Abstract
We investigate how hardware specifications can impact the final run time and the required number of physical qubits to achieve a quantum advantage in the fault tolerant regime. Within a particular time frame, both the code cycle time and the number of achievable physical qubits may vary by orders of magnitude between different quantum hardware designs. We start with logical resource requirements corresponding to a quantum advantage for a particular chemistry application, simulating the FeMo-co molecule, and explore to what extent slower code cycle times can be mitigated by using additional qubits. We show that in certain situations, architectures with considerably slower code cycle times will still be able to reach desirable run times, provided enough physical qubits are available. We utilize various space and time optimization strategies that have been previously considered within the field of error-correcting surface codes. In particular, we compare two distinct methods of parallelization: Game of Surface Code's Units and AutoCCZ factories. Finally, we calculate the number of physical qubits required to break the 256-bit elliptic curve encryption of keys in the Bitcoin network within the small available time frame in which it would actually pose a threat to do so. It would require 317 × 106 physical qubits to break the encryption within one hour using the surface code, a code cycle time of 1 μs, a reaction time of 10 μs, and a physical gate error of 10−3. To instead break the encryption within one day, it would require 13 × 106 physical qubits.
Highlights
With the advent of quantum computers, the race to a quantum computational advantage has gained serious traction in both the academic and commercial sectors
Within a particular time frame, both the code cycle time and the number of achievable physical qubits may vary by orders of magnitude between different quantum hardware designs
To calculate the results presented we use various surface code strategies, including the Game of Surface Codes scheme, which uses units to parallelize layers of T gates[18] and AutoCCZ factories,[31,32] which are both highlighted in Sec
Summary
With the advent of quantum computers, the race to a quantum computational advantage has gained serious traction in both the academic and commercial sectors. Entanglement distribution may be a viable method of enabling distant connectivity for large-scale devices with limited physical connectivity in the limit of large reserves of quantum memory.[12] Higher dimensional error correction codes can have access to a greater range of transversal gates,[13,14] which may considerably improve final runtimes, where transversal implies that each qubit in a code block is acted on by at most a single physical gate, and each code block is corrected independently when an error occurs Realizing this 3D (or greater) physical connectivity could be challenging for many of the current quantum platforms; photonic-interconnected modules may be the most flexible architecture with regard to its possible connectivity graph;[15] currently, achieved connection speeds would present a considerable bottleneck.[16] A variant of the 3D surface code may still be realizable with hardware that is only scalable in two dimensions because the thickness (extra dimension) can be made relatively small and independent of code distance.[17] In Sec. II, we highlight the surface code in more detail and include relevant considerations for physical resource estimation. We calculate the number of physical qubits that are required to break the elliptic curve encryption of Bitcoin keys within the time frame that it poses a threat to do, as a function of the code cycle time
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