Efficient magic state factories with a catalyzed $|CCZ\rangle$ to $2|T\rangle$ transformation

Craig Gidney; Austin G. Fowler

doi:10.22331/q-2019-04-30-135

Efficient magic state factories with a catalyzed $|CCZ\rangle$ to $2|T\rangle$ transformation

Craig Gidney and Austin G. Fowler

Google Inc., Santa Barbara, California 93117, USA

Published:	2019-04-30, volume 3, page 135
Eprint:	arXiv:1812.01238v3
Doi:	https://doi.org/10.22331/q-2019-04-30-135
Citation:	Quantum 3, 135 (2019).

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Abstract

We present magic state factory constructions for producing $|CCZ\rangle$ states and $|T\rangle$ states. For the $|CCZ\rangle$ factory we apply the surface code lattice surgery construction techniques described in [15] to the fault-tolerant Toffoli [21,12]. The resulting factory has a footprint of $12d \times 6d$ (where $d$ is the code distance) and produces one $|CCZ\rangle$ every $5.5d$ surface code cycles. Our $|T\rangle$ state factory uses the $|CCZ\rangle$ factory's output and a catalyst $|T\rangle$ state to exactly transform one $|CCZ\rangle$ state into two $|T\rangle$ states. It has a footprint $25\%$ smaller than the factory in [15] but outputs $|T\rangle$ states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the case $\theta=22.5^\circ$ produces a particularly efficient circuit for producing $|\sqrt{T}\rangle$ states. Compared to using the $12d \times 8d \times 6.5d$ $|T\rangle$ factory of [15], our $|CCZ\rangle$ factory can quintuple the speed of algorithms that are dominated by the cost of applying Toffoli gates, including Shor's algorithm [31] and the chemistry algorithm of Babbush et al. [1]. Assuming a physical gate error rate of $10^{-3}$, our CCZ factory can produce $\sim 10^{10}$ states on average before an error occurs. This is sufficient for classically intractable instantiations of the chemistry algorithm, but for more demanding algorithms such as Shor's algorithm the mean number of states until failure can be increased to $\sim 10^{12}$ by increasing the factory footprint $\sim 20\%$.

Featured image: Comparison between new factories and old factories.

► BibTeX data

@article{Gidney2019efficientmagicstate,
  doi = {10.22331/q-2019-04-30-135},
  url = {https://doi.org/10.22331/q-2019-04-30-135},
  title = {Efficient magic state factories with a catalyzed {$|CCZ\rangle$} to {$2|T\rangle$} transformation},
  author = {Gidney, Craig and Fowler, Austin G.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {135},
  month = apr,
  year = {2019}
}

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