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Moore's law states that performance of classical computers doubles every two years (later revised to 18 months) and a price for computing resources halves during same time period.

Currently, it seems that similar statement can be made for quantum annealers in term of number of qubits (D-Wave). Maybe, also for universal quantum gate-based computers. IBM plans to introduce 53 qubit processor, but so far the highest number of qubits is 20 in Tokyo processor.

My question: Is there any paper/article/business analysis dealing with Moore's law for quantum computer?

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Moore's law deals with the number of transistors in an integrated circuit, which is used as a proxy for computational power. In a quantum computing device the analogy would be the number of qubits. However, this by itself would be a poor benchmark, namely because it is easy to build lots of qubits. Building many qubits with properties such as long coherence times, low crosstalk, good readout fidelity etc... is what is actually difficult to achieve. So in the quantum world, number of qubits is not a proxy for computational power. Instead, it is the number of number of qubits combined with the quality of those qubits, as measured by their error rates and coherence times, that determines the overall computational power of a quantum device.

The notion of "computational power" is the next question. To quantify this we need some kind of benchmark that we can use to quantify the performance of a give quantum device, and compare it to others. One such benchmark for gate-based quantum devices is the Quantum Volume, developed by IBM [1] (disclosure, I am on this paper). This is a benchmark using random quantum circuits to mimic the performance of an average quantum circuit on a given device. It takes into account both the number of qubits and their quality as is necessary for quantum devices (at least noisy ones). IBM uses this as their benchmark and have seen exponential scaling of this value every year[2]; they aim to double the Quantum Volume every year. There are of course infinitely many benchmarks possible, and a suite of benchmarks is likely at some point in the future, however Quantum Volume seems to be gaining traction. See for example Refs.[3,4].

Note also that the 53-qubit Rochester device from IBM has been out for some time, and can be accessed by members of the IBM Quantum Network. All of the devices, along with their Quantum Volume, can be see on the IBM Quantum Experience website [5].

[1] https://arxiv.org/pdf/1811.12926.pdf

[2] https://www.ibm.com/blogs/research/2019/03/power-quantum-device/

[3] https://www.honeywell.com/en-us/newsroom/news/2020/03/quantum-volume-the-power-of-quantum-computers

[4] https://www.honeywell.com/content/dam/honeywell/files/HQS-QCCD-Demonstration.pdf

[5] https://quantum-computing.ibm.com/docs/cloud/backends/systems/

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The article of Devoret and Schoelkopf [1] and an update provided in Section 7.1 of Reagor [2] makes a comparison between Moore's law and an observed trend of exponentially improving $T_1$ and $T_2$ times for superconducting qubits. The trend they present shows a roughly exponential improvement from $10^0$ to $10^6$ nanoseconds for $T_2$ between various superconducting qubit architectures over the course of 20 years (up until around 2015). I've seen this referred to as "Shoelkopf's Law" in a few places, though I don't know if that label is broadly recognized in the industry.

As the other answer points out, there is much more to factor into the performance of a quantum computer than isolated coherence times alone, but the trend in performance improvement is very suggestive. Its worth considering that these reported times are likely the best case single qubit performances taken from many fabrication attempts, so one would expect the lifetime of a typical qubit on a larger processor to fall short of this observed trendline.


[1] Devoret, Michel H., and Robert J. Schoelkopf. "Superconducting circuits for quantum information: an outlook." Science 339.6124 (2013): 1169-1174. URL: https://science.sciencemag.org/content/339/6124/1169 (temporary URL, no paywall: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.679.2113&rep=rep1&type=pdf)

[2] Reagor, Matthew James. Superconducting cavities for circuit quantum electrodynamics. Yale University, 2016. (temporary URL: https://rsl.yale.edu/sites/default/files/files/RSL_Theses/reagor-thesis-20151202.pdf)

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