1,688 research outputs found
Quantum Networks for Concentrating Entanglement
If two parties, Alice and Bob, share some number, n, of partially entangled
pairs of qubits, then it is possible for them to concentrate these pairs into
some smaller number of maximally entangled states. We present a simplified
version of the algorithm for such entanglement concentration, and we describe
efficient networks for implementing these operations.Comment: 15 pages, 2 figure
Optimal Direct Sum Results for Deterministic and Randomized Decision Tree Complexity
A Direct Sum Theorem holds in a model of computation, when solving some k
input instances together is k times as expensive as solving one. We show that
Direct Sum Theorems hold in the models of deterministic and randomized decision
trees for all relations. We also note that a near optimal Direct Sum Theorem
holds for quantum decision trees for boolean functions.Comment: 7 page
Quantum Bounded Query Complexity
We combine the classical notions and techniques for bounded query classes
with those developed in quantum computing. We give strong evidence that quantum
queries to an oracle in the class NP does indeed reduce the query complexity of
decision problems. Under traditional complexity assumptions, we obtain an
exponential speedup between the quantum and the classical query complexity of
function classes.
For decision problems and function classes we obtain the following results: o
P_||^NP[2k] is included in EQP_||^NP[k] o P_||^NP[2^(k+1)-2] is included in
EQP^NP[k] o FP_||^NP[2^(k+1)-2] is included in FEQP^NP[2k] o FP_||^NP is
included in FEQP^NP[O(log n)] For sets A that are many-one complete for PSPACE
or EXP we show that FP^A is included in FEQP^A[1]. Sets A that are many-one
complete for PP have the property that FP_||^A is included in FEQP^A[1]. In
general we prove that for any set A there is a set X such that FP^A is included
in FEQP^X[1], establishing that no set is superterse in the quantum setting.Comment: 11 pages LaTeX2e, no figures, accepted for CoCo'9
Quantum Zero-Error Algorithms Cannot be Composed
We exhibit two black-box problems, both of which have an efficient quantum
algorithm with zero-error, yet whose composition does not have an efficient
quantum algorithm with zero-error. This shows that quantum zero-error
algorithms cannot be composed. In oracle terms, we give a relativized world
where ZQP^{ZQP}\=ZQP, while classically we always have ZPP^{ZPP}=ZPP.Comment: 7 pages LaTeX. 2nd version slightly rewritte
Communication Complexity Lower Bounds by Polynomials
The quantum version of communication complexity allows the two communicating
parties to exchange qubits and/or to make use of prior entanglement (shared
EPR-pairs). Some lower bound techniques are available for qubit communication
complexity, but except for the inner product function, no bounds are known for
the model with unlimited prior entanglement. We show that the log-rank lower
bound extends to the strongest model (qubit communication + unlimited prior
entanglement). By relating the rank of the communication matrix to properties
of polynomials, we are able to derive some strong bounds for exact protocols.
In particular, we prove both the "log-rank conjecture" and the polynomial
equivalence of quantum and classical communication complexity for various
classes of functions. We also derive some weaker bounds for bounded-error
quantum protocols.Comment: 16 pages LaTeX, no figures. 2nd version: rewritten and some results
adde
Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication
We study nondeterministic multiparty quantum communication with a quantum
generalization of broadcasts. We show that, with number-in-hand classical
inputs, the communication complexity of a Boolean function in this
communication model equals the logarithm of the support rank of the
corresponding tensor, whereas the approximation complexity in this model equals
the logarithm of the border support rank. This characterisation allows us to
prove a log-rank conjecture posed by Villagra et al. for nondeterministic
multiparty quantum communication with message-passing.
The support rank characterization of the communication model connects quantum
communication complexity intimately to the theory of asymptotic entanglement
transformation and algebraic complexity theory. In this context, we introduce
the graphwise equality problem. For a cycle graph, the complexity of this
communication problem is closely related to the complexity of the computational
problem of multiplying matrices, or more precisely, it equals the logarithm of
the asymptotic support rank of the iterated matrix multiplication tensor. We
employ Strassen's laser method to show that asymptotically there exist
nontrivial protocols for every odd-player cyclic equality problem. We exhibit
an efficient protocol for the 5-player problem for small inputs, and we show
how Young flattenings yield nontrivial complexity lower bounds
A generalized Grothendieck inequality and entanglement in XOR games
Suppose Alice and Bob make local two-outcome measurements on a shared
entangled state. For any d, we show that there are correlations that can only
be reproduced if the local dimension is at least d. This resolves a conjecture
of Brunner et al. Phys. Rev. Lett. 100, 210503 (2008) and establishes that the
amount of entanglement required to maximally violate a Bell inequality must
depend on the number of measurement settings, not just the number of
measurement outcomes. We prove this result by establishing the first lower
bounds on a new generalization of Grothendieck's constant.Comment: Version submitted to QIP on 10-20-08. See also arxiv:0812.1572 for
related results, obtained independentl
Lower Bounds for Quantum Search and Derandomization
We prove lower bounds on the error probability of a quantum algorithm for
searching through an unordered list of N items, as a function of the number T
of queries it makes. In particular, if T=O(sqrt{N}) then the error is lower
bounded by a constant. If we want error <1/2^N then we need T=Omega(N) queries.
We apply this to show that a quantum computer cannot do much better than a
classical computer when amplifying the success probability of an RP-machine. A
classical computer can achieve error <=1/2^k using k applications of the
RP-machine, a quantum computer still needs at least ck applications for this
(when treating the machine as a black-box), where c>0 is a constant independent
of k. Furthermore, we prove a lower bound of Omega(sqrt{log N}/loglog N)
queries for quantum bounded-error search of an ordered list of N items.Comment: 12 pages LaTeX. Submitted to CCC'99 (formerly Structures
Substituting a qubit for an arbitrarily large number of classical bits
We show that a qubit can be used to substitute for an arbitrarily large
number of classical bits. We consider a physical system S interacting locally
with a classical field phi(x) as it travels directly from point A to point B.
The field has the property that its integrated value is an integer multiple of
some constant. The problem is to determine whether the integer is odd or even.
This task can be performed perfectly if S is a qubit. On the otherhand, if S is
a classical system then we show that it must carry an arbitrarily large amount
of classical information. We identify the physical reason for such a huge
quantum advantage, and show that it also implies a large difference between the
size of quantum and classical memories necessary for some computations. We also
present a simple proof that no finite amount of one-way classical communication
can perfectly simulate the effect of quantum entanglement.Comment: 8 pages, LaTeX, no figures. v2: added result on entanglement
simulation with classical communication; v3: minor correction to main proof,
change of title, added referenc
Better Non-Local Games from Hidden Matching
We construct a non-locality game that can be won with certainty by a quantum
strategy using log n shared EPR-pairs, while any classical strategy has winning
probability at most 1/2+O(log n/sqrt{n}). This improves upon a recent result of
Junge et al. in a number of ways.Comment: 11 pages, late
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