171 research outputs found
Estimating Jones and HOMFLY polynomials with One Clean Qubit
The Jones and HOMFLY polynomials are link invariants with close connections
to quantum computing. It was recently shown that finding a certain
approximation to the Jones polynomial of the trace closure of a braid at the
fifth root of unity is a complete problem for the one clean qubit complexity
class. This is the class of problems solvable in polynomial time on a quantum
computer acting on an initial state in which one qubit is pure and the rest are
maximally mixed. Here we generalize this result by showing that one clean qubit
computers can efficiently approximate the Jones and single-variable HOMFLY
polynomials of the trace closure of a braid at any root of unity.Comment: 22 pages, 11 figures, revised in response to referee comment
Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit
The Turaev-Viro invariants are scalar topological invariants of
three-dimensional manifolds. Here we show that the problem of estimating the
Fibonacci version of the Turaev-Viro invariant of a mapping torus is a complete
problem for the one clean qubit complexity class (DQC1). This complements a
previous result showing that estimating the Turaev-Viro invariant for arbitrary
manifolds presented as Heegaard splittings is a complete problem for the
standard quantum computation model (BQP). We also discuss a beautiful analogy
between these results and previously known results on the computational
complexity of approximating the Jones polynomial.Comment: 16 pages, 8 figures, presented at TQC '11. Added reference
Experimental approximation of the Jones polynomial with DQC1
We present experimental results approximating the Jones polynomial using 4
qubits in a liquid state nuclear magnetic resonance quantum information
processor. This is the first experimental implementation of a complete problem
for the deterministic quantum computation with one quantum bit model of quantum
computation, which uses a single qubit accompanied by a register of completely
random states. The Jones polynomial is a knot invariant that is important not
only to knot theory, but also to statistical mechanics and quantum field
theory. The implemented algorithm is a modification of the algorithm developed
by Shor and Jordan suitable for implementation in NMR. These experimental
results show that for the restricted case of knots whose braid representations
have four strands and exactly three crossings, identifying distinct knots is
possible 91% of the time.Comment: 5 figures. Version 2 changes: published version, minor errors
corrected, slight changes to improve readabilit
Power of Quantum Computation with Few Clean Qubits
This paper investigates the power of polynomial-time quantum computation in
which only a very limited number of qubits are initially clean in the |0>
state, and all the remaining qubits are initially in the totally mixed state.
No initializations of qubits are allowed during the computation, nor
intermediate measurements. The main results of this paper are unexpectedly
strong error-reducible properties of such quantum computations. It is proved
that any problem solvable by a polynomial-time quantum computation with
one-sided bounded error that uses logarithmically many clean qubits can also be
solvable with exponentially small one-sided error using just two clean qubits,
and with polynomially small one-sided error using just one clean qubit. It is
further proved in the case of two-sided bounded error that any problem solvable
by such a computation with a constant gap between completeness and soundness
using logarithmically many clean qubits can also be solvable with exponentially
small two-sided error using just two clean qubits. If only one clean qubit is
available, the problem is again still solvable with exponentially small error
in one of the completeness and soundness and polynomially small error in the
other. As an immediate consequence of the above result for the two-sided-error
case, it follows that the TRACE ESTIMATION problem defined with fixed constant
threshold parameters is complete for the classes of problems solvable by
polynomial-time quantum computations with completeness 2/3 and soundness 1/3
using logarithmically many clean qubits and just one clean qubit. The
techniques used for proving the error-reduction results may be of independent
interest in themselves, and one of the technical tools can also be used to show
the hardness of weak classical simulations of one-clean-qubit computations
(i.e., DQC1 computations).Comment: 44 pages + cover page; the results in Section 8 are overlapping with
the main results in arXiv:1409.677
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