# Fidelity in Quantum Computing

## Table of Contents

## Introduction

According to a recent MIT article, IBM aims to build a 100,000 qubit quantum computer within a decade. Google is aiming even higher, aspiring to release a million qubit computer by by the end of the decade. We witness a continuous push towards larger quantum processors with increasing numbers of qubits. IBM is expected to release a 1,000-qubit processor sometime this year.

Quantum computing is on the brink of revolutionizing complex problem-solving. However, the practical implementation of quantum algorithms faces significant challenges due to the error-prone nature and hardware limitations of near-term quantum devices. Focusing solely on the number of qubits, as the media and marketing departments continue to do, is a bit of a red herring. Number of qubits is an easily quantifiable metric that keeps increasing every few months suggesting a straightforward path to quantum supremacyâ€”the point at which quantum computers can solve problems beyond the reach of classical supercomputers. However, this emphasis on quantity overlooks the quality of the computational process itself. A qubit, or quantum bit, is the quantum version of the classical binary bit. It is the basic unit of quantum information, capable of representing and processing complex data through states of superposition and entanglement. At first glance, it seems logical to assume that more qubits mean a more powerful quantum computer. However, this perspective neglects a critical factor that is equally, if not more, important: fidelity.

## The Fidelity Imperative

Fidelity in quantum computing measures the accuracy of quantum operations, including how effectively a quantum computer can perform calculations without errors. In quantum systems, noise and decoherence can degrade the coherence of quantum states, leading to errors and reduced computational accuracy. Errors are not just common; they’re expected. Quantum states are delicate, easily disturbed by external factors like temperature fluctuations, electromagnetic fields, and even stray cosmic rays.

This is where fidelity comes into play. A high-fidelity quantum processor performs operations with minimal errors, ensuring that the results of computations are reliable and accurate. Conversely, a processor with low fidelity, regardless of its qubit count, is prone to errors that can render its computational advantages moot. Today’s quantum systems, known as Noisy Intermediate-Scale Quantum (NISQ) machines are plagued by errors that significantly limit their value. A machine boasting thousands of qubits but only achieving 99% fidelity is arguably less useful than a smaller quantum computer with a 99.9% fidelity rate. Systems like Quantinuum, which utilize trapped-ion qubit technology, demonstrate that a smaller number of high-fidelity qubits can be more effective than a larger array of less stable ones.

## The Error Correction Challenge

Error correction is a significant challenge in quantum computing. Classical computers use redundancy to correct errorsâ€”storing multiple copies of data and using majority rules to fix discrepancies. Quantum error correction is far more complex due to the principles of superposition and entanglement. Effective quantum error correction requires additional qubits, further complicating the balance between qubit count and fidelity. Some of the key challenges of quantum error correction include:

### Quantum Superposition and Entanglement

Qubits can exist in a state of superposition, meaning they can represent both 0 and 1 simultaneously, and they can be entangled with other qubits, creating a complex correlation that doesn’t exist in classical computing. These properties are fundamental for quantum computing’s power but make error correction exceedingly difficult. Measuring a quantum state to check for errors can collapse its superposition or disrupt entanglement, thereby destroying the information you’re trying to protect.

### Quantum Decoherence

Qubits are extremely sensitive to their environment. Any interaction with the external world can cause them to lose their quantum properties through a process called decoherence. This includes interactions necessary for error correction. The challenge is to correct errors without accelerating decoherence, requiring sophisticated techniques that can detect and correct errors without directly measuring the quantum state.

### Error Types Are More Complex

In classical computing, errors can be simplified to bit flips (0 becomes 1, and vice versa). Quantum computing introduces a second type of error, phase flips, due to the nature of superposition. A qubit can experience both types of errors simultaneously, complicating the error correction process. Designing quantum error correction codes that can identify and correct both types of errors without disturbing the quantum state is a significant challenge.

### Resource Requirements

Quantum error correction schemes, such as the surface code, require a large number of physical qubits to encode a single logical qubit that is protected from errors. The overhead can be enormous, with current estimates suggesting thousands of physical qubits might be needed for one fault-tolerant logical qubit. Given the difficulty of maintaining even a small number of qubits in a coherent quantum state, scaling up to the numbers required for error correction is a daunting task.

### No Cloning Theorem

The no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. This theorem is a fundamental principle of quantum mechanics and poses a significant challenge for quantum error correction. In classical computing, error correction often involves copying data to backup or redundant systems. The no-cloning theorem means that quantum error correction must find ways to protect information without the ability to make backups.

### Other Technical Limitations

Current quantum computers are in the Noisy Intermediate-Scale Quantum (NISQ) era, characterized by machines with a limited number of qubits that can only maintain their quantum state for very short periods. The technical limitations of NISQ devices, including noise and operational errors, make implementing effective quantum error correction codes a massive challenge.

A quantum processor with a robust error correction mechanism can mitigate the impact of errors, enhancing its fidelity. This, however, is still beyond the current capability of quantum computers. It makes a strong case for prioritizing the development of high-fidelity quantum processors, even if it means having fewer qubits.

## Error Mitigation?

Of course, in the complex world of quantum, even the above statement is not fully correct. In a recent paper [PDF], IBM demonstrated that quantum computers, even in the current pre-fault tolerant stage, begin to demonstrate real-world utility by using error mitigation rather than error correction. The research challenges the prevailing notion that quantum computers must achieve fault tolerance before being of practical use. This efficiency does not imply a broad superiority across all computational tasks but signifies a targeted advantage in problems well-suited to the quantum architecture.

## Conclusion

In conclusion, while the number of qubits in a quantum processor is an important metric, it doesn’t tell the whole story. Fidelity and error correction are equally, if not more, significant in determining a quantum computer’s practical utility. As the field of quantum computing matures, understanding and improving these factors will be key to unlocking its full potential.