100 research outputs found
Counterfactual Computation
Suppose that we are given a quantum computer programmed ready to perform a
computation if it is switched on. Counterfactual computation is a process by
which the result of the computation may be learnt without actually running the
computer. Such processes are possible within quantum physics and to achieve
this effect, a computer embodying the possibility of running the computation
must be available, even though the computation is, in fact, not run. We study
the possibilities and limitations of general protocols for the counterfactual
computation of decision problems (where the result r is either 0 or 1). If p(r)
denotes the probability of learning the result r ``for free'' in a protocol
then one might hope to design a protocol which simultaneously has large p(0)
and p(1). However we prove that p(0)+p(1) never exceeds 1 in any protocol and
we derive further constraints on p(0) and p(1) in terms of N, the number of
times that the computer is not run. In particular we show that any protocol
with p(0)+p(1)=1-epsilon must have N tending to infinity as epsilon tends to 0.
These general results are illustrated with some explicit protocols for
counterfactual computation. We show that "interaction-free" measurements can be
regarded as counterfactual computations, and our results then imply that N must
be large if the probability of interaction is to be close to zero. Finally, we
consider some ways in which our formulation of counterfactual computation can
be generalised.Comment: 19 pages. LaTex, 2 figures. Revised version has some new sections and
expanded explanation
Quantum Effects in Algorithms
We discuss some seemingly paradoxical yet valid effects of quantum physics in
information processing. Firstly, we argue that the act of ``doing nothing'' on
part of an entangled quantum system is a highly non-trivial operation and that
it is the essential ingredient underlying the computational speedup in the
known quantum algorithms. Secondly, we show that the watched pot effect of
quantum measurement theory gives the following novel computational possibility:
suppose that we have a quantum computer with an on/off switch, programmed ready
to solve a decision problem. Then (in certain circumstances) the mere fact that
the computer would have given the answer if it were run, is enough for us to
learn the answer, even though the computer is in fact not run.Comment: 10 pages, Latex. For Proceedings of First NASA International
Conference on Quantum Computation and Quantum Communication (Palm Springs,
February 1998
Universal quantum information compression and degrees of prior knowledge
We describe a universal information compression scheme that compresses any
pure quantum i.i.d. source asymptotically to its von Neumann entropy, with no
prior knowledge of the structure of the source. We introduce a diagonalisation
procedure that enables any classical compression algorithm to be utilised in a
quantum context. Our scheme is then based on the corresponding quantum
translation of the classical Lempel-Ziv algorithm. Our methods lead to a
conceptually simple way of estimating the entropy of a source in terms of the
measurement of an associated length parameter while maintaining high fidelity
for long blocks. As a by-product we also estimate the eigenbasis of the source.
Since our scheme is based on the Lempel-Ziv method, it can be applied also to
target sequences that are not i.i.d.Comment: 17 pages, no figures. A preliminary version of this work was
presented at EQIS '02, Tokyo, September 200
Matchgates and classical simulation of quantum circuits
Let G(A,B) denote the 2-qubit gate which acts as the 1-qubit SU(2) gates A
and B in the even and odd parity subspaces respectively, of two qubits. Using a
Clifford algebra formalism we show that arbitrary uniform families of circuits
of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines,
can be classically efficiently simulated. This reproduces a result originally
proved by Valiant using his matchgate formalism, and subsequently related by
others to free fermionic physics. We further show that if the n.n. condition is
slightly relaxed, to allowing the same gates to act only on n.n. and next-n.n.
qubit lines, then the resulting circuits can efficiently perform universal
quantum computation. From this point of view, the gap between efficient
classical and quantum computational power is bridged by a very modest use of a
seemingly innocuous resource (qubit swapping). We also extend the simulation
result above in various ways. In particular, by exploiting properties of
Clifford operations in conjunction with the Jordan-Wigner representation of a
Clifford algebra, we show how one may generalise the simulation result above to
provide further classes of classically efficiently simulatable quantum
circuits, which we call Gaussian quantum circuits.Comment: 18 pages, 2 figure
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