Conditions for advantageous quantum Bitcoin mining

Our aim is to determine the conditions for quantum computing technology to give rise to the 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 with a small hash rate compared to 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 ​min before measuring, which is surprising, as the network mines blocks every 10 ​min 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.