3,154 research outputs found
Quantum and Classical Tradeoffs
We propose an approach for quantifying a quantum circuit's quantumness as a
means to understand the nature of quantum algorithmic speedups. Since quantum
gates that do not preserve the computational basis are necessary for achieving
quantum speedups, it appears natural to define the quantumness of a quantum
circuit using the number of such gates. Intuitively, a reduction in the
quantumness requires an increase in the amount of classical computation, hence
giving a ``quantum and classical tradeoff''.
In this paper we present two results on this direction. The first gives an
asymptotic answer to the question: ``what is the minimum number of
non-basis-preserving gates required to generate a good approximation to a given
state''. This question is the quantum analogy of the following classical
question, ``how many fair coins are needed to generate a given probability
distribution'', which was studied and resolved by Knuth and Yao in 1976. Our
second result shows that any quantum algorithm that solves Grover's Problem of
size n using k queries and l levels of non-basis-preserving gates must have
k*l=\Omega(n)
Quantum-classical tradeoffs and multi-controlled quantum gate decompositions in variational algorithms
Quantum algorithms for unconstrained optimization problems, such as the
Quantum Approximate Optimization Algorithm (QAOA), have been proposed as
interesting near-term algorithms which operate under a hybrid quantum-classical
execution model. Recent work has shown that the QAOA can also be applied to
constrained combinatorial optimization problems by incorporating the problem
constraints within the design of the variational ansatz - often resulting in
quantum circuits containing many multi-controlled gate operations. This paper
investigates potential resource tradeoffs for the QAOA when applied to the
particular constrained optimization problem of Maximum Independent Set. We
consider three variants of the QAOA which make different tradeoffs between the
number of classical parameters, quantum gates, and iterations of classical
optimization. We also study the quantum cost of decomposing the QAOA circuits
on hardware which may support different qubit technologies and native gate
sets, and compare the different algorithms using the gate decomposition score
which combines the fidelity of the gate operations with the efficiency of the
decomposition into a single metric. We find that all three QAOA variants can
attain similar performance but the classical and quantum resource costs may
vary greatly between them.Comment: 17 pages, 8 figures, 5 table
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
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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