#qldpc

QLDPC Codes

Quantum Low-Density Parity-Check (QLDPC) codes prevent noise and decoherence problems in fragile quantum data. They may be more efficient than surface codes for fault-tolerant quantum computing.

A full explanation of QLDPC codes:

History

The 1960s work of Robert Gallager on classical coding theory led to LDPC codes. Scholars like David MacKay and Daniel Gottesman proposed changing quantum codes in the 1990s. Kitaev's Toric Code demonstrated quantum codes with local checks in the early 2000s, but QLDPC code families with provably enhanced parameters aroused current attention.

They Work

QLDPC codes work by encoding a limited number of logical qubits with protected, useful data into a larger number of physical qubits with unprocessed, noisy hardware.

Only a few “parity-check” measurements, or stabilisers, are necessary to verify each physical qubit, making the structure “low-density”. QLDPC codes can be visualised using a Tanner graph or sparse parity-check matrix. This sparse structure is necessary for fast, error-finding decoding methods.

Correcting Errors

A logical qubit encodes more physical qubits.

Physical qubits can go wrong with bit flips and phase flips.

To measure syndrome, stabiliser operators are measured. These measurements establish a “syndrome” that displays errors without destroying quantum information.

Traditional computers use decoding algorithms to diagnose the syndrome and find the most likely error.

A repair operation on the physical qubits reverses the error and restores the logical qubit.

Construction and Types

QLDPC stabiliser codes are generally built using the Calderbank-Shor-Steane (CSS) architecture, which combines two compatible conventional LDPC codes. Notable types are:

Hypergraph Product (HGP) codes: Two classical codes are combined. HGP-built high-rate LDPC codes include the “La-cross codes” analysed. Their stabilisers resemble long-armed cross-stitch.

Bivariate Bicycle (BB) codes: Low qubit overhead and high error robustness.

Advantages

Low overhead: QLDPC codes can protect a specified quantity of quantum information with fewer physical qubits than surface codes due to their constant encoding rate and linear distance. This is crucial to building big, usable quantum computers. The “La-cross codes” family outperforms surface codes in encoding rate and qubit overhead for the same number of logical qubits and distance.

Some QLDPC codes have a high theoretical error threshold, allowing them to tolerate a high number of physical errors while maintaining a low logical error rate.

Sparse algorithms enable fast decoding for real-time error correction.

Drawbacks and Issues

In contrast to topological codes, which generally use nearest-neighbor interactions, many powerful QLDPC codes require connections between physically separated qubits, which is a hardware issue. Neutral atom arrays may physically change qubits, making them a suitable platform. However, the “La-cross codes” family leverages Rydberg blockade interactions for long-range connections and seeks a static implementation without qubit shuttling.

Creating accurate and efficient decoders for huge codes that operate in real time is difficult, even though decoding can be done quickly. In “quantum degeneracy,” several error patterns produce the same symptom, making decoding harder. High-speed decoders are being developed, however Belief Propagation with Ordered Statistics Decoder (BP+OSD) is being used.

Research is underway to identify practical techniques to implement all fault-tolerant quantum logic gates in QLDPC scripts.

Applications

Uses QLDPC codes are mostly utilised in fault-tolerant quantum computing. They enable enormous quantum computers that can calculate practically despite hardware noise. Specifically, they are used for:

Long-term quantum memory.

Universal Quantum Computation: Protecting logical qubits and enabling fault-tolerant operations to execute quantum algorithms correctly.

“La-cross codes” show Neutral Atom Qubit implementation.

Without qubit shuttling, “La-cross codes” can be implemented in neutral atom registers quickly and statically.

Rydberg-blockade interactions provide natural long-range communication. Coupled qubits to highly excited electronic Rydberg states occur when one laser-stimulated atom shifts the Rydberg states of an adjacent atom off-resonance. Interatomic distance affects gate fidelity because mistakes increase with distance.

These codes outperform surface codes when the error probability of the two-qubit nearest-neighbor gate is less than 0.1%, according to circuit-level simulations. Long-range gates reduce logical error probability for small physical error probabilities notwithstanding their penalty.

IBM Quantum introduced Relay-BP, a revolutionary heuristic decoder that promises real-time quantum error correction for large-scale quantum computers. This breakthrough advances fault-tolerant quantum computing.

Urgent Need for Advanced Quantum Computing Decoders

Noise can disrupt qubits' delicate quantum states, causing problems for quantum computers. Quantum Low-Density Parity Check (qLDPC) codes have lower qubit overhead than surface codes, making them appealing. Quantum information is protected from noise by QEC codes. For QEC codes to be practical, the quantum computer's classical hardware decoder algorithms must be fast enough to prevent backlog, create low logical error rates, and be affordable.

Traditional methods sometimes fail to meet these challenging goals. Decoders must be:

Flexible: Decodes many qLDPC circuits.

Accurate: Quantum calculations demand low logical error rates.

Compact: Ideal for ASIC and FPGA implementations due to its small footprint.

Process symptoms at production speed to avoid backlogs.

Standard Belief Propagation (BP) works well for fast and tiny FPGA implementations, but oscillating error beliefs and trapping sets allow it to fail to converge for qLDPC codes, resulting in low logical error rates. BP+Ordered Statistics Decoding (OSD), the “gold standard” for qLDPC decoding accuracy, is too resource-intensive for real-time FPGA implementation. Matching and Union-Find decode surface codes well, but not generic qLDPC codes.

Relay-BP: A New Decoding Method

By changing the BP algorithm, “Relay-ensembling with locally-averaged memory,” or Relay-BP, solves these issues. The unique approach to blood pressure is using impaired memory strengths. Reduces oscillations and breaks symmetries that confine standard BP algorithms, improving convergence.

Relay-BP relies on the “relay” mechanism, which chains together subsequent Disordered Memory Belief Propagation (DMem-BP) runs. Each DMem-BP instance, or “leg,” initialises the final marginals (error beliefs) from the previous run, improving convergence rate and solution quality throughout the relay. Relay-BP can enhance decoding accuracy without restarting by encountering several valid ensembling technique corrections.

Importantly, Relay-BP can lower memory strengths. This “forgetting” capacity allows the algorithm move on from bad solutions when trapped in trapping sets or unable to find one. Originally a coding error, this discovery was significant for performance.

Outstanding Results in All Metrics

IBM specialists say Relay-BP is the best real-time qLDPC decoder due to its adaptability, compactness, speed, and accuracy.

In circuit-noise decoding, Relay-BP is more accurate. For bivariate-bicycle codes (gross and two-gross), it outperforms BP+OSD+CS-10 by 1-2 orders of magnitude (p = 3 × 10^-3). It works like surface code min-weight-matching decoders. As error probability declines, logical error rates are several orders of magnitude smaller than in regular BP.

Due to its parallel nature, Relay-BP is a lightweight message-passing decoder optimised for low-footprint, quick decoding with FPGA or ASIC real-time implementations. Decoders in FPGAs or ASICs are needed for superconducting qubits with microsecond QEC cycle periods. Relay-BP meets real-time decoding needs with error rates orders of magnitude lower than BP and Mem-BP in 600 iterations.

Flexibility: Unlike surface code-specific decoders, Relay-BP can decode any qLDPC code and many circuits.

A small FPGA implementation is ensured by its message-passing design.

For bivariate-bicycle codes, Relay-BP's ability to add negative memory strengths improves accuracy. The “relay” structure, where data is exchanged across DMem-BP runs, also improves convergence rates and results compared to independent ensembling.

Journey to Innovation

Tristan Müller, the study's principal author, modified BP by adding “cheap and impactful” components from previous studies, creating Relay-BP. His expertise in many-body physics, where memory strength concerns are common, helped him understand the need of altering memory strengths, including the coincidental finding of negative memories' value.

Relay-BP's success is due to IBM's electrical engineering, many-body physics, software development, systems, and number theory teams. This complex innovation relied on this collaborative atmosphere that encourages cross-disciplinary learning.

In Search of Fault-Tolerant Quantum Computing

Even while the current study focused on error correction in quantum memory keeping qubit stability, the team is currently investigating ways to expand Relay-BP's capabilities to quantum processing, which requires larger issue sizes and more complex logical operations.

Library qLDPC

Infleqtion researchers and JPMorgan Chase introduced a new open-source research software library today to speed up quantum application efficiency efforts. The Economist Commercialising Quantum conference in London on May 13–14 will provide more details on the news.

In conjunction with JPMorgan Chase, Infleqtion researchers created the open-source qLDPC library. Available on GitHub.

The main reason for this package was to help build and analyse quantum low density parity check (qLDPC) codes. However, the library tools also work for general error-correcting stabiliser and subsystem codes.

Reducing the number of physical qubits needed for quantum error correcting is one of the qLDPC library's biggest benefits. This library reduces fault-tolerant quantum computing hardware requirements by 10–100x. A single logical, error-corrected qubit used to need 1,500 physical qubits to work reliably. This new library may lower the requirement to 15–150 physical qubits per logical qubit, depending on implementation. This breakthrough fixes a scaling quantum system bottleneck.

The qLDPC library tools suit Infleqtion's neutral atom-based quantum computing technology. Infleqtion's hardware enables for extremely customisable qubit layouts, enabling the library's more efficient error-correcting codes.

The open-source library qLDPC is meant for collaboration. Developers, academics, and hardware partners can directly interact with the codebase to find new error correction and quantum workload optimisation approaches across platforms.

Important qLDPC library features include:

ClassicalCode: A class for classical linear error-correcting codes over finite fields, featuring pre-defined families and GAP/GUAVA package communication for more codes.

A class for creating stabiliser and subsystem Galois-qudit codes. Get_logical_ops, concatenate, and get_distance perform nontrivial logical Pauli operator construction, code concatenation, and code distance computation, respectively.

CSSCode: QuditCode subclass for building quantum CSS codes from two suitable ClassicalCodes. It uses research paper approaches to estimate code distance upper bounds in get_distance_bound.

Special quantum code constructs and family classes:

Two-block quantum codes.

BBCode, bivariate bicycle codes, including toric layout identification (like long-distance checks) and neutral atom qubit layouts that minimise communication distance. Several arXiv papers mention these constructs.

Product hypergraph codes.

Product codes for subsystem hypergraphs.

Subsystem hypergraph product codes simplex.

Lifted product codes.

Quantum Tanner codes.

decoders.py: BP-OSD, BP-LSD, belief-field, minimum-weight perfect matching, and additional error decoding modules.An interface for custom decoders is included.

Abstract algebra (groups, algebras, representations) module in Python. It communicates with GAP and GroupNames.org and uses SymPy pre-defined groups.

objects.py: A package for building quantum code auxiliary objects like Cayley and chain complexes.

qldpc.circuits.get_transversal_ops: A qubit code subroutine that constructs all SWAP-transversal logical Clifford gates in one code block, but it has exponential complexity and is more suitable for small-to-moderate codes.

Package requires Python >= 3.10 and can be installed via PyPI or source. C compilers for Windows and cvxpy for macOS may be required.