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arXiv:2002.06257 · IEEE International Conference on Quantum Computing and Engineering (QCE)

A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes

Muyuan Li, Theodore J. Yoder

Published Oct 1, 2020• 12 citations
arXivHTMLPDF

Abstract

We provide a numerical investigation of two families of subsystem quantum codes that are related to hypergraph product codes by gauge-fixing. The first family consists of the Bravyi-Bacon-Shor (BBS) codes which have optimal code parameters for subsystem quantum codes local in 2-dimensions. The second family consists of the constant rate “generalized Shor” codes of Bacon and Cassicino [1], which we re-brand as subsystem hypergraph product (SHP) codes. We show that any hypergraph product code can be obtained by entangling the gauge qubits of two SHP codes. To evaluate the performance of these codes, we simulate both small and large examples. For circuit noise, a [21], [4], [3] BBS code and a [49], [16], [3] SHP code have pseudthresholds of 2 × 10-3and 8 × 10-4, respectively. Simulations for phenomenological noise show that large BBS and SHP codes start to outperform surface codes with similar encoding rate at physical error rates 1 × 10-6and 4 × 10-4, respectively.

DOI
10.1109/qce49297.2020.00024

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We provide a numerical investigation of two families of subsystem quantum codes that are related to hypergraph product codes by gauge-fixing. The first family consists of the Bravyi-Bacon-Shor (BBS) codes which have optimal code parameters for subsystem quantum codes local in 2-dimensions. The second family consists of the constant rate “generalized Shor” codes of Bacon and Cassicino [1], which we re-brand as subsystem hypergraph product (SHP) codes. We show that any hypergraph product code can be obtained by entangling the gauge qubits of two SHP codes. To evaluate the performance of these codes, we simulate both small and large examples. For circuit noise, a [21], [4], [3] BBS code and a [49], [16], [3] SHP code have pseudthresholds of 2 × 10-3and 8 × 10-4, respectively. Simulations for phenomenological noise show that large BBS and SHP codes start to outperform surface codes with similar encoding rate at physical error rates 1 × 10-6and 4 × 10-4, respectively.
Given any two classical codes with parameters [n1,k,d1][{n}_{1},k,{d}_{1}][n1​,k,d1​] and [n2,k,d2][{n}_{2},k,{d}_{2}][n2​,k,d2​], we show how to construct a quantum subsystem code in two dimensions with parameters [[N,K,D]][[N,K,D]][[N,K,D]] satisfying $N\ensuremath{\le}2{n}_{1}{n}_{2}, K=k$, and D=min(d1,d2)D=min({d}_{1},{d}_{2})D=min(d1​,d2​). These quantum codes are in the class of generalized Bacon-Shor codes introduced by Bravyi [Phys. Rev. A 83, 012320 (2011)]. We note that constructions of good classical codes can be used to construct quantum codes that saturate Bravyi's bound KD=O(N)KD=O(N)KD=O(N) on the code parameters of two-dimensional subsystem codes. One of these good constructions uses classical expander codes. This construction has the additional advantage of a linear time quantum decoder based on the classical Sipser-Spielman flip decoder. Finally, while the subsystem codes we create do not have asymptotic thresholds, we show how they can be gauge fixed to certain hypergraph product codes that do.
Hypergraph product codes introduced by Tillich and Zémor are a class of quantum LDPC codes with constant rate and distance scaling with the square-root of the block size. Quantum expander codes, a subclass of these codes, can be decoded using the linear time small-set-flip algorithm of Leverrier, Tillich and Zémor. In this paper, we numerically estimate the performance for the hypergraph product codes under independent bit and phase flip noise. We focus on two families of hypergraph product codes. The first family has rate 1/61∼1.6%1/61 \sim 1.6\%1/61∼1.6%, has qubits of weight 101010 or 121212 and stabilizers of weight 111111. We report a threshold near 4.6%4.6\%4.6% for the small-set-flip decoder. We also show that for similar rate, the performance of the hypergraph product is better than the performance of the toric code as soon as we deal with more than 500500500 logical qubits and that for 360036003600 logical qubits, the logical error rate for the hypergraph product code is several orders of magnitude smaller. The second family has rate 0.20.20.2, qubits of weight 101010 and 202020 and stabilizers of weight 151515. We report a threshold near 2%2\%2% for the small-set-flip decoder.
Hypergraph product (HGP) codes are one of the most popular family of quantum low-density parity-check (LDPC) codes. Circuit-level simulations show that they can achieve the same logical error rate as surface codes with a reduced qubit overhead. They have been extensively optimized by importing classical techniques such as the progressive edge growth, or through random search, simulated annealing or reinforcement learning techniques. In this work, instead of machine learning (ML) algorithms that improve the code performance through local transformations, we impose additional global symmetries, that are hard to discover through ML, and we perform an exhaustive search. Precisely, we focus on the hypergraph product of two cyclic codes, which we call CxC codes and we study C2 codes which are the product a cyclic code with itself and CxR codes which are the product of a cyclic codes with a repetition code. We discover C2 codes and CxR codes that significantly outperform previously optimized HGP codes, achieving better parameters and a logical error rate per logical qubit that is up to three orders of magnitude better. Moreover, some C2 codes achieve simultaneously a lower logical error rate and a smaller qubit overhead than state-of-the-art LDPC codes such as the bivariate bicycle codes, at the price of a larger block length. Finally, leveraging the cyclic symmetry imposed on the codes, we design an efficient planar layout for the QCCD architecture, allowing for a trapped ion implementation of the syndrome extraction circuit in constant depth.
Topological subsystem codes can combine the advantages of both topological codes and subsystem codes. Suchara et al. proposed a framework based on hypergraphs for construction of such codes. They also studied the performance of some subsystem codes. Later Bravyi et al. proposed a subsystem surface code. Building upon these works, we propose efficient decoding algorithms for large classes of subsystem codes on hypergraphs and surfaces. We also propose a construction of the subsystem surface codes that includes the code proposed by Bravyi et al. Our simulations for the subsystem code on the square octagon lattice resulted in a noise threshold of 1.75%. This is comparable to previous result of 2% by Bombin et al. who used a different algorithm.
The hypergraph product creates a quantum stabilizer code from two input classical linear codes; a paradigmatic example being the surface code as a hypergraph product of two classical repetition codes. Many properties of the hypergraph product code can be inherited from those of the classical codes such as the code dimension, minimum distance and certain fault-tolerant gadgets. We investigate ways to reduce the number of physical qubits in hypergraph product codes while maintaining some of their useful properties for fault tolerance. We show that the code dimension, canonical logical basis, and minimum distances of the hypergraph product code are preserved through this reduction. We also provide syndrome-measurement schedules that preserve the effective distance as well as examples of reduced hypergraph product codes with parameter improvements such as [ ⁣[610,64,6] ⁣]→[ ⁣[441,64,6] ⁣][\![610,64,6]\!] \rightarrow [\![441,64,6]\!][[610,64,6]]→[[441,64,6]] and [ ⁣[1225,49,11] ⁣]→[ ⁣[931,49,11] ⁣][\![1225,49,11]\!] \rightarrow [\![931,49,11]\!][[1225,49,11]]→[[931,49,11]]. In memory simulations with circuit-level depolarizing noise, the reduced codes can exhibit logical error rates comparable to their unreduced versions at the sampled low physical-error rates, while using fewer physical qubits. Finally, we show how overhead reduction can be compatible with homomorphic measurement gadgets, fold-transversal gates and automorphisms, which extends the savings to logical computation.
We study the performance of Bacon-Shor codes, quantum subsystem codes which are well suited for applications to fault-tolerant quantum memory because the error syndrome can be extracted by performing two-qubit measurements. Assuming independent noise, we find the optimal block size in terms of the bit-flip error probability pXp_XpX​ and the phase error probability pZp_ZpZ​, and determine how the probability of a logical error depends on pXp_XpX​ and pZp_ZpZ​. We show that a single Bacon-Shor code block, used by itself without concatenation, can provide very effective protection against logical errors if the noise is highly biased (pZ/pX≫1)p_Z/p_X\gg 1)pZ​/pX​≫1) and the physical error rate pZp_ZpZ​ is a few percent or below. We also derive an upper bound on the logical error rate for the case where the syndrome data is noisy.