Quantum Computing Breakthrough: Reducing Errors in Stabilizer Codes

Stabilizer codes are a class of quantum error-correcting codes that form the basis of most proposals for fault-tolerant quantum computers. These codes are defined by a set of parity-check operators, which are measured to infer information about errors that may have occurred on the physical qubits of the code. In typical settings, measuring these operators is itself a noisy process, and the noise strength scales with the number of qubits involved in a given parity check or its weight.

How Do Stabilizer Codes Work?

A stabilizer code encodes logical qubits into a larger set of physical qubits. The code then measures parity-check operators on these physical qubits to detect errors. If an error occurs, it will change the measurement outcome of one or more parity checks. By analyzing these changes, the code can determine and correct which physical qubits are affected by the error.

Problems in Stabilizer Code

One major problem with stabilizer codes is that they often have high-weight parity checks, which means that a large number of qubits need to be measured simultaneously. This makes the measurement process noisy and prone to errors. Furthermore, many hardware platforms require long-range connectivity between qubits, which can be difficult to implement.

Quantum low-density parity-check (qLDPC) codes are stabilizer codes with low-weight parity checks and qubit degrees. These codes are designed to minimize the number of qubits involved in each parity check, which reduces the noise associated with measurement.

The surface code is an example of a qLDPC code that has gained significant attention in recent years. It encodes logical qubits into a two-dimensional grid of physical qubits and measures parity checks on these qubits to detect errors. The surface code has low-weight parity checks (weight four) and qubit degrees (degree two), making it an attractive candidate for practical implementation.

Despite its advantages, the surface code has some significant limitations. It always encodes a fixed number of logical qubits for a given topology, which means that it has a low encoding rate. This low encoding rate leads to estimates of millions of physical qubits being required for large-scale quantum algorithms.

There exist other families of qLDPC codes with higher encoding rates than the surface code. These codes often have slightly higher parity-check weights and require long-range connectivity between qubits. However, they offer improved performance under realistic noise assumptions.

Given a qLDPC code with favorable parameters (e.g., high encoding rate) but parity-check weights that are too high to limit its performance under realistic noise assumptions, one can ask if there is any way to reduce the weights of the parity checks while mostly retaining the favorable parameters of the code. Hastings has provided a method known as weight reduction, which takes an input Calderbank-Shor-Steane (CSS) code and outputs a qLDPC code with O(1) parity-check weights while increasing the number of physical qubits by a constant factor and reducing the code distance by a constant factor.

Weight reduction is a technique that reduces the weights of parity checks in a stabilizer code. This is achieved by adding additional qubits to the code and modifying the parity checks accordingly. The resulting code has lower-weight parity checks, which reduces the noise associated with measurement.

We can apply weight reduction to practical codes by first providing a self-contained presentation of Hastings’ weight-reduction procedure. This involves optimizing the procedure to improve its overhead and then applying it to small- to medium-sized qLDPC codes that could potentially be implemented on hardware in the short to medium term.

By applying weight reduction to practical codes, we can construct qLDPC codes with low-weight parity checks. These codes have better parameters in the finite-size regime of interest, which makes them more suitable for practical implementation. In particular, our method gives codes with improved encoding rates and lower parity-check weights compared to existing codes.

Stabilizer codes are an essential component of quantum computing, but they often suffer from high-weight parity checks that limit their performance under realistic noise assumptions. Weight reduction is a technique that can reduce the weights of parity checks in these codes, making them more suitable for practical implementation. By applying weight reduction to small- to medium-sized qLDPC codes, we can construct codes with improved parameters and better performance under realistic noise assumptions.