656 research outputs found
A New Quantum Lower Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs
We give a new version of the adversary method for proving lower bounds on
quantum query algorithms. The new method is based on analyzing the eigenspace
structure of the problem at hand. We use it to prove a new and optimal strong
direct product theorem for 2-sided error quantum algorithms computing k
independent instances of a symmetric Boolean function: if the algorithm uses
significantly less than k times the number of queries needed for one instance
of the function, then its success probability is exponentially small in k. We
also use the polynomial method to prove a direct product theorem for 1-sided
error algorithms for k threshold functions with a stronger bound on the success
probability. Finally, we present a quantum algorithm for evaluating solutions
to systems of linear inequalities, and use our direct product theorems to show
that the time-space tradeoff of this algorithm is close to optimal.Comment: 16 pages LaTeX. Version 2: title changed, proofs significantly
cleaned up and made selfcontained. This version to appear in the proceedings
of the STOC 06 conferenc
A strong direct product theorem for quantum query complexity
We show that quantum query complexity satisfies a strong direct product
theorem. This means that computing copies of a function with less than
times the quantum queries needed to compute one copy of the function implies
that the overall success probability will be exponentially small in . For a
boolean function we also show an XOR lemma---computing the parity of
copies of with less than times the queries needed for one copy implies
that the advantage over random guessing will be exponentially small.
We do this by showing that the multiplicative adversary method, which
inherently satisfies a strong direct product theorem, is always at least as
large as the additive adversary method, which is known to characterize quantum
query complexity.Comment: V2: 19 pages (various additions and improvements, in particular:
improved parameters in the main theorems due to a finer analysis of the
output condition, and addition of an XOR lemma and a threshold direct product
theorem in the boolean case). V3: 19 pages (added grant information
Quantum and classical strong direct product theorems and optimal time-space tradeoffs
A strong direct product theorem says that if we want to compute
independent instances of a function, using less than times
the resources needed for one instance, then our overall success
probability will be exponentially small in .
We establish such theorems for the classical as well as quantum
query complexity of the OR-function. This implies slightly
weaker direct product results for all total functions.
We prove a similar result for quantum communication
protocols computing instances of the disjointness function.
Our direct product theorems imply a time-space tradeoff
T^2S=\Om{N^3} for sorting items on a quantum computer, which
is optimal up to polylog factors. They also give several tight
time-space and communication-space tradeoffs for the problems of
Boolean matrix-vector multiplication and matrix multiplication
Symmetry-assisted adversaries for quantum state generation
We introduce a new quantum adversary method to prove lower bounds on the
query complexity of the quantum state generation problem. This problem
encompasses both, the computation of partial or total functions and the
preparation of target quantum states. There has been hope for quite some time
that quantum state generation might be a route to tackle the {\sc Graph
Isomorphism} problem. We show that for the related problem of {\sc Index
Erasure} our method leads to a lower bound of which matches
an upper bound obtained via reduction to quantum search on elements. This
closes an open problem first raised by Shi [FOCS'02].
Our approach is based on two ideas: (i) on the one hand we generalize the
known additive and multiplicative adversary methods to the case of quantum
state generation, (ii) on the other hand we show how the symmetries of the
underlying problem can be leveraged for the design of optimal adversary
matrices and dramatically simplify the computation of adversary bounds. Taken
together, these two ideas give the new result for {\sc Index Erasure} by using
the representation theory of the symmetric group. Also, the method can lead to
lower bounds even for small success probability, contrary to the standard
adversary method. Furthermore, we answer an open question due to \v{S}palek
[CCC'08] by showing that the multiplicative version of the adversary method is
stronger than the additive one for any problem. Finally, we prove that the
multiplicative bound satisfies a strong direct product theorem, extending a
result by \v{S}palek to quantum state generation problems.Comment: 35 pages, 5 figure
Randomized vs. Deterministic Separation in Time-Space Tradeoffs of Multi-Output Functions
We prove the first polynomial separation between randomized and deterministic
time-space tradeoffs of multi-output functions. In particular, we present a
total function that on the input of elements in , outputs
elements, such that: (1) There exists a randomized oblivious algorithm with
space , time and one-way access to randomness, that
computes the function with probability ; (2) Any deterministic
oblivious branching program with space and time that computes the
function must satisfy . This implies that
logspace randomized algorithms for multi-output functions cannot be black-box
derandomized without an overhead in time.
Since previously all the polynomial time-space tradeoffs of multi-output
functions are proved via the Borodin-Cook method, which is a probabilistic
method that inherently gives the same lower bound for randomized and
deterministic branching programs, our lower bound proof is intrinsically
different from previous works. We also examine other natural candidates for
proving such separations, and show that any polynomial separation for these
problems would resolve the long-standing open problem of proving
time lower bound for decision problems with
space.Comment: 15 page
Quantum Reverse Shannon Theorem
Dual to the usual noisy channel coding problem, where a noisy (classical or
quantum) channel is used to simulate a noiseless one, reverse Shannon theorems
concern the use of noiseless channels to simulate noisy ones, and more
generally the use of one noisy channel to simulate another. For channels of
nonzero capacity, this simulation is always possible, but for it to be
efficient, auxiliary resources of the proper kind and amount are generally
required. In the classical case, shared randomness between sender and receiver
is a sufficient auxiliary resource, regardless of the nature of the source, but
in the quantum case the requisite auxiliary resources for efficient simulation
depend on both the channel being simulated, and the source from which the
channel inputs are coming. For tensor power sources (the quantum generalization
of classical IID sources), entanglement in the form of standard ebits
(maximally entangled pairs of qubits) is sufficient, but for general sources,
which may be arbitrarily correlated or entangled across channel inputs,
additional resources, such as entanglement-embezzling states or backward
communication, are generally needed. Combining existing and new results, we
establish the amounts of communication and auxiliary resources needed in both
the classical and quantum cases, the tradeoffs among them, and the loss of
simulation efficiency when auxiliary resources are absent or insufficient. In
particular we find a new single-letter expression for the excess forward
communication cost of coherent feedback simulations of quantum channels (i.e.
simulations in which the sender retains what would escape into the environment
in an ordinary simulation), on non-tensor-power sources in the presence of
unlimited ebits but no other auxiliary resource. Our results on tensor power
sources establish a strong converse to the entanglement-assisted capacity
theorem.Comment: 35 pages, to appear in IEEE-IT. v2 has a fixed proof of the Clueless
Eve result, a new single-letter formula for the "spread deficit", better
error scaling, and an improved strong converse. v3 and v4 each make small
improvements to the presentation and add references. v5 fixes broken
reference
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