Magic state distillation with low overhead (1209.2426v1)

Abstract: We propose a new family of error detecting stabilizer codes with an encoding rate 1/3 that permit a transversal implementation of the pi/8-rotation $T$ on all logical qubits. The new codes are used to construct protocols for distilling high-quality `magic' states $T|+>$ by Clifford group gates and Pauli measurements. The distillation overhead has a poly-logarithmic scaling as a function of the output accuracy, where the degree of the polynomial is $\log_2{3}\approx 1.6$. To construct the desired family of codes, we introduce the notion of a triorthogonal matrix --- a binary matrix in which any pair and any triple of rows have even overlap. Any triorthogonal matrix gives rise to a stabilizer code with a transversal $T$-gate on all logical qubits, possibly augmented by Clifford gates. A powerful numerical method for generating triorthogonal matrices is proposed. Our techniques lead to a two-fold overhead reduction for distilling magic states with output accuracy $10^{-12}$ compared with the best previously known protocol.

• The paper introduces a new family of stabilizer codes based on triorthogonal matrices, enabling more efficient protocols for magic state distillation.
• The proposed protocols significantly reduce distillation overhead, achieving a scaling exponent $\gamma \approx 1.6$ compared to previous methods with higher exponents.
• These novel techniques lead to a substantial reduction in the cost of producing high-fidelity magic states, improving the practical feasibility of fault-tolerant quantum computing.

An Overview of "Magic State Distillation with Low Overhead"

The paper "Magic state distillation with low overhead" by Bravyi and Haah addresses a crucial challenge in fault-tolerant quantum computation: the high cost associated with implementing non-Clifford gates, specifically the $T$-gate, an essential component for achieving a universal set of quantum gates. The authors propose a new family of stabilizer codes that enable more efficient protocols for distilling magic states, which are necessary resources for such computations.

Magic state distillation is a method that leverages error-detecting codes to produce highly-accurate ancillae needed for non-Clifford operations. The main contribution of the paper is the introduction of stabilizer codes characterized by a unique structure called triorthogonal matrices. These matrices enable transversal $T$-gates across all logical qubits. By utilizing these codes, the distillation overhead—defined as the number of raw magic states required to produce a single high-fidelity magic state—is significantly reduced.

Key Contributions

1. Triorthogonal Matrices: At the heart of the proposed method is the concept of a triorthogonal matrix, which is a binary matrix that satisfies specific orthogonality conditions concerning pairs and triples of its rows. The authors show that such matrices can be transformed into CSS stabilizer codes conducive to transversal $T$-gates, facilitating efficient distillation.
2. Low Overhead Protocols: The authors develop distillation protocols where the overhead scales as $O(\log^\gamma(1/\epsilon))$, with $\gamma \approx 1.6$. This is a reduction compared to previous protocols where $\gamma$ was much larger. For instance, the best previous protocol by Meier et al. had a scaling exponent $\gamma \approx 2.32$. The reduction in overhead implies fewer operations and resources, which is a significant advantage for practical quantum computing applications.
3. Numerical Generation of Triorthogonal Matrices: The work also outlines a numerical technique for generating triorthogonal matrices, laying a foundation for systematically constructing stabilizer codes that are both efficient in terms of the yield (number of logical qubits per physical qubit) and resilient (able to correct more errors).
4. Implementation and Cost Advantages: Using raw ancillas with an initial error rate and targeting an output error rate, the authors provide numerical evidence for their protocol, reporting a two-fold reduction in distillation cost for outputs with accuracy similar to $10^{-12}$ compared to the best prior methods.

Implications and Future Directions

The implications of this work are both theoretical and practical. Theoretically, it advances our understanding of resource-efficient quantum error correction and facilitates a deeper insight into the structural properties of quantum codes. Practically, it lays a pathway toward developing quantum technology with viable non-Clifford gate implementations, reducing the high resource demands typically associated with them.

The authors speculate that the scaling exponent $\gamma$ cannot be reduced below $1$ with any concatenated distillation protocol, presenting an open question and a potential boundary for future research in quantum code optimization. Additionally, exploring triorthogonal matrices might yield further reductions in overhead and enable more resilient codes.

In conclusion, Bravyi and Haah's paper makes a significant contribution to the magic state distillation literature by providing novel methods to streamline the production of high-fidelity magic states with reduced overhead, advancing the practical realization of efficient quantum computing. The results underscore the potential for continuing development in error-correcting codes and fault-tolerant quantum computation strategies.