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Error correction vs. Error mitigation

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December 11, 2023
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min read

Today’s quantum computers have error rates that are too high for their calculations to be considered of any commercial use. Error Mitigation (EM) and Quantum Error Correction (QEC) are two of the three strategies being employed to reduce these error rates to tolerable levels; the third is quantum error suppression, which aims to prevent errors from occurring in the first place. When comparing Error Mitigation vs. Quantum Error Correction, the major differences between the two are:

  • QEC refers to error detection and correction during the execution of the algorithm, whereas EM compensates for errors after execution has been completed and after measurement results have been returned
  • QEC significantly increases the size and complexity of quantum circuits, whereas EM involves classical post-processing that does not affect the quantum circuits at all
  • QEC lowers error rates in direct proportion to the additional overhead they introduce, whereas EM slows the increase in error rates as circuits scale in width and depth
  • QEC protects quantum information by distributing it across the multiple physical qubits that encode a single logical qubit, whereas EM is neither preventative nor protective
  • QEC is restricted to transversal gate operations on logical qubits, whereas EM does not impose any runtime-related restrictions

The single most important difference between QEC and EM is their long-term applicability. QEC is the strategy that will lower error rates to acceptable tolerances and usher in the era of Fault-Tolerant Quantum Computing (FTQC). In contrast, EM has scalability issues that limit the quantum corrections that can be done with existing methods even in the near-term.


What is Error Mitigation

Quantum error mitigation is classical computation that takes the measurement results from a quantum circuit and adjusts them based on the noise model of the target quantum computer. This noise model characterizes the device after errors have already been suppressed, detected, and corrected to the extent possible. Various methods can determine where and how errors manifest, those errors can then be assumed to have occurred, and then compensation for those errors can be classically applied to the measurement results.  

It is important to note that EM does not magically correct all the residual errors. What happens is that overall failure rates increase as quantum circuits scale in width and depth. EM effectively reduces the rate of that increase. However, since the failure rates continue to increase, albeit at a slower rate, EM eventually loses its applicability to large-scale quantum computing.

Another way to define EM is through an explanation of what it does not do. Specifically, EM does not impose conditions of any kind on the quantum circuits. For a few examples: physical qubits do not need to be encoded as logical qubits, operations are not restricted to transversal gates, and mid-circuit measurements do not need to be performed on stabilizer qubits.

Key Components of Error Mitigation

Multiple methods of EM have been demonstrated experimentally, and more are probably being researched. Some of the characteristics that these methods have in common include:

  • Despite best efforts to suppress, detect, and correct errors, errors have nonetheless occurred.
  • Compensation for these residual errors occurs during classical post-processing, and does not affect circuit design or implementation in any way.
  • The overall probability of failure increases at a slower rate, as circuits scale in width and depth, than if the method was not implemented.  
  • Gate operations are not restricted in any way, although the fidelity of implementing the selected gates will influence the noise model.
  • The “qubits” in the quantum circuit can refer to either physical or logical qubits since the method itself does not have any preventative or protective role.

At a high level, EM starts with measurement results and a proposed solution to a problem. It essentially looks at how the noise of the quantum computer may have affected the measurements, and it adjusts the results to compensate for that noise. The proposed solution might now be more convincing, or the proposed solution may have changed. But the goal is to reduce the failure rate of the quantum computer and to arrive at a correct solution for the problem.

Advantages of Error Mitigation

Despite the limitations of EM heading into the era of large-scale Fault-Tolerant Quantum Computing (FTQC), it is nonetheless advantageous while we remain in the Noisy Intermediate Scale Quantum (NISQ) era. The advantages over its companion strategies include:

  • Error suppression, detection, and correction methods do not address the errors that can arise as final measurements are taken; only mitigation methods can.
  • Even if errors and failures occur in parts of a circuit, EM might be able to consider what success should look like and adjust the measurement results accordingly.
  • The catching of individual errors diminishes in significance, because the errors become somewhat statistical in nature.  

In summary, by reducing the rate of increase of failure, EM allows larger circuits to be demonstrated on NISQ hardware than would otherwise be possible.  

Core Principles of Quantum Error Correction

Despite the considerable variety of implementations, called quantum error correction codes (QECC), QEC only has a handful of core principles. At a high level, these principles are:

  • Quantum information is distributed across multiple physical qubits, which are then encoded into logical qubits designed to protect this quantum information.
  • This quantum information can be measured in a way that preserves the quantum information on the data qubits while the quantum information on the auxiliary qubits is destroyed.
  • These mid-circuit measurements, as they are called, on the auxiliary qubits detect the presence of errors, as well as the types of errors, on the undisturbed data qubits.
  • The encoding of the physical qubits within the logical qubit determines which of the physical qubits have been affected by the errors.
  • Classical logic is used to apply corrective operations on the affected qubit(s), thus correcting the error(s) before the error(s) can propagate throughout the system.

The underlying theme, therefore, is preservation. Errors that are not suppressed, are going to occur. If the quantum information is encoded onto only one qubit, an error could have a cascading effect throughout the entire system. QEC essentially contains an error, prevents its spread, and preserves the correct quantum information as a logical qubit. Transversal operations then act upon the preserved, correct quantum information only.  

Advantages of Quantum Error Correction

References to FTQC always refer to the use of logical qubits, which means that QEC codes are essential. As a long-term strategy, the advantages of QEC include:

  • The detection and correction of both bit-flip and phase-flip errors, thus allowing calculations to be executed accurately.
  • The revelation of insights into the causes of errors, which can in turn lead to less-noisy hardware, as well as in more effective error suppression methods.
  • The revelation of insights into the propagation of errors throughout a system, which can lead to new and revised codes.
  • The co-design of hardware and QECC such that the hardware optimally implements the code, and the code is optimized for the hardware.
  • The lessons of classical error correction, while not all applicable to quantum information, are time-tested.

In summary, QEC is a long-term strategy. It is more theory than practice at the moment, although demonstrations are scaling up as quantum computers become larger.  

Comparing EM and QEC

Error Mitigation (EM) and Quantum Error Correction (QEC) are different-but-complementary strategies. For starters, QEC is implemented during runtime, whereas EM is implemented after execution has completed. Some of the other ways in which they compare and contrast include:

  • Because of limited numbers of physical qubits, QEC cannot be demonstrated at scale today, whereas EM is more effective at the smaller scales of today.
  • QEC can lower error rates to tolerable levels, whereas EM slows down the rate at which failure becomes more likely as circuits scale in width and depth.
  • QEC implements both quantum and classical operations, whereas EM focuses solely on classical operations.
  • QEC does not address errors that arise while taking and returning final measurements, whereas EM can.  
  • QEC codes are defined, and sometimes fabricated, whereas the noise models used by EM may be constantly changing.

The key takeaway is that QEC is the long-term strategy while EM is a near-term strategy. The goal of QEC is to reduce error rates, which can be brought down through the encoding of additional physical qubits per logical qubit. Research continues on both strategies, however, so only time will tell what FTQC will look like.

For more information, “Introduction to Quantum Error Correction” summarizes an article that discusses the need for QEC, the types of errors, Error Mitigation vs. Quantum Error Correction, logical qubits, prototypical QEC codes, the current state of the art, and concerns about scalability. Another option is the Q-CTRL Advanced article “Differentiating quantum error correction, suppression, and mitigation,” which discusses suppression, correction, and mitigation and the implementation of these three strategies as an ensemble.

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