Abstract
Magic states are fundamental building blocks on the road to fault-tolerant quantum computing. Calderbank-Shor-Steane (CSS) codes play a crucial role in the construction of magic distillation protocols. Previous work has cast quantum computing with magic states for odd dimension within a phase-space setting in which universal quantum computing is described by the statistical mechanics of quasiprobability distributions. Here we extend this framework to the important qubit case and show that we can exploit common structures in CSS circuits to obtain distillation bounds capable of outperforming previous monotone bounds in regimes of practical interest. Moreover, in the case of CSS-code projections, we arrive at a novel cutoff result on the code length of the CSS code in terms of parameters characterizing a desired distillation, which implies that for fixed target error rate and acceptance probability, one needs to consider only CSS codes below a threshold number of qubits. These entropic constraints are not due simply to the data-processing inequality but rely explicitly on the stochastic representation of such protocols.
- Received 5 January 2023
- Accepted 30 May 2023
DOI:https://doi.org/10.1103/PRXQuantum.4.020359
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
The current leading approach to building a fault-tolerant quantum computer is magic state injection. In this scheme, highly nonclassical states, known as magic states, produce quantum speedups when fed into a classically simulable but fault-tolerant subset of quantum circuits. In order to achieve a desired speedup, magic states typically need to undergo distillation by this subset of circuits, which refines many noisy copies of a magic state into fewer, less noisy copies. Existing protocols for magic-state distillation are extremely costly, and so a better understanding of the resource requirements at play is of paramount importance. Our work addresses qubit magic distillation, which is, interestingly, the friendliest to implement experimentally but the hardest to describe theoretically. We develop a framework that studies the thermodynamics of magic states as they are processed by a quantum computer, which allows us to derive constraints on magic distillation that resemble thermodynamic second laws.
Within our framework, magic states can be rigorously viewed as “negative-free-energy” resources that are consumed under stochastic processing in a quantum computer, which thereby produces “useful work” in the form of computational advantages that cannot be achieved by classical computers. We further define entropies that order the usefulness of these resources relative to classical states. By considering how this order changes throughout the distillation process, we are able to assess the capabilities of a large and important family of distillation protocols based on quantum error-correcting codes. In particular, we are able to provide trade-off relations between various performance metrics on these protocols, which constrain the regimes in which they are useful for magic distillation.
Our research paves the way for a better understanding of how and why qubit magic states produce quantum computational advantage. We anticipate that our methods can be extended to identify similar thermodynamic resource constraints on all qubit distillation protocols.
