Constant-Overhead Magic State Distillation (2408.07764v2)

Abstract: Magic state distillation is a crucial yet resource-intensive process in fault-tolerant quantum computation. The protocol's overhead, defined as the number of input magic states required per output magic state with an error rate below $\epsilon$, typically grows as $\mathcal{O}(\log^\gamma(1/\epsilon))$. Achieving smaller overheads, i.e., smaller exponents $\gamma$, is highly desirable; however, all existing protocols require polylogarithmically growing overheads with some $\gamma > 0$, and identifying the smallest achievable exponent $\gamma$ for distilling magic states of qubits has remained challenging. To address this issue, we develop magic state distillation protocols for qubits with efficient, polynomial-time decoding that achieve an $\mathcal{O}(1)$ overhead, meaning the optimal exponent $\gamma = 0$; this improves over the previous best of $\gamma \approx 0.678$ due to Hastings and Haah. In our construction, we employ algebraic geometry codes to explicitly present asymptotically good quantum codes for $2^{10}$-dimensional qudits that support transversally implementable logical gates in the third level of the Clifford hierarchy. The use of asymptotically good codes with non-vanishing rate and relative distance leads to the constant overhead. These codes can be realised by representing each $2^{10}$-dimensional qudit as a set of $10$ qubits, using stabiliser operations on qubits. The $10$-qubit magic states distilled with these codes can be converted to and from conventional magic states for the controlled-controlled-$Z$ ($CCZ$) and $T$ gates on qubits with only a constant overhead loss, making it possible to achieve constant-overhead distillation of such standard magic states for qubits. These results resolve the fundamental open problem in quantum information theory concerning the construction of magic state distillation protocols with the optimal exponent.

• The paper introduces a magic state distillation protocol with constant overhead (γ = 0) that overcomes traditional resource scaling challenges in quantum error correction.
• It employs algebraic geometry codes to construct triorthogonal matrices over finite fields, facilitating transversal non-Clifford gate implementation and efficient error correction via a polynomial-time decoder.
• The results promise scalable, fault-tolerant quantum computation by significantly reducing resource demands and sustaining low error rates in universal quantum systems.

Constant-Overhead Magic State Distillation: An Advanced Quantum Code Construction

The paper discusses a novel approach to magic state distillation, a pivotal process in fault-tolerant quantum computation. Magic state distillation allows for the purification of unreliable quantum states into high-fidelity states necessary for universal quantum computation. Historically, the challenge has been the resource overhead, which, for some protocols, scales as $\mathcal{O}(\log^\gamma(1/\epsilon))$, with $\gamma > 0$. The primary contribution of this paper is the construction of a magic state distillation protocol achieving constant overhead, represented by $\gamma = 0$. This breakthrough is built on developing a new family of quantum codes using algebraic geometry codes, and adapting a unique protocol that eliminates the need for post-selection.

A core component of the paper is the construction of a triorthogonal matrix over finite fields of order $2^s$, used to define quantum codes that support transversal implementation of non-Clifford gates. These codes are asymptotically good, meaning they have linearly scaling dimension and distance with respect to the code length. A significant result is the establishment of a magic state distillation protocol using such codes, which maintains error rates below a determined threshold while achieving an overhead that remains constant regardless of the desired target error rate.

The researchers employed algebraic geometry codes due to their favorable parameters on themselves and their duals. This choice was strategic, as traditional Reed-Muller codes, previously used, do not share these properties when applied to duals, leading to only moderate reductions in $\gamma$. By leveraging algebraic geometry codes in dimensions that are fixed, as opposed to growing, the authors constructed new quantum codes capable of supporting a transversal non-Clifford gate, which is instrumental in magic state distillation. These developments rely on the Ihara bound-based asymptotic properties of function fields, which provide sufficient rational places relative to genus, allowing the construction of codes with advantageous properties.

An essential technical contribution of the paper is the establishment of an efficient polynomial-time decoder for $Z$ errors, pivotal for practical implementations. This decoder's existence is assured by the linear minimum distance of the code, essential for robust error correction within the protocol. The authors provide a succinct theoretical treatment of this decoder, ensuring the practical feasibility of the constant-overhead distillation process.

Furthermore, this work involves intricate algebra over prime power fields, constructing a magic state defined over $2^{10}$-dimensional qudits. This constructs a framework where each magic state can be reversibly converted between a multi-qubit state and a qudit state, allowing efficient transition between qubit protocols and higher-dimensional constructs with minimal resource loss. The careful algebraic structure allows for the decomposition of high-dimensional gates into sequences of established qubit-level operations, cementing the protocol's practicality.

The paper's theoretical framework is rigorous, addressing an open problem in quantum error correction by demonstrating that constant overhead can be practically and theoretically achieved. It posits the potential for more reliable and resource-efficient fault-tolerant quantum computation, shifting the landscape for implementing quantum algorithms on fault-tolerant architectures.

In conclusion, the implications of this research extend beyond just the theoretical bounds — into the practical domain of quantum computing where resources are constrained, making it an invaluable tool in the progression towards scalable, universal quantum computation. As researchers further explore these codes and distillation techniques, the prospect of practical quantum computing becomes increasingly tangible. Future inquiries may focus on optimizing constant factors, exploring alternative algebraic structures, and adapting these protocols for emerging quantum architectures.