Overview of Quantum Error Mitigation in NISQ through QML
This article is part of our research under the guidance of Professor N.Geethanjali, Department of Computer Science & Technology, Sri Krishnadevaraya University, Andhra Pradesh State, INDIA.
QEM is the recent advancement in Quantum Computing NISQ devices in order to reduce the impact of noise caused by Quantum Hardware during computation.
This article is covered in three sections. One is to briefly introduce Quantum Errors and Noise, the second is QEM techniques, and the final one is to QEM vs QML.
I — Introduction to Quantum Errors and Noise
Noise and Errors are the two bottlenecks in quantum computing that need to solve and get better and more accurate computation results. These two lead more research in order to perfect our quantum models in Quantum Computing and Quantum Machine learning. The below diagram gives a clear distinction between these two.
Near Term Devices
The word “Near Term” refers to quantum computation and has been coined to cluster quantum algorithms to be run on current quantum computing hardware which could be developed in the next few years.
NISQ Devices
NISQ is a hardware-focused definition and does not necessarily imply a temporal connotation.
NISQ devices will be useful tools for exploring many-body quantum physics and may have other useful applications, but the 100-qubit quantum computer will not change the world right away — we should regard it as a significant step toward the more powerful quantum technologies of the future. Prototypes of quantum computers, also known as noisy intermediate-scale quantum (NISQ) computers[5].
NISQ implements the model of quantum circuits, in which all quantum gates bind to the topology of graph G, and nodes in the graph represent qubits and edges as linear operators between the qubits.
Quantum Noise
Let us define Noise and its related concepts that deal with Quantum Information.
Noise can be defined as multiple factors that affect the accuracy of calculations in NISQ devices. Quantum Computers can easily be susceptible to noise from sources of various disruptions causing the quantum information an idle qubit holds to fade away. Another way of causing errors in quantum information is a quantum logical operation on qubits. In all the ways, the final quantum state of the quantum computer is not the expected precise state. To avoid these difficulties of causing noise, the quantum information becomes randomized or totally erased. This physical process is known as “ Decoherence”.
The below example gives a clear picture of Noise.
Noise is caused by decoherence and imperfect control.
Noise occurring in a quantum processor. It is difficult to sufficiently isolate the qubits from the effects of external noise, errors during quantum computation are inevitable.
The impact of embedding noise in data causes to spoil the computation process and leads to incorrect expected results.
Quantum Error Vs Quantum Noise
Quantum Errors may occur during storing and transformation or moving quantum information in qubits, whereas noise may occur during the calculations, state preparation, or the final measurement of the information in Quantum Hardware. Kindly note that during each quantum gate operation makes a certain amount of noise.
Types of Errors and Quantum Computers
Quantum Noise makes an influence on qubits which leads to errors in basis states. Primarily there are two types of flip errors in Quantum Computing: bit-flip errors and phase-flip errors. Bit-flip errors occur during the state of the qubit changes from |0> to |1> and vice versa. These are also known as X-errors. On the other hand, phase-flip errors may occur during |1> to changes to -|1> but |0> remains |0>. Both errors can interact with each other and raise complex errors in the quantum system.
The errors that arise in Quantum computers are Gate errors, Decoherence errors, Measurement errors, and cross-talk errors. These errors are more explored in the upcoming Quantum Error Correction article.
Mitigation
Mitigation can be treated as a transformation that maps the noise quantum function to a new mitigated function, and is mathematically defined as follows:
Current Quantum Computing hardware is subject to various sources of noise, and QEM over these difficulties using statistical methods.
The difference between Error and Noise needs to understand before applying the respective technique in order to reduce their effect.
Quantum Hardware — Bias — Noise — QEM
Quantum Hardware is key to making noise in quantum operations. For perfect and imperfect quantum systems in order to deal with noise. In perfect hardware, there would be zero bias and noiseless, whereas in imperfect hardware there would be finite bias and QEM entered to solve finite bias through its protocols. QEM reduces the computational errors and then evaluates accurate results from noisy quantum circuits.
From the above diagram, it is clear that in the absence of QEM, the noise increases output information degrades, in the presence of QEM the impact of noise is reduced and exponentially growth in algorithms output.
QEM is NISQ Technique
QEM is collaborating with other NISQ Techniques [1] like Variational Quantum Algorithms, Quantum Error Mitigation, Quantum Circuit Compilation, and Benchmarking protocols. The relationship between NISQ techniques is described below the picture. The collaboration of these techniques to execute various complex computational tasks in a wide range of domains such as Mathematical calculations, Machine Learning Algorithms, Cryptoanalysis, chemical or drug discovery, optimization problems, etc.,
QEM techniques need no extra qubit and can crush errors in finding expectation values with simple classical post-processing and different runs of quantum circuits.
Why QEM, not QEC?
QEC promises to enable Quantum Computation with arbitrary levels of noise, it is out of reach for near-term Quantum processes. To overcome this difficulty QEM enters into the picture.
QEM especially deals with near-term quantum processes. QEM provides us a feasible alternative to mitigate errors of near-term processes, and the continuous path that will take us from today’s quantum hardware to tomorrow’s fault-tolerant quantum computers.
QEM is an excellent tool for improving the performance of NISQ computing based on the reduce the impact of Noise.
QEM is a Postprocessing Approach
The below figure depicts the scope of QEM after executing the quantum circuit and dealing with noisy output. It is considered a post-processing approach due to the completion of the quantum circuit and starting over to work on the output.
II — Quantum Error Mitigation Techniques
There are certain concepts that need to be aware of to implement Quantum Error Mitigation techniques. These are related to the implementation of Circuits and other required parts covered in the respective article. These concepts are noise channel, inverse noisy channel, noisy state, noisy circuit, and noisy model.
In a nutshell, the below diagram describes the required circuit-level components for QEM. Each component is integrated with a high component and all these are easily tractable in QEM techniques.
Let us define the concepts mentioned in the above diagram, which are important for doing quantum programming in major frameworks. Sometimes, we need to simulate or create noise-based programs in order to see the effect or prototypes in real-world applications.
The general noise on a quantum device is called Noise Channel.
The Quantum State yields noise after operations applied to it is called Noise State.
There is no concept called Noise Gate, the Quantum gates operate on noise information or make noise after gate operation.
The Noise model encapsulates the assumptions on noise channels and acts on quantum circuits. These quantum circuits are called Noise Circuits.
Inverse Noisy Channel
What does QEM require?
Fewer qubits and gate resources and is, therefore, more suitable for practical NISQ devices.
There are certainly desirable features for error mitigation techniques or protocols or methods.
- QEM protocol ideally requires qubit overhead on current NISQ Computers.
- QEM protocol works at varying levels of Noise.
- QEM protocol requires a few assumptions (or no assumptions) about the final state that is prepared by the computation.
Methods or Techniques
The most common methods of QEM are Probabilistic Error Cancellation, Measurement Error Mitigation, Symmetry Constraints, Purity Constraints, Subspace Expansions, N-Representability, and Learning—Based. Each method or technique has its own specialization. Out of these techniques, there are various individual methods developed for the specific purpose and a few of them are …
Kindly note that we give an outline of the QEM technique in order to introduce techniques from a high-level perspective. The reader who is interested in the respective protocol or technique can refer to the research paper provided reference or in the references list. Required concepts, terminology, and notations have to be introduced before understanding QEM protocols, everything cannot be covered in a single article, will do the needful in a separate article on QEM methods.
1. Zero-Noise Extrapolation (ZNE) [10]
ZNE is a typical error mitigation protocol using classical post-processing, it collects at different error rates to fit the function of expectation values with respect to the collected error rates and then extrapolates to the zero noise limit; otherwise will make use of noisy states gathered from different circuit fault rates. Calculating expectation value for Circuit fault rates. ZNE is depicted in the below diagram.
ZNE can be defined as a function. A function ‘f’ accepts different error rates
and transform to zero noise limit is called “Zero Noise Extrapolation”.
2. Learning-Based Protocol [7]
PEC completely depends on the full tomography of the noise channels.
A function ‘f’ accepts the primary circuit and a set of parameters and outputs the error-mitigated expectation value. i.e.,
The Learning-based Protocol process is depicted in the below diagram
ZNE and Learning based protocols are considered to be Data-driven approaches that collect expectation values of circuits with different rates, or circuits with similar structures.
3. Probabistic Error Cancel (PEC) [9]
PEC estimates the noiseless expectation value using a linear combination of expectation values with different noise terms, and it is completely on Quasi-Decomposition. The below diagram depicts the PEC process.
The main intuition of PEC is to apply the distribution techniques Quasi-probability decomposition of the inverse noise processes, leading to a linear combination of a set of noisy circuits.
4. Measurement Error Mitigation
This protocol aims to improve the accuracy of the measurement results obtained from noisy quantum devices. The error may be encountered at the final measurement stage of the Quantum Algorithm introducing an additional bias in the expectation value. When we perform a measurement to obtain the output binary string, ideally we want to perform projective measurement into some positive operator-value measures yield to another output statistic.
5. Quantum Subspace expansion
This technique uses post-processing to mitigate errors for some VQE algorithms. Subspace expansion mitigates coherent errors due to imperfect variational optimization. Consider an example where a noisy ground state is prepared on the quantum computer and then target Hamiltonian, and variational subspace spanned by group state as
The quantum subspace is spanned as per the noise levels of the quantum states. Subspace expansion is one of the strategies for mitigation techniques.
The key idea is to extend the notion of quantum subspaces to include general operators that are related to the target’s noisy quantum states, it allows to extract of the state into an error-mitigated eigenstate of the target.
6. Symmetry Constraints [11]
Symmetry expansion and Symmetry verification frameworks play vital roles in dealing with noisy output states produced by Quantum circuits.
Symmetry verification is a QEM protocol that projects the noisy output quantum state back into the symmetry subspace.
Symmetry verification removes the quantum circuit execution and costs additional measurement. This approach leads to wrong results and produces a post-selected state. Symmetry verification is a special case of the QSE protocol. This method is especially used for low-cost error mitigation[12].
On the negative side, the symmetry verification circuit applied in the noisy state may lead to extra errors and reduces the reliability of verification.
Symmetry Expansion
Symmetry Expansion is a uniform expansion using a subset of symmetry operators
Symmetry Expansion mechanisms have various symmetry operators and mechanisms to deal with noise output states and are well explained in this article [11]. The popular symmetry expansion mechanisms are Standard behavior, Bias Cancellation, and Negative infidelity.
III — Integrating QEM to the QAI/QML
Let us consider the research perspective under noise and decohere mitigation and the close relationship of QEM and QML can assist each other in the Quantum Computing domain.
There are factors for integrating QEM into QAI/QML in terms of Noise, Bias, and variance tradeoffs. QEM and AI/ML domain has the underlying relationship of noise handling in the respective algorithms and this may cause us to easily integrate QEM into QAI/QML.
Machine Learning /Deep Learning algorithms reduce noise-based and decoherence issues and are considered as the tool for dealing with QEM and can be implemented in Quantum Neural Networks [6] for various reasons. One of the research examples shows that deep-learning-based protocol for reducing readout errors on quantum hardware.
QEM has a strong relationship with AI/ML/DL due to noise reduction and it does not require more qubits to solve the issues of noise. QEM protocols can be applied in state preparation, and final measurement stages in quantum algorithms, and these two tasks are very important in the Learning area.
Noise and decoherence may even be of advantage in training, as it helps correct overfitting issues in the Machine Learning domain [6]. Neural Networks play a vital role in order to reduce noise and have various techniques dealing in this regard.
First approach QML for QEM
Noise is the main problem in QEM and QML, QML/QNN methods are being used as a tool for reducing the noise for the QEM technique. One of the best examples using the QEM technique implemented is Learning-Based through Machine Learning protocol using Clifford Circuits[7]. The below diagrams give a clear picture of using QEM and QML together while working with NISQ computers.
Second approach QEM for QML
So far, our discussion goes to QEM being able to handle noise data with the help of QML in NISQ. Let us consider QEM as a postprocessing approach to correcting the noise data and revert to the QML algorithm for training in NISQ computers. In this scenario, QEM is a postprocessing approach along with preprocessing step for correcting the data for QML Algorithms. In this scenario, the computation model is depicted as follows.
In this scenario, QML cannot directly correct the data in NISQ rather it has to be assisted by QEM protocols (even though QML is used as a tool for noise reduction), but QML is trained in NISQ, there has to be dealt with noise data through QEM only, due to this QEM acts as a preprocessing step for QML algorithms in NISQ. This scenario comes across in production systems.
Conclusion
We have explored Noise related concepts in Quantum Computing. Have more elaborate types of noises that can handle in QC along with QEM protocols in further articles. This article defined more in terms of research perspective to get a glance at QEM and its strong relationship with AI or Machine Learning domain.
Thanks for reading our article, appreciated your feedback, comments, sharing, and mistakes if any.
References
- Near-Term Quantum Computing Techniques https://arxiv.org/abs/2211.08737
2. Quantum readout error mitigation via deep learning https://arxiv.org/abs/2112.03585
3. Quantum Error Mitigation https://arxiv.org/abs/2210.00921
4. Independent state and measurement characterization for quantum computers:https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.033285
5. Quantum Computing in the NISQ Era and Beyond by John Preskill https://arxiv.org/abs/1801.00862
6. Quantum Learning with Noise and Decoherence: https://arxiv.org/ftp/arxiv/papers/1612/1612.07593.pdf
7. Learning-based quantum error mitigation: https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.2.040330
8. Impact of quantum noise on the training of quantum Generative Adversarial Networks: https://arxiv.org/abs/2203.01007
9. Error mitigation for short-depth quantum circuits: https://arxiv.org/abs/1612.02058
10. Practical Quantum Error Mitigation for Near-Future Applications: https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.031027
11. Quantum Error Mitigation using Symmetry Expansion: https://quantum-journal.org/papers/q-2021-09-21-548/
12. Low-cost error mitigation by symmetry verification: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.062339
