  Quantum key distribution (QKD) systems provide a method for two users to
exchange a provably secure key. Synchronizing the users' clocks is an essential
step before a secure key can be distilled. Qubit-based synchronization
protocols directly use the transmitted quantum states to achieve
synchronization and thus avoid the need for additional classical
synchronization hardware. Previous qubit-based synchronization protocols
sacrifice secure key either directly or indirectly, and all known qubit-based
synchronization protocols do not efficiently use all publicly available
information published by the users. Here, we introduce a Bayesian probabilistic
algorithm that incorporates all published information to efficiently find the
clock offset without sacrificing any secure key. Additionally, the output of
the algorithm is a probability, which allows us to quantify our confidence in
the synchronization. For demonstration purposes, we present a model system with
accompanying simulations of an efficient three-state BB84 prepare-and-measure
protocol with decoy states. We use our algorithm to exploit the correlations
between Alice's published basis and mean photon number choices and Bob's
measurement outcomes to probabilistically determine the most likely clock
offset. We find that we can achieve a 95 percent synchronization confidence in
only 4,140 communication bin widths, meaning we can tolerate clock drift
approaching 1 part in 4,140 in this example when simulating this system with a
dark count probability per communication bin width of 8e-4 and a received mean
photon number of 0.01.
