50,404 research outputs found
Separating Quantum Communication and Approximate Rank
One of the best lower bound methods for the quantum communication complexity of a function H (with or without shared entanglement) is the logarithm of the approximate rank of the communication matrix of H. This measure is essentially equivalent to the approximate gamma-2 norm and generalized discrepancy, and subsumes several other lower bounds. All known lower bounds on quantum communication complexity in the general unbounded-round model can be shown via the logarithm of approximate rank, and it was an open problem to give any separation at all between quantum communication complexity and the logarithm of the approximate rank.
In this work we provide the first such separation: We exhibit a total function H with quantum communication complexity almost quadratically larger than the logarithm of its approximate rank. We construct H using the communication lookup function framework of Anshu et al. (FOCS 2016) based on the cheat sheet framework of Aaronson et al. (STOC 2016). From a starting function F, this framework defines a new function H=F_G. Our main technical result is a lower bound on the quantum communication complexity of F_G in terms of the discrepancy of F, which we do via quantum information theoretic arguments. We show the upper bound on the approximate rank of F_G by relating it to the Boolean circuit size of the starting function F
Energy-constrained LOCC-assisted quantum capacity of bosonic dephasing channel
We study the LOCC-assisted quantum capacity of bosonic dephasing channel with
energy constraint on input states. We start our analysis by focusing on the
energy-constrained squashed entanglement of the channel, which is an upper
bound for the energy-constrained LOCC-assisted quantum capacity. As computing
energy-constrained squashed entanglement of the channel is challenging due to a
double optimization (over the set of density matrices and the isometric
extensions of a squashing channel), we first derive an upper bound for it, and
then we discuss how tight that bound is for energy-constrained LOCC-assisted
quantum capacity of bosonic dephasing channel. We prove that the optimal input
state is diagonal in the Fock basis. Furthermore, we prove that for a generic
channel, the optimal squashing channel belongs to the set of symmetric quantum
Markov chain inducer (SQMCI) channels of the channel system-environment output,
provided that such a set is non-empty. With supporting arguments, we conjecture
that this is instead the case for the bosonic dephasing channel. Hence, for it
we analyze two explicit examples of squashing channels which are not SQMCI, but
are symmetric. Through them, we derive explicit upper and lower bounds for the
energy-constrained LOCC-assisted quantum capacity of the bosonic dephasing
channel in terms of its quantum capacity with different noise parameters. As
the difference between upper and lower bounds is at most of the order
, we conclude that the bounds are tight. Hence we provide a very good
estimation of the LOCC-assisted quantum capacity of the bosonic dephasing
channel
Randomized adaptive quantum state preparation
We develop an adaptive method for quantum state preparation that utilizes
randomness as an essential component and that does not require classical
optimization. Instead, a cost function is minimized to prepare a desired
quantum state through an adaptively constructed quantum circuit, where each
adaptive step is informed by feedback from gradient measurements in which the
associated tangent space directions are randomized. We provide theoretical
arguments and numerical evidence that convergence to the target state can be
achieved for almost all initial states. We investigate different randomization
procedures and develop lower bounds on the expected cost function change, which
allows for drawing connections to barren plateaus and for assessing the
applicability of the algorithm to large-scale problems
- …