Composable Logical Gate Error Quantifies Approximate Quantum Error Correction in Gottesman-Kitaev-Preskill Codes

Quantifying errors in quantum gates represents a critical challenge in building practical quantum computers, particularly when perfect operation is unattainable. Lukas Brenner, Beatriz Dias, and Robert Koenig, all from Technical University of Munich, now present a new method for assessing these errors, introducing a single value they term ‘composable logical gate error’. This quantity captures both how closely a gate achieves its intended function and the extent to which information escapes the protected quantum code, and importantly, it simplifies the analysis of complex quantum circuits. Their work demonstrates how to calculate this error based on the fundamental operations of a quantum system, avoiding complex mathematical requirements often found in continuous-variable quantum computing, and reveals that established understandings of error correction in ideal systems do not always hold true when dealing with the approximate, physically realistic quantum codes needed for practical applications.

Perfect implementations are often unachievable with the available set of physical operations. To address this, the researchers introduce a single scalar quantity, termed the composable logical gate error. This quantity captures both the deviation of the logical action from the desired target gate, as well as leakage out of the code space. It is subadditive under successive application of gates, providing a simple means for analysing circuits. The team demonstrates how to bound the composable logical gate error in terms of matrix elements of physical unitaries between approximate logical computational basis states. In the continuous-variable context, this sidesteps the need for computing energy-bounded norms.

Continuous Variable Quantum Error Correction Schemes

This document represents a significant body of work in the field of continuous variable quantum information, exploring the potential of using oscillators as qubits and the challenges of building fault-tolerant quantum computers based on this approach. The research focuses on continuous variable (CV) quantum computation, where information is encoded using the properties of oscillators rather than traditional spin-based qubits, offering advantages for hardware implementation using superconducting circuits. A central theme is quantum error correction (QEC), which aims to protect fragile quantum information from noise, investigating several QEC codes specifically designed for CV systems, including Gottesman-Kitaev-Preskill (GKP) codes, which encode qubits in the coherent states of an oscillator, and grid state codes, which utilize a grid-like structure in phase space for encoding. Stabilizer codes, a general framework for QEC, are also explored.

The research relies on a strong mathematical foundation, employing operator theory, convex geometry, matrix analysis, functional analysis, and lattice algorithms to analyze and optimize CV QEC. The work is motivated by the practical challenges of building quantum computers and references work on superconducting circuits and other hardware platforms. Quantum optimal transport, a mathematical framework for comparing and transforming quantum states, is used to analyze the performance of QEC codes. Key concepts include GKP states, which encode a qubit into the position and momentum of an oscillator, and grid states, which are similar to GKP states but use a grid-like structure.

The stabilizer formalism provides a powerful way to describe QEC codes, while the diamond norm and energy-constrained diamond norm measure the distance between quantum channels and quantify the performance of QEC codes. Specific research directions include improving GKP code performance through better decoding algorithms and optimized parameters, developing fault-tolerant quantum computation protocols for CV systems, combining oscillators and qubits in hybrid quantum systems, and exploring the use of oscillators for solving computational problems. Researchers are also applying CV quantum computation to machine learning tasks and developing new mathematical techniques for analyzing and optimizing QEC codes. The document highlights the complexity of preparing and manipulating GKP states and investigates its computational demands.

Important contributions come from Gottesman, Kitaev, and Preskill, who introduced the GKP code in 2001, and Ralph et al., who conducted early work on CV quantum computation. Recent work by Brenner, Caha, Coiteux-Roy, and Koenig focuses on the complexity of GKP states and factoring with oscillators, while Koenig and Rouzé investigate limitations of local update recovery in stabilizer-GKP codes. Matsuura, Menicucci, and Yamasaki have contributed to continuous-variable fault-tolerant quantum computation, and Campagne-Ibarcq et al. have demonstrated quantum error correction with GKP states. Eickbusch et al. have achieved fast universal control of an oscillator with weak dispersive coupling to a qubit.

Subadditive Logical Gate Error Quantifies Circuit Reliability

Scientists have developed a new method for quantifying errors in quantum computations, focusing on the accuracy of logical gates, the fundamental building blocks of quantum circuits. This work introduces the “composable logical gate error,” a single value that captures both how closely a gate performs its intended function and the extent to which errors leak out of the protected quantum information. The method provides a means to analyze the reliability of complex quantum circuits by offering a simple, quantifiable measure of error accumulation. Researchers demonstrated that this logical gate error is “subadditive,” meaning that the error in a longer circuit grows more slowly than the sum of errors in individual gates, offering a more realistic assessment of overall circuit performance.

Researchers established that the logical gate error remains consistent even when additional quantum systems are added, simplifying the analysis of increasingly complex computations. Measurements reveal that the accuracy of implementing fundamental quantum operations, known as Paulis, improves directly with increased “squeezing” of the quantum state, a technique for reducing noise. However, the team discovered that certain more complex operations, called Cliffords, which function perfectly in ideal conditions, exhibit a constant error rate even with infinite squeezing when implemented in approximate quantum systems. This finding highlights a critical distinction between theoretical predictions and the realities of physically realizable quantum hardware. The research establishes a robust method for quantifying and managing errors in quantum computations, paving the way for more accurate and reliable quantum technologies.

Composable Gate Errors in Approximate GKP Codes

This work introduces a new metric, the composable logical gate error, to quantify the accuracy of logical gates in approximate error correction, acknowledging that perfect implementations are often unattainable. The researchers demonstrate that this metric effectively captures both deviations from the ideal gate operation and leakage from the code space, and importantly, it simplifies analysis of complex circuits due to its subadditivity. By applying this metric to linear implementations of quantum gates within approximate Gottesman-Kitaev-Preskill (GKP) codes, the team found that the accuracy of implementing Pauli gates improves consistently with increased squeezing. However, the study also reveals a crucial distinction between ideal and approximate GKP codes; certain Clifford gates, which function perfectly in ideal conditions, exhibit a constant logical gate error even with infinite squeezing when implemented in approximate codes.

This finding highlights that results derived for ideal GKP codes do not automatically translate to physically realistic, approximate systems. The authors acknowledge that their analysis relies on specific parameter choices and approximations within the GKP codes, and further research is needed to explore the behavior of more complex gate implementations and different code parameters. They also established relationships between various approximate GKP states, providing quantitative bounds on their overlap and establishing a foundation for future investigations into the properties of these states.