@article {3499, title = {Encoding Majorana codes}, year = {2024}, month = {2/12/2024}, abstract = {

To implement a quantum error correction protocol, we first need a scheme to prepare our state in the correct subspace of the code, and this can be done using a unitary encoding circuit. Majorana codes are special since any gates that transform such codes must preserve fermionic parity. In this paper, we present an algorithm that uses the stabilizer matrix to compute unitary encoding circuits for Majorana codes. We present two approaches, both of which use a version of Gaussian elimination with row operations replaced with elementary fermionic Clifford operations. One approach uses an additional ancilla mode and works for all Majorana stabilizer codes, while the second approach does not use ancilla but does not work if the total parity is inside the stabilizer group.

}, url = {https://arxiv.org/abs/2402.07829}, author = {Maryam Mudassar and Riley W. Chien and Daniel Gottesman} } @article {3494, title = {Error Correction in Dynamical Codes}, year = {2024}, month = {3/7/2024}, abstract = {

We ask what is the general framework for a quantum error correcting code that is defined by a sequence of measurements. Recently, there has been much interest in Floquet codes and space-time codes. In this work, we define and study the distance of a dynamical code. This is a subtle concept and difficult to determine: At any given time, the system will be in a subspace which forms a quantum error-correcting code with a given distance, but the full error correction capability of that code may not be available due to the schedule of measurements associated with the code. We address this challenge by developing an algorithm that tracks information we have learned about the error syndromes through the protocol and put that together to determine the distance of a dynamical code, in a non-fault-tolerant context. We use the tools developed for the algorithm to analyze the initialization and masking properties of a generic Floquet code. Further, we look at properties of dynamical codes under the constraint of geometric locality with a view to understand whether the fundamental limitations on logical gates and code parameters imposed by geometric locality for traditional codes can be surpassed in the dynamical paradigm. We find that codes with a limited number of long range connectivity will not allow non-Clifford gates to be implemented with finite depth circuits in the 2D setting.

}, url = {https://arxiv.org/abs/2403.04163}, author = {Xiaozhen Fu and Daniel Gottesman} } @article {3353, title = {Complexity and order in approximate quantum error-correcting codes}, year = {2023}, month = {10/7/2023}, abstract = {

We establish rigorous connections between quantum circuit complexity and approximate quantum error correction (AQEC) properties, covering both all-to-all and geometric scenarios including lattice systems. To this end, we introduce a type of code parameter that we call subsystem variance, which is closely related to the optimal AQEC precision. Our key finding is that if the subsystem variance is below an O(k/n) threshold then any state in the code subspace must obey certain circuit complexity lower bounds, which identify nontrivial {\textquoteleft}{\textquoteleft}phases\&$\#$39;\&$\#$39; of codes. Based on our results, we propose O(k/n) as a boundary between subspaces that should and should not count as AQEC codes. This theory of AQEC provides a versatile framework for understanding the quantum complexity and order of many-body quantum systems, offering new insights for wide-ranging physical scenarios, in particular topological order and critical quantum systems which are of outstanding importance in many-body and high energy physics. We observe from various different perspectives that roughly O(1/n) represents a common, physically significant {\textquoteleft}{\textquoteleft}scaling threshold\&$\#$39;\&$\#$39; of subsystem variance for features associated with nontrivial quantum order.

}, url = {https://arxiv.org/abs/2310.04710}, author = {Jinmin Yi and Weicheng Ye and Daniel Gottesman and Zi-Wen Liu} } @article {3364, title = {Partial Syndrome Measurement for Hypergraph Product Codes}, year = {2023}, month = {9/26/2023}, abstract = {

Hypergraph product codes are a promising avenue to achieving fault-tolerant quantum computation with constant overhead. When embedding these and other constant-rate qLDPC codes into 2D, a significant number of nonlocal connections are required, posing difficulties for some quantum computing architectures. In this work, we introduce a fault-tolerance scheme that aims to alleviate the effects of implementing this nonlocality by measuring generators acting on spatially distant qubits less frequently than those which do not. We investigate the performance of a simplified version of this scheme, where the measured generators are randomly selected. When applied to hypergraph product codes and a modified small-set-flip decoding algorithm, we prove that for a sufficiently high percentage of generators being measured, a threshold still exists. We also find numerical evidence that the logical error rate is exponentially suppressed even when a large constant fraction of generators are not measured.

}, url = {https://arxiv.org/abs/2306.17122}, author = {Noah Berthusen and Daniel Gottesman} } @article {3203, title = {Opportunities and Challenges in Fault-Tolerant Quantum Computation}, year = {2022}, month = {10/28/2022}, abstract = {

I will give an overview of what I see as some of the most important future directions in the theory of fault-tolerant quantum computation. In particular, I will give a brief summary of the major problems that need to be solved in fault tolerance based on low-density parity check codes and in hardware-specific fault tolerance. I will then conclude with a discussion of a possible new paradigm for designing fault-tolerant protocols based on a space-time picture of quantum circuits.

}, keywords = {FOS: Physical sciences, Quantum Physics (quant-ph)}, doi = {10.48550/ARXIV.2210.15844}, url = {https://arxiv.org/abs/2210.15844}, author = {Daniel Gottesman} }