Implementing the quantum fanout operation with simple pairwise interactions (pp1081-1090)
Stephen Fenner and Rabins Wosti
doi:
https://doi.org/10.26421/QIC23.13-14-1
Abstracts:
It has been shown that, for even $n$,
evolving $n$
qubits
according to a Hamiltonian that is the sum of pairwise interactions
between the particles, can be used to exactly implement an
$(n+1)$-qubit
fanout
gate using a particular constant-depth circuit~[\href{https://arXiv.org/abs/quant-ph/0309163}{arXiv:quant-ph/0309163}].
However, the coupling coefficients in the Hamiltonian considered in that
paper are assumed to be all equal. In this paper, we generalize these
results and show that for all $n$,
including odd $n$,
one can exactly implement an $(n+1)$-qubit
parity gate and hence, equivalently in constant depth an
$(n+1)$-qubit
fanout
gate, using a similar Hamiltonian but with unequal couplings, and we
give an exact characterization of the constraints that the couplings
must satisfy in order for them to be adequate to implement
fanout
via the same circuit.}{In particular, we show the following:Letting
$J_{ij}$
be the coupling strength between the
$\ordth{i}$
and $\ordth{j}$
qubits,
the set of couplings $\{J_{ij}\}$
is adequate to implement
fanout
via the circuit above if and only if there exists
$J>0$
such that 1. each
$J_{ij}$
is an odd integer multiple of $J$,
and 2. for each
$i$,
there are an even number of $j\ne i$
such that $J_{ij}/J \equiv
3\pmod{4}$.
Key Words:
constant-depth quantum circuit;
quantum
fanout gate; Hamiltonian; pairwise
interactions; spin-exchange interaction; Heisenberg interaction; modular
arithmetic |