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An Arbitrary Two-qubit Computation In 23 Elementary Gates
Quantum circuits currently constitute a dominant model for quantum
computation. Our work addresses the problem of constructing quantum circuits to
implement an arbitrary given quantum computation, in the special case of two
qubits. We pursue circuits without ancilla qubits and as small a number of
elementary quantum gates as possible. Our lower bound for worst-case optimal
two-qubit circuits calls for at least 17 gates: 15 one-qubit rotations and 2
CNOTs. To this end, we constructively prove a worst-case upper bound of 23
elementary gates, of which at most 4 (CNOT) entail multi-qubit interactions.
Our analysis shows that synthesis algorithms suggested in previous work,
although more general, entail much larger quantum circuits than ours in the
special case of two qubits. One such algorithm has a worst case of 61 gates of
which 18 may be CNOTs. Our techniques rely on the KAK decomposition from Lie
theory as well as the polar and spectral (symmetric Shur) matrix decompositions
from numerical analysis and operator theory. They are related to the canonical
decomposition of a two-qubit gate with respect to the ``magic basis'' of
phase-shifted Bell states, published previously. We further extend this
decomposition in terms of elementary gates for quantum computation.Comment: 18 pages, 7 figures. Version 2 gives correct credits for the GQC
"quantum compiler". Version 3 adds justification for our choice of elementary
gates and adds a comparison with classical library-less logic synthesis. It
adds acknowledgements and a new reference, adds full details about the 8-gate
decomposition of topC-V and stealthily fixes several minor inaccuracies.
NOTE: Using a new technique, we recently improved the lower bound to 18 gates
and (tada!) found a circuit decomposition that requires 18 gates or less.
This work will appear as a separate manuscrip
Quantum Physics and Computers
Recent theoretical results confirm that quantum theory provides the
possibility of new ways of performing efficient calculations. The most striking
example is the factoring problem. It has recently been shown that computers
that exploit quantum features could factor large composite integers. This task
is believed to be out of reach of classical computers as soon as the number of
digits in the number to factor exceeds a certain limit. The additional power of
quantum computers comes from the possibility of employing a superposition of
states, of following many distinct computation paths and of producing a final
output that depends on the interference of all of them. This ``quantum
parallelism'' outstrips by far any parallelism that can be thought of in
classical computation and is responsible for the ``exponential'' speed-up of
computation.
This is a non-technical (or at least not too technical) introduction to the
field of quantum computation. It does not cover very recent topics, such as
error-correction.Comment: 27 pages, LaTeX, 8 PostScript figures embedded. A bug in one of the
postscript files has been fixed. Reprints available from the author. The
files are also available from
http://eve.physics.ox.ac.uk/Articles/QC.Articles.htm
Quantum computers can search rapidly by using almost any transformation
A quantum computer has a clear advantage over a classical computer for
exhaustive search. The quantum mechanical algorithm for exhaustive search was
originally derived by using subtle properties of a particular quantum
mechanical operation called the Walsh-Hadamard (W-H) transform. This paper
shows that this algorithm can be implemented by replacing the W-H transform by
almost any quantum mechanical operation. This leads to several new applications
where it improves the number of steps by a square-root. It also broadens the
scope for implementation since it demonstrates quantum mechanical algorithms
that can readily adapt to available technology.Comment: This paper is an adapted version of quant-ph/9711043. It has been
modified to make it more readable for physicists. 9 pages, postscrip
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