Quantum Codes Gain Stability through Measurement and Feedback Techniques

Researchers are developing novel methods to construct non-Abelian quantum low-density parity check (qLDPC) codes and, crucially, generate non-Clifford operations, representing a significant step towards fault-tolerant quantum computation. Maine Christos from the California Institute of Technology, Chiu Fan Bowen Lo from Harvard University, Vedika Khemani from Stanford University, and Rahul Sahay from Harvard University detail constructions for these codes achieved by gauging transversal Clifford gates through measurement and feedback. This collaborative work identifies two distinct approaches to gauging qLDPC codes, one leveraging Poincaré duality to create a unique Clifford stabilizer code and another employing ancilla qubits to gauge single transversal gates for broader applicability. The resulting gauged codes exhibit properties akin to two-dimensional non-Abelian topological order, and importantly, the procedures demonstrated allow for the preparation of logical Clifford gates, offering a promising protocol for implementing non-Clifford operations on any qLDPC code.

Quantum computers promise to revolutionise fields from medicine to materials science, but building robust and scalable machines remains a formidable challenge, with qLDPC codes considered promising candidates due to their potential advantages over existing approaches. This work generalizes the pioneering measurement-based gauging protocol of Tantivasadakarn et al. originally developed to prepare 2D non-Abelian order by gauging a transversal CZ gate between two copies of the toric code. Two qualitatively different approaches to gauging qLDPC codes were identified, both of which reduce to one another in the 2D case. The first, termed “homological gauging”, applies to codes with an algebraic structure analogous to Poincaré duality, resulting in a qLDPC non-Abelian Clifford stabilizer code with stabilizers reminiscent of a Type-III twisted quantum double. The second, “graph gauging”, applies to general qLDPC codes, introducing a graph of ancilla qubits tailored to the input code’s properties to gauge a single transversal gate. Both constructions yield codes exhibiting characteristics analogous to two-dimensional non-Abelian topological order, including the behaviour of a single anyon confined to a torus, a fundamental concept in topological quantum computation. Researchers have shown that logical operators can trap the qLDPC analogue of a single anyon on a torus, confirming the persistence of non-Abelian braiding, a key feature of topological quantum computation, within this novel code structure. The structure of ground states and logical operators within these non-Abelian qLDPC codes was determined, demonstrating that the notion of non-Abelian braiding persists even without an underlying spatial manifold. By enabling the preparation of magic states through the measurement of logical Clifford gates, the gauging procedures offer a pathway to perform non-Clifford operations, a crucial step towards realising the full potential of qLDPC codes for universal quantum computation. Measuring transversal Clifford gates of a qLDPC code constitutes a non-Clifford operation, such as measuring the +1 eigenvalue of CZ on the state |++⟩ to yield a magic state, generalizing the work of Williamson and Yoder, which previously showed gauging could measure logical Pauli operators to the case of general Clifford gates. Transversal Clifford gates, implemented through measurement and feedback, underpin the construction of these codes. Homological gauging applies to codes exhibiting a generalised form of Poincaré duality, linking the code’s stabilizer structure to its logical Clifford gates, effectively mirroring a two-dimensional construction by repeating the gauging procedure independently across layers. Alternatively, graph gauging offers a more versatile technique applicable to general qLDPC codes, though it necessitates considering how the code’s structure can be embedded within an underlying graph geometry. A toy model employed stacks of N decoupled two-dimensional toric codes, designated B and C, to demonstrate how the same input code could be gauged using structurally different ancilla qubit arrangements. One configuration utilised a stack of N Lieb lattices, each layer realising a 2D cluster state with an independent Z2 0-form symmetry, enabling independent gauging of CZ symmetries between layers of the B and C codes. Conversely, a three-dimensional cubic lattice of ancilla qubits was also explored, possessing a single Z2 0-form symmetry capable of gauging a single transversal CZ gate acting simultaneously between all layers of the B and C codes. This choice, while resulting in a more complex theory with 3D gauge fields exhibiting non-Abelian statistics, demonstrates the flexibility of the graph gauging approach. This research expands the landscape of known qLDPC phases of matter while simultaneously providing new avenues towards universal quantum computation. Explicit lattice examples were achieved from both forms of gauging, including hybrid Abelian non-Abelian codes where one part of a qLDPC code takes on a non-Abelian character while the rest remains Abelian upon gauging an addressable logical gate. The team’s approach builds upon earlier work involving the toric code, successfully scaling the concept beyond two dimensions and demonstrating a greater degree of flexibility. The importance of this lies in the potential to unlock more powerful and versatile quantum computers, as qLDPC codes are considered frontrunners in the race to build robust and scalable quantum devices, but their inherent limitations in performing non-Clifford operations have been a major hurdle. While the theoretical framework is sound, the complexity of implementing these gauged codes, particularly the management of ancilla qubits and the precise control required for transversal gates, remains substantial. Further research is needed to explore the overhead associated with this approach and to determine its feasibility in the presence of real-world noise, likely inspiring new investigations into the design of qLDPC codes specifically tailored for non-Abelian gauging and optimising the ancilla qubit graphs. Ultimately, this work broadens the scope of what’s computationally possible with quantum machines, inching us closer to a future where complex quantum algorithms can be reliably executed.