Analog information decoding of bosonic quantum LDPC codes

TL;DR

This paper addresses decoding of concatenated bosonic-LDPC codes by leveraging the inherently analog syndrome information from bosonic qubits. It introduces analog Tanner graph decoding (ATD), which embeds analog readouts into belief-propagation-based decoders, and demonstrates strong improvements for single-shot decoding of a 3D surface code using cat-qubit inner codes. The authors furthermore develop a quasi-single-shot protocol (w-QSS) that uses analog information to drastically reduce the number of repetition rounds needed in time-domain decoding, achieving competitive thresholds (e.g., near $1.66\%$ non-single-shot threshold for 3D SC) while limiting overhead. They provide an open-source software toolkit (MQT) and validate their approach on a phenomenological cat-qubit noise model, highlighting significant gains in sustainable thresholds and reduced measurement overhead, thus advancing fault-tolerant prospects for concatenated bosonic-QLDPC codes. The work lays a foundation for general decoding with analog information and points to practical architectures such as three-dimensional concatenated cat codes for scalable quantum fault tolerance.

Abstract

Quantum error correction is crucial for scalable quantum information processing applications. Traditional discrete-variable quantum codes that use multiple two-level systems to encode logical information can be hardware-intensive. An alternative approach is provided by bosonic codes, which use the infinite-dimensional Hilbert space of harmonic oscillators to encode quantum information. Two promising features of bosonic codes are that syndrome measurements are natively analog and that they can be concatenated with discrete-variable codes. In this work, we propose novel decoding methods that explicitly exploit the analog syndrome information obtained from the bosonic qubit readout in a concatenated architecture. Our methods are versatile and can be generally applied to any bosonic code concatenated with a quantum low-density parity-check (QLDPC) code. Furthermore, we introduce the concept of quasi-single-shot protocols as a novel approach that significantly reduces the number of repeated syndrome measurements required when decoding under phenomenological noise. To realize the protocol, we present a first implementation of time-domain decoding with the overlapping window method for general QLDPC codes, and a novel analog single-shot decoding method. Our results lay the foundation for general decoding algorithms using analog information and demonstrate promising results in the direction of fault-tolerant quantum computation with concatenated bosonic-QLDPC codes.

Figures (15)

• Figure 1: Overview of our main techniques. (a) We investigate QLDPC codes concatenated with cat qubits encoded in coherent states of a harmonic oscillator. (b) An important property of cat qubits, in addition to their biased noise model, is that the syndrome information obtained from qubit readout is intrinsically analog-valued, as the wavefunction of a coherent state is a Gaussian centered at $\alpha$. (c) Depending on the measured value $x_m$ during (quadrature) readout, we can assign an outcome-dependent error probability $p(x_m)$ that is a function of the size of the cat qubit $\alpha^2$. (d) We incorporate the analog information obtained during the syndrome measurements into a Tanner graph construction that we refer to as an analog Tanner graph (ATG). The ATG stores the analog syndrome information directly in the factor graph used for decoding. We show that the ATG construction can be adapted to work with decoding strategies such as (e) single-shot shot decoding and (f) overlapping window time-domain decoding.
• Figure 2: Tanner graph of the Hamming code. The square nodes represent the checks $V_C$ (rows of the check matrix $H$) and the circles represent data nodes $V_D$ (columns of $H$).
• Figure 3: three-dimensional surface codes with periodic boundaries indicated by additional edges on the sides of the lattice. (a) A vertex check (of weight six). (b) A face check (of weight four). (c) A volume check (of weight 12). (d) three-dimensional lattice $\Lambda$ with open boundaries. Rough boundaries are indicated with open edges. A single qubit X error gives a pair-like syndrome at the endpoints of the error string (indicated by red vertices). A single qubit Z error produces a loop-like syndrome at adjacent faces (indicated by blue faces). The loop can be readily seen in the dual lattice pictures whose dual edges are indicated with dashed lines. (e) Logical operators of the three-dimensional surface codes. A loop-like string corresponds to a logical X operator. A logical Z operator corresponds to a loop of faces in the dual lattice, i.e., forming a dual sheet wrapping across the lattice along two axes.
• Figure 4: (a) Illustration of the tanner graph for SSMSA decoding. The analog information taken into account by SSMSA can be illustrated similarly to ATD (although in SSMSA the analog information is not directly incorporated in the factor graph), however, because of the cutoff parameter, it may happen that SSMSA discards the analog information, which corresponds to ignoring the virtual nodes, indicated by dashed edges. (b) Sketch of the analog Tanner graph (ATG). The subgraph colored in black corresponds to the Tanner graph of the code. The subgraph highlighted in yellow corresponds to the virtual nodes that are used to incorporate the analog information. Their union is the ATG.
• Figure 5: Comparison of the SSMSA decoder with ATD for increasing syndrome noise $\sigma$ on a family of lifted product codes with distances $d\in \set{12,16,20}$, cf. Section \ref{['sec:app_code_constructions']}. The data error rate of the unbiased depolarizing noise model is fixed at $p=0.05$. (a) Comparison of the original SSMSA implementation and ATD using BP (without OSD). (b) Comparison of SSMSA+OSD and ATD using BP+OSD. The legend is shared between both panels and $\Gamma$ indicates the cutoff value of the soft information (SI) SSMSA decoder. Note that the case $\Gamma = 0$ corresponds to ignoring the analog information and, therefore, is equivalent to ordinary hard syndrome decoding.

Theorems & Definitions (7)

• Example 2.1: Tanner graph of the Hamming code
• Definition 4.1: Analog check matrix
• Definition 5.1: Multi-round parity-check matrix and Tanner graph
• Proposition 5.2: Informal statement
• Example A.1: Three dimensional surface code from repetition codes
• Proposition B.1: Formal version of Proposition \ref{['prop:informal']}
• proof