What It Takes for Quantum Computers to Mine Bitcoin Efficiently

Table of Links Abstract and I. Introduction A. Quantum Bitcoin Mining B. Our Contribution C. Comparison with Related Works D. Conventions II. Background A. Bitcoin Basics B. Bitcoin Security C. Grover’s Search Algorithm D. Quantum Attacks III. Approach A. Algorithm B. Markov Chain C. Assumptions and Approximations IV. Results A. Probability of Success B. Performance Measures C. Example Application V. Discussion, Acknowledgments, and References C. Example Application In this subsection we demonstrate an application of our results by calculating numerical estimates for a quantum miner’s performance. These calculation demonstrate how to use our results to evaluate the feasibility of a given quantum computer for Bitcoin mining. We give estimates for both effective hash rate and energy efficiency required for advantegous mining. The quantum computer we consider is described in [4]. First, the computer has a gate speed of 66.7 MHz, which is the speed achievable on current devices. Aggarwal et al. also show that a single Grover iteration (for the Bitcoin search problem) would take a circuit of depth 297784 to perform if we assume no overhead from error correction. We make this assumption for simplicity as the error correction overhead has a non-trivial relationship with the number of sequential Grover iterations used. For this quantum computer, This means the quantum computer would comprise only a small fraction of the mining power of the Bitcoin network. Finally, we calculate the effective hash rate to be Efficiency Requirement Next we turn to the efficiency the quantum computer would need to outperform a classical computer at Bitcoin mining. Recall the condition for this outperformance is given by Eq. 63. Plugging into this equation we find the condition for advantageous quantum mining. If we instead use Eq. 69 which employs an additional approximation, then we get same result, up to three significant figures. Authors: (1) Robert R. Nerem, Institute for Quantum Science and Technology, University of Calgary, Alberta T2N 1N4, Canada (riley.nerem@gmail.com); (2) Daya R. Gaur, Department of Mathematics and Computer Science, University of Lethbridge, Alberta T1K 3M4, Canada. This paper is available on arxiv under CC BY 4.0 DEED license. Table of Links Abstract and I. Introduction Abstract and I. Introduction Abstract and I. Introduction A. Quantum Bitcoin Mining A. Quantum Bitcoin Mining B. Our Contribution B. Our Contribution C. Comparison with Related Works C. Comparison with Related Works D. Conventions D. Conventions II. Background II. Background A. Bitcoin Basics A. Bitcoin Basics B. Bitcoin Security B. Bitcoin Security C. Grover’s Search Algorithm C. Grover’s Search Algorithm D. Quantum Attacks D. Quantum Attacks III. Approach III. Approach III. Approach A. Algorithm A. Algorithm B. Markov Chain B. Markov Chain C. Assumptions and Approximations C. Assumptions and Approximations IV. Results IV. Results A. Probability of Success A. Probability of Success B. Performance Measures B. Performance Measures C. Example Application C. Example Application V. Discussion, Acknowledgments, and References V. Discussion, Acknowledgments, and References V. Discussion, Acknowledgments, and References C. Example Application In this subsection we demonstrate an application of our results by calculating numerical estimates for a quantum miner’s performance. These calculation demonstrate how to use our results to evaluate the feasibility of a given quantum computer for Bitcoin mining. We give estimates for both effective hash rate and energy efficiency required for advantegous mining. The quantum computer we consider is described in [4]. First, the computer has a gate speed of 66.7 MHz, which is the speed achievable on current devices. Aggarwal et al. also show that a single Grover iteration (for the Bitcoin search problem) would take a circuit of depth 297784 to perform if we assume no overhead from error correction. We make this assumption for simplicity as the error correction overhead has a non-trivial relationship with the number of sequential Grover iterations used. For this quantum computer, This means the quantum computer would comprise only a small fraction of the mining power of the Bitcoin network. Finally, we calculate the effective hash rate to be Efficiency Requirement Next we turn to the efficiency the quantum computer would need to outperform a classical computer at Bitcoin mining. Recall the condition for this outperformance is given by Eq. 63. Plugging into this equation we find the condition Efficiency Requirement for advantageous quantum mining. If we instead use Eq. 69 which employs an additional approximation, then we get same result, up to three significant figures. Authors: (1) Robert R. Nerem, Institute for Quantum Science and Technology, University of Calgary, Alberta T2N 1N4, Canada (riley.nerem@gmail.com); (2) Daya R. Gaur, Department of Mathematics and Computer Science, University of Lethbridge, Alberta T1K 3M4, Canada. Authors: Authors: (1) Robert R. Nerem, Institute for Quantum Science and Technology, University of Calgary, Alberta T2N 1N4, Canada (riley.nerem@gmail.com); (2) Daya R. Gaur, Department of Mathematics and Computer Science, University of Lethbridge, Alberta T1K 3M4, Canada. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv available on arxiv