17 research outputs found
Parallel approximation of non-interactive zero-sum quantum games
This paper studies a simple class of zero-sum games played by two competing
quantum players: each player sends a mixed quantum state to a referee, who
performs a joint measurement on the two states to determine the players'
payoffs. We prove that an equilibrium point of any such game can be
approximated by means of an efficient parallel algorithm, which implies that
one-turn quantum refereed games, wherein the referee is specified by a quantum
circuit, can be simulated in polynomial space.Comment: 18 page
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
Efficient Online Quantum Generative Adversarial Learning Algorithms with Applications
The exploration of quantum algorithms that possess quantum advantages is a
central topic in quantum computation and quantum information processing. One
potential candidate in this area is quantum generative adversarial learning
(QuGAL), which conceptually has exponential advantages over classical
adversarial networks. However, the corresponding learning algorithm remains
obscured. In this paper, we propose the first quantum generative adversarial
learning algorithm-- the quantum multiplicative matrix weight algorithm
(QMMW)-- which enables the efficient processing of fundamental tasks. The
computational complexity of QMMW is polynomially proportional to the number of
training rounds and logarithmically proportional to the input size. The core
concept of the proposed algorithm combines QuGAL with online learning. We
exploit the implementation of QuGAL with parameterized quantum circuits, and
numerical experiments for the task of entanglement test for pure state are
provided to support our claims
Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver
We study the design of polylogarithmic depth algorithms for approximately
solving packing and covering semidefinite programs (or positive SDPs for
short). This is a natural SDP generalization of the well-studied positive LP
problem.
Although positive LPs can be solved in polylogarithmic depth while using only
parallelizable iterations, the best known
positive SDP solvers due to Jain and Yao require parallelizable iterations. Several alternative solvers have
been proposed to reduce the exponents in the number of iterations. However, the
correctness of the convergence analyses in these works has been called into
question, as they both rely on algebraic monotonicity properties that do not
generalize to matrix algebra.
In this paper, we propose a very simple algorithm based on the optimization
framework proposed for LP solvers. Our algorithm only needs iterations, matching that of the best LP solver. To surmount
the obstacles encountered by previous approaches, our analysis requires a new
matrix inequality that extends Lieb-Thirring's inequality, and a
sign-consistent, randomized variant of the gradient truncation technique
proposed in
Epsilon-net method for optimizations over separable states
We give algorithms for the optimization problem: \max_\rho \ip{Q}{\rho},
where is a Hermitian matrix, and the variable is a bipartite {\em
separable} quantum state. This problem lies at the heart of several problems in
quantum computation and information, such as the complexity of QMA(2). While
the problem is NP-hard, our algorithms are better than brute force for several
instances of interest. In particular, they give PSPACE upper bounds on promise
problems admitting a QMA(2) protocol in which the verifier performs only
logarithmic number of elementary gate on both proofs, as well as the promise
problem of deciding if a bipartite local Hamiltonian has large or small ground
energy. For , our algorithm runs in time exponential in . While
the existence of such an algorithm was first proved recently by Brand{\~a}o,
Christandl and Yard [{\em Proceedings of the 43rd annual ACM Symposium on
Theory of Computation}, 343--352, 2011], our algorithm is conceptually simpler.Comment: 21 pages. Comments are welcom
No-Regret Learning and Equilibrium Computation in Quantum Games
As quantum processors advance, the emergence of large-scale decentralized
systems involving interacting quantum-enabled agents is on the horizon. Recent
research efforts have explored quantum versions of Nash and correlated
equilibria as solution concepts of strategic quantum interactions, but these
approaches did not directly connect to decentralized adaptive setups where
agents possess limited information. This paper delves into the dynamics of
quantum-enabled agents within decentralized systems that employ no-regret
algorithms to update their behaviors over time. Specifically, we investigate
two-player quantum zero-sum games and polymatrix quantum zero-sum games,
showing that no-regret algorithms converge to separable quantum Nash equilibria
in time-average. In the case of general multi-player quantum games, our work
leads to a novel solution concept, (separable) quantum coarse correlated
equilibria (QCCE), as the convergent outcome of the time-averaged behavior
no-regret algorithms, offering a natural solution concept for decentralized
quantum systems. Finally, we show that computing QCCEs can be formulated as a
semidefinite program and establish the existence of entangled (i.e.,
non-separable) QCCEs, which cannot be approached via the current paradigm of
no-regret learning