Towards Practical Classical Processing for the Surface Code

Abstract

The surface code is unarguably the leading quantum error correction code for 2-D nearest neighbor architectures, featuring a high threshold error rate of approximately 1%, low overhead implementations of the entire Clifford group, and flexible, arbitrarily long-range logical gates. These highly desirable features come at the cost of significant classical processing complexity. We show how to perform the processing associated with an n × n lattice of qubits, each being manipulated in a realistic, fault-tolerant manner, in O(n 2 ) average time per round of error correction. We also describe how to parallelize the algorithm to achieve O(1) average processing per round, using only constant computing resources per unit area and local communication. Both of these complexities are optimal.Quantum computing promises exponentially faster processing of certain problems, including factoring [1] and simulating quantum physics [2]. Many quantum algorithms are now known [3]. The primary challenges are to mitigate and cope with the imperfections of quantum devices. The surface code [4,5] supports a powerful quantum computing scheme [6][7][8] featuring an experimentally realistic threshold error rate of approximately 1% [9] and requiring only a 2-D square lattice of qubits with nearest neighbor interactions. In this work, we describe how to perform the complex classical processing associated with the full, fault-tolerant scheme in a complexity-optimal manner. Using the current version of our code, we can simulate the fault-tolerant operation of millions of qubits, four orders of magnitude more than in any previous work.

Majorana Fermion Surface Code for Universal Quantum Computation

“…More precisely, when a stabilizer eigenvalue changes in a surface code cycle, it is efficient to store the location of that stabilizer and wait several code cycles, accumulating a spacetime diagram of stabilizer errors as additional errors occur [30,59,60]. After sufficiently many code cycles, the spacetime diagram may be used to determine the most likely configuration of Wilson lines that could have generated those errors [28,29] using a minimum-weight perfect matching algorithm [64,65]. Errors may be subsequently corrected by software when performing logical qubit manipulations and readouts [28].…”

Towards practical classical processing for the surface code: Timing analysis

“…It is also computationally expensive to obtain data at high error rates and high distances as the minimum weight perfect matching problem becomes more difficult around and above the threshold error rate (0.9% [5]). The raw data used to generate Figs.…”

“…We have two operational versions of extended minimum weight perfect matching -complete match [4] which firstly constructs explicit edges between all pairs of vertices no more than approximately d rounds of error correction apart, and edges on demand match [5] which only constructs a small number of local edges and adds further edges to the problem as required. The graphs and matchings generated by complete match (cmatch) and edges on demand match (eodmatch) given Fig.…”

“…This software is by orders of magnitude the fastest currently available. We will review the necessary aspects of the surface code [6,7], fault-tolerant schemes built on the surface code [8][9][10], and our classical processing algorithm [5] as required. Our goal is to analyze in detail the performance and correctness of our implementation of this algorithm.…”

“…The primary barrier to the realization of a quantum computer is the physical realization of quantum gates with sufficiently low error to enable quantum error correction to be used. Topological quantum error correction (TQEC) codes can tolerate error rates of order 1% [4,5] and require only 2-D nearest neighbor interactions, both physically reasonable targets, however the classical processing associated with the error correction is highly nontrivial. Without significant future effort, the classical processing will almost certainly limit the speed of any quantum computer, particularly one with intrinsically fast quantum gates.…”