Towards Fault-Tolerant Implementation of Holographic Quantum Error Correction

Abstract

Constructing and operating a large-scale, fault-tolerant quantum computer remains one of the most arduous challenges for the field of quantum information science, inasmuch from the theoretical and practical standpoints. Much progress is still required in the development of new, high-rate quantum error correction codes, without which the dream of commercially-viable quantum computing is not achievable. On the other hand, the specific details of implementation with regards to novel quantum codes remain equally as important and challenging. This dissertation presents an extensive practical study for a relatively new class of quantum error correction codes, known as holographic quantum codes. First studied exclusively in the context of toy-model simulations for the famous AdS/CFT correspondence, one of the leading theoretical proposals for the emergence of gravity in the quantum regime, we study the error-correction properties of holographic quantum codes for real-world quantum computing, as well as addressing specific implementation issues.

Firstly, we present a new subclass of holographic quantum codes that has been discovered, which we name Evenbly codes. The particular code construction presented demonstrates many aspects of the gauge-gravity correspondence that previous tensor-network models of holography lacked. Among these are: state-dependent operator reconstruction, hailing from a novel gauge-fixing picture; quantum corrections to the Ryu-Takayanagi formula, which are expected in the finite-𝑁 regime of AdS/CFT; and an analytical derivation of bona fide scaling dimensions from a conformal field theory defined on the boundary of AdS, in agreement with what is known from holographic renormalization group theory. Additionally, the gauge-fixing picture allows for Evenbly codes to be viewed as novel holographic subsystem codes, in which the gauge degree of freedom chosen permits markedly different quantum error correction properties to emerge, ranging from very high thresholds to low-weight transversal logical operations, to distance scaling of logical qubits that bests the most well-studied mainstream quantum error correction codes, such as topological codes. Finally, we show that asymptotically zero-rate versions of these codes not only attain and exceed the zero-rate hashing bound at various bias points, but that several holographic codes supersede the current state-of-the-art record, beating the hashing bound as we move towards the 2-Pauli noise regime.

Secondly, we consider the practical usability of holographic codes for universal quantum computation. Utilizing a novel connection to code concatenation, we construct heterogeneous holographic codes which allow for universal fault-tolerant logic, thus circumventing the Eastin-Knill theorem. We pinpoint the thresholds of these codes under the quantum erasure channel, showing that they exceed the thresholds given by traditional code concatenation, all while conferring significant savings in space overhead. Also, we consider fault-tolerant syndrome extraction for precursor seed codes of holographic codes, showing that, by considering the entire stabilizer group of 2𝑛-𝑘 elements, such syndrome extraction protocols are amenable to large gate reductions if the flag fault tolerance protocol is utilized.

Finally, we investigate engineering-level dilemmas associated with executing quantum algorithms and error-correction codes on real devices. Penultimately, we derive and demonstrate a lower bound for the number of SWAP gates needed to realize an algorithm on a finite-connectivity quantum device, permitting future algorithmic strategies to be fairly compared. This lower bound is derived using insights from quantum information theory, graph theory, and quantum circuit complexity theory. In particular, we show that the use of entropic divergences allows us to lower-bound the number of SWAP gates needed via a relationship with the quantum Fisher information metric. Lastly, we investigate several examples of near-term spin-qubit architectures, and utilize the multipartite maximally-entangled states as benchmark measures, thus aiding in the design of future quantum devices. By utilizing benchmarks known for characterizing multipartite quantum entanglement, we establish a framework for efficiently diagnosing and differentiating architectural connectivity features under realistic noise models and compilation features. Our results include a trade-off evaluation regarding the utility of advanced local connectivity for a spin-qubit device versus the amount of crosstalk present. Our study shows that limitations exist in spin-qubit architectures concerning the relative amount of local connectivity.

At the end of this dissertation, we provide concluding comments, as well as ideas for future directions in the field of holographic quantum error correction.