3,480 research outputs found
Quantum de Finetti Theorems under Local Measurements with Applications
Quantum de Finetti theorems are a useful tool in the study of correlations in
quantum multipartite states. In this paper we prove two new quantum de Finetti
theorems, both showing that under tests formed by local measurements one can
get a much improved error dependence on the dimension of the subsystems. We
also obtain similar results for non-signaling probability distributions. We
give the following applications of the results:
We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the
exponential time hypothesis.
We show that the maximum winning probability of free games can be estimated
in polynomial time by linear programming. We also show that 3-SAT with m
variables can be reduced to obtaining a constant error approximation of the
maximum winning probability under entangled strategies of O(m^{1/2})-player
one-round non-local games, in which the players communicate O(m^{1/2}) bits all
together.
We show that the optimization of certain polynomials over the hypersphere can
be performed in quasipolynomial time in the number of variables n by
considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy
of semidefinite programs. As an application to entanglement theory, we find a
quasipolynomial-time algorithm for deciding multipartite separability.
We consider a result due to Aaronson -- showing that given an unknown n qubit
state one can perform tomography that works well for most observables by
measuring only O(n) independent and identically distributed (i.i.d.) copies of
the state -- and relax the assumption of having i.i.d copies of the state to
merely the ability to select subsystems at random from a quantum multipartite
state.
The proofs of the new quantum de Finetti theorems are based on information
theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor
improvements. v3: added some explanations, mostly about Theorem 1 and
Conjecture 5. STOC version. v4, v5. small improvements and fixe
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
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
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
The next few years will be exciting as prototype universal quantum processors
emerge, enabling implementation of a wider variety of algorithms. Of particular
interest are quantum heuristics, which require experimentation on quantum
hardware for their evaluation, and which have the potential to significantly
expand the breadth of quantum computing applications. A leading candidate is
Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates
between applying a cost-function-based Hamiltonian and a mixing Hamiltonian.
Here, we extend this framework to allow alternation between more general
families of operators. The essence of this extension, the Quantum Alternating
Operator Ansatz, is the consideration of general parametrized families of
unitaries rather than only those corresponding to the time-evolution under a
fixed local Hamiltonian for a time specified by the parameter. This ansatz
supports the representation of a larger, and potentially more useful, set of
states than the original formulation, with potential long-term impact on a
broad array of application areas. For cases that call for mixing only within a
desired subspace, refocusing on unitaries rather than Hamiltonians enables more
efficiently implementable mixers than was possible in the original framework.
Such mixers are particularly useful for optimization problems with hard
constraints that must always be satisfied, defining a feasible subspace, and
soft constraints whose violation we wish to minimize. More efficient
implementation enables earlier experimental exploration of an alternating
operator approach to a wide variety of approximate optimization, exact
optimization, and sampling problems. Here, we introduce the Quantum Alternating
Operator Ansatz, lay out design criteria for mixing operators, detail mappings
for eight problems, and provide brief descriptions of mappings for diverse
problems.Comment: 51 pages, 2 figures. Revised to match journal pape
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
Testing product states, quantum Merlin-Arthur games and tensor optimisation
We give a test that can distinguish efficiently between product states of n
quantum systems and states which are far from product. If applied to a state
psi whose maximum overlap with a product state is 1-epsilon, the test passes
with probability 1-Theta(epsilon), regardless of n or the local dimensions of
the individual systems. The test uses two copies of psi. We prove correctness
of this test as a special case of a more general result regarding stability of
maximum output purity of the depolarising channel. A key application of the
test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain
several structural results that had been previously conjectured, including the
fact that efficient soundness amplification is possible and that two Merlins
can simulate many Merlins: QMA(k)=QMA(2) for k>=2. Building on a previous
result of Aaronson et al, this implies that there is an efficient quantum
algorithm to verify 3-SAT with constant soundness, given two unentangled proofs
of O(sqrt(n) polylog(n)) qubits. We also show how QMA(2) with log-sized proofs
is equivalent to a large number of problems, some related to quantum
information (such as testing separability of mixed states) as well as problems
without any apparent connection to quantum mechanics (such as computing
injective tensor norms of 3-index tensors). As a consequence, we obtain many
hardness-of-approximation results, as well as potential algorithmic
applications of methods for approximating QMA(2) acceptance probabilities.
Finally, our test can also be used to construct an efficient test for
determining whether a unitary operator is a tensor product, which is a
generalisation of classical linearity testing.Comment: 44 pages, 1 figure, 7 appendices; v6: added references, rearranged
sections, added discussion of connections to classical CS. Final version to
appear in J of the AC
Quantum XOR Games
We introduce quantum XOR games, a model of two-player one-round games that
extends the model of XOR games by allowing the referee's questions to the
players to be quantum states. We give examples showing that quantum XOR games
exhibit a wide range of behaviors that are known not to exist for standard XOR
games, such as cases in which the use of entanglement leads to an arbitrarily
large advantage over the use of no entanglement. By invoking two deep
extensions of Grothendieck's inequality, we present an efficient algorithm that
gives a constant-factor approximation to the best performance players can
obtain in a given game, both in case they have no shared entanglement and in
case they share unlimited entanglement. As a byproduct of the algorithm we
prove some additional interesting properties of quantum XOR games, such as the
fact that sharing a maximally entangled state of arbitrary dimension gives only
a small advantage over having no entanglement at all.Comment: 43 page
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
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