21 research outputs found
Computational depth complexity of measurement-based quantum computation
We prove that one-way quantum computations have the same computational power
as quantum circuits with unbounded fan-out. It demonstrates that the one-way
model is not only one of the most promising models of physical realisation, but
also a very powerful model of quantum computation. It confirms and completes
previous results which have pointed out, for some specific problems, a depth
separation between the one-way model and the quantum circuit model. Since
one-way model has the same computational power as unbounded quantum fan-out
circuits, the quantum Fourier transform can be approximated in constant depth
in the one-way model, and thus the factorisation can be done by a polytime
probabilistic classical algorithm which has access to a constant-depth one-way
quantum computer. The extra power of the one-way model, comparing with the
quantum circuit model, comes from its classical-quantum hybrid nature. We show
that this extra power is reduced to the capability to perform unbounded
classical parity gates in constant depth.Comment: 12 page
Bounds on the Power of Constant-Depth Quantum Circuits
We show that if a language is recognized within certain error bounds by
constant-depth quantum circuits over a finite family of gates, then it is
computable in (classical) polynomial time. In particular, our results imply
EQNC^0 is contained in P, where EQNC^0 is the constant-depth analog of the
class EQP. On the other hand, we adapt and extend ideas of Terhal and
DiVincenzo (quant-ph/0205133) to show that, for any family F of quantum gates
including Hadamard and CNOT gates, computing the acceptance probabilities of
depth-five circuits over F is just as hard as computing these probabilities for
circuits over F. In particular, this implies that NQNC^0 = NQACC = NQP = coC=P
where NQNC^0 is the constant-depth analog of the class NQP. This essentially
refutes a conjecture of Green et al. that NQACC is contained in TC^0
(quant-ph/0106017)
Small Depth Quantum Circuits
Small depth quantum circuits have proved to be unexpectedly powerful in comparison to their classical counterparts. We survey some of the recent work on this and present some open problems.National Security Agency; Advanced Research and Development Agency under Army Research Office (DAAD 19-02-1-0058
Binary Matroids and Quantum Probability Distributions
We characterise the probability distributions that arise from quantum
circuits all of whose gates commute, and show when these distributions can be
classically simulated efficiently. We consider also marginal distributions and
the computation of correlation coefficients, and draw connections between the
simulation of stabiliser circuits and the combinatorics of representable
matroids, as developed in the 1990s.Comment: 24 pages (inc appendix & refs
Universal Quantum Circuits
We define and construct efficient depth-universal and almost-size-universal
quantum circuits. Such circuits can be viewed as general-purpose simulators for
central classes of quantum circuits and can be used to capture the
computational power of the circuit class being simulated. For depth we
construct universal circuits whose depth is the same order as the circuits
being simulated. For size, there is a log factor blow-up in the universal
circuits constructed here. We prove that this construction is nearly optimal.Comment: 13 page
Depth-2 QAC circuits cannot simulate quantum parity
We show that the quantum parity gate on qubits cannot be cleanly
simulated by a quantum circuit with two layers of arbitrary C-SIGN gates of any
arity and arbitrary 1-qubit unitary gates, regardless of the number of allowed
ancilla qubits. This is the best known and first nontrivial separation between
the parity gate and circuits of this form. The same bounds also apply to the
quantum fanout gate. Our results are incomparable with those of Fang et al.
[3], which apply to any constant depth but require a sublinear number of
ancilla qubits on the simulating circuit.Comment: 21 pages, 2 figure
A New Lower Bound Technique for Quantum Circuits without Ancillae
We present a technique to derive depth lower bounds for quantum circuits. The technique is based on the observation that in circuits without ancillae, only a few input states can set all the control qubits of a Toffoli gate to 1. This can be used to selectively remove large Toffoli gates from a quantum circuit while keeping the cumulative error low. We use the technique to give another proof that parity cannot be computed by constant depth quantum circuits without ancillæ