Doubled Color Codes (1509.03239v1)

Abstract: We show how to perform a fault-tolerant universal quantum computation in 2D architectures using only transversal unitary operators and local syndrome measurements. Our approach is based on a doubled version of the 2D color code. It enables a transversal implementation of all logical gates in the Clifford+T basis using the gauge fixing method proposed recently by Paetznick and Reichardt. The gauge fixing requires six-qubit parity measurements for Pauli operators supported on faces of the honeycomb lattice with two qubits per site. Doubled color codes are promising candidates for the experimental demonstration of logical gates since they do not require state distillation. Secondly, we propose a Maximum Likelihood algorithm for the error correction and gauge fixing tasks that enables a numerical simulation of logical circuits in the Clifford+T basis. The algorithm can be used in the online regime such that a new error syndrome is revealed at each time step. We estimate the average number of logical gates that can be implemented reliably for the smallest doubled color code and a toy noise model that includes depolarizing memory errors and syndrome measurement errors.

• The paper presents a novel fault tolerance scheme using doubled color codes that employ transversal logical gates and gauge fixing.
• It introduces a maximum likelihood decoding algorithm to correct correlated errors and sustain logical operations in the Clifford+T basis.
• The approach eliminates the overhead of state distillation, promising simpler and more scalable experimental quantum computation.

An In-Depth Analysis of Doubled Color Codes

The paper presented in “Doubled Color Codes” by Sergey Bravyi and Andrew Cross addresses the design and implementation of fault-tolerant universal quantum computation using two-dimensional (2D) architectures. This work introduces a novel coding scheme named "doubled color codes" which fundamentally relies on transversal logical gates and local syndrome measurements for ensuring fault tolerance, potentially bypassing the need for more resource-intensive methods such as state distillation.

Overview of the Doubled Color Code Approach

At the core of the proposed approach are doubled color codes, which leverage a doubled version of the 2D color code, along with the recently introduced gauge fixing method by Paetznick and Reichardt. The gauge fixing method enables implementing all logical gates in the Clifford+$T$ basis using transversal operations. Intriguingly, the doubled color codes support transversal logical gates in these bases without requiring state distillation, thereby significantly reducing implementation complexity.

In essence, the paper constructs a universal set of logical gates by toggling between two different error-correcting codes through gauge fixing: the $C$-code with a transversal Clifford group and the $T$-code supporting a transversal $T$-gate. This switching requires the measurement of six-qubit parity checks corresponding to Pauli operators housed on the honeycomb lattice, with two qubits at each site.

Error Correction and Maximum Likelihood Decoding

The authors introduce a Maximum Likelihood (ML) decoding algorithm for error correction tailored to logical circuits in the Clifford+$T$ basis. This decoder can function in an online regime, assimilating new error syndromes as they appear in real-time, independent of circuit length. This technique extends to the scenario where transversal $T$-gates introduce interdependencies among errors, demanding an error correction routine that accommodates these correlations with minimal computational overhead.

For empirical validation, the authors have simulated a model with the smallest doubled color code, involving depolarizing memory errors and erroneous syndrome measurements in a controlled noise model. This simulation estimates the average number of logical gates that can be implemented reliably and reveals how doubled color codes can sustain a reasonable rate of logical error per gate under certain noise conditions.

Implications and Future Directions

Practically, doubled color codes hold promise in experimental quantum computing by providing a pathway to demonstrate logical gates efficiently. Without state distillation, implementing non-Clifford gates like the $T$-gate becomes feasible with significantly lower overhead. Theoretical implications also underline a distinct strategy of utilizing gauge fixing within codes to circumvent expectations rooted in no-go theorems concerning transversal non-Clifford gates in two-dimensional geography.

Looking ahead, this work suggests several intriguing avenues for advancing quantum code development. Achieving exceedingly reliable computation close to the error threshold elucidated (approximately 0.55% logical error rate) invites further exploration in broadening the constructions to accommodate medium-scale and higher distance codes. Extensions could also explore integrating this method in more intricate quantum systems such as the three-dimensional color codes while retaining efficient error correction mechanisms.

In conclusion, the introduction of doubled color codes profoundly enriches the landscape of quantum error correction through a fresh lens that balances experimental practicality with theoretical robustness. The potential to perform fault-tolerant quantum computations with reduced resource demands positions this work as a critical stepping stone towards realizing scalable quantum computers.