the uncomputability of the entangled value of multiprover games (Forum of Mathematics, Pi 2019). Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP∗ contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson’s problem on the relation between the commuting operator and tensor product models for quantum correlations
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