  Our aim is to determine conditions for quantum computing technology to give
rise to security risks associated with quantum Bitcoin mining. Specifically, we
determine the speed and energy efficiency a quantum computer needs to offer an
advantage over classical mining. We analyze the setting in which the Bitcoin
network is entirely classical except for a single quantum miner who has small
hash rate compared to that of the network. We develop a closed-form
approximation for the probability that the quantum miner successfully mines a
block, with this probability dependent on the number of Grover iterations the
quantum miner applies before making a measurement. Next, we show that, for a
quantum miner that is "peaceful", this success probability is maximized if the
quantum miner applies Grover iterations for 16 minutes before measuring, which
is surprising as the network mines blocks every 10 minutes on average. Using
this optimal mining procedure, we show that the quantum miner outperforms a
classical computer in efficiency (cost per block) if the condition $Q < Crb$ is
satisfied, where $Q$ is the cost of a Grover iteration, $C$ is the cost of a
classical hash, $r$ is the quantum miner's speed in Grover iterations per
second, and $b$ is a factor that attains its maximum if the quantum miner uses
our optimal mining procedure. This condition lays the foundation for
determining when quantum mining, and the known security risks associated with
it, will arise.
