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What are the most advanced statistical methods for quantum computing R&D?

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Quantum computing is a rapidly evolving field that promises to revolutionize many domains, such as cryptography, artificial intelligence, and physics. However, developing and testing quantum algorithms and devices requires sophisticated statistical methods that can handle the inherent uncertainty and noise of quantum systems. In this article, you will learn about some of the most advanced statistical methods for quantum computing R&D, and how they can help you design, optimize, and validate your quantum solutions.

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1 Quantum Bayesian inference

One of the main challenges of quantum computing R&D is to infer the parameters and states of quantum systems from noisy and incomplete measurements. Quantum Bayesian inference is a framework that combines Bayesian statistics and quantum mechanics to update the beliefs about quantum systems based on the data and prior knowledge. Quantum Bayesian inference can be used to estimate the fidelity and performance of quantum devices, to design optimal quantum experiments, and to learn quantum models and algorithms.

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Georgios Korpas, Ph.D., MSCS

Senior Research Scientist @ HSBC | Artificial Intelligence, Quantum Computing, Optimization and Applied Mathematics
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This, to the right, is a very badly informed paragraph. To learn the parameters of quantum systems, given their delicacy, much more precise techniques must be used. Firstly, full quantum state tomography is not scalable to large systems. For smaller subsystems one technique I particularly like because it comes with convergence guarantees is that of moment/SOS relaxations as used in the context of polynomial optimization. In this case it can essentially be seen as minimizing the Frobenius norm of the measured vs expected state subject to physical constraints. This is a nonconvex nonlinear polynomial optimization problem and can be solved well. More generally, the current state-of-the-art methods involve classical shadows which are quite good

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2 Quantum machine learning

Quantum machine learning is a branch of quantum computing that applies machine learning techniques to quantum data and systems. Quantum machine learning can leverage the advantages of quantum computation, such as parallelism, entanglement, and interference, to enhance the speed, accuracy, and complexity of machine learning tasks. Quantum machine learning can be used to analyze and classify quantum data, to generate and optimize quantum states and circuits, and to implement quantum neural networks and reinforcement learning agents.

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Georgios Korpas, Ph.D., MSCS

Senior Research Scientist @ HSBC | Artificial Intelligence, Quantum Computing, Optimization and Applied Mathematics
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The paragraph on the left was written by ChatGPT (seriously?). If anybody suggests to use QML to learn about the parameters of the qubits or quantum states they must be delusional. Again, sure, one can use ML techniques but overall the best approaches are linear dynamical systems, open quantum systems specifically. For example, we can take master equations modelling these systems and map them, after proper discretization, to LTS and use all sort of well known DSP-inspired methods. Sampling methods and frequencies are crucial as well as the learning algorithms which, in my view, must be coming with certificates. For example, see my previous response. People have also tried deep learning approaches but no serious scientist would use QML here.

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3 Quantum error correction

Quantum error correction is a technique that protects quantum information from the effects of noise and decoherence, which can cause errors and loss of coherence in quantum systems. Quantum error correction uses redundant encoding of quantum bits (qubits) into larger quantum states, and applies error detection and correction procedures to identify and correct errors. Quantum error correction is essential for achieving scalable and reliable quantum computing, and requires advanced statistical methods to design and optimize the error correction codes and protocols.

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Georgios Korpas, Ph.D., MSCS

Senior Research Scientist @ HSBC | Artificial Intelligence, Quantum Computing, Optimization and Applied Mathematics
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Ok, how is this related to the previous questions is unclear. But yes, QEC is of fundamental importance and topological quantum error correction is quite cool in my view. Generally, new more efficient quantum error correction codes, syndrome extraction circuits, decoders, and fault-tolerant operations are part of the whole R&D roadmap startups should be focusing on. Particularly, it is already interesting (Nov 2023) that certain LDPC codes with a high encoding rate can achieve error threshold of 0.8% for the (standard) circuit-based noise model which definitely affirms the progress on the make.

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4 Quantum hypothesis testing

Quantum hypothesis testing is a technique that evaluates the validity of hypotheses about quantum systems or processes based on the observed data. Quantum hypothesis testing can be used to compare and discriminate between different quantum states, operations, or channels, to verify the properties or functionality of quantum devices or algorithms, and to measure the quantum resources or advantages of quantum computing. Quantum hypothesis testing involves calculating the probability of observing the data under different hypotheses, and choosing the most likely or optimal one.

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5 Quantum Monte Carlo methods

Quantum Monte Carlo methods are a class of computational methods that use random sampling to approximate the solutions of complex quantum problems. Quantum Monte Carlo methods can be used to simulate quantum systems or processes, to optimize quantum functions or parameters, and to estimate quantum quantities or expectations. Quantum Monte Carlo methods can overcome some of the limitations of classical computation, such as the exponential growth of the Hilbert space or the sign problem, by exploiting the stochastic nature of quantum mechanics.

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6 Quantum information geometry

Quantum information geometry is a branch of mathematics that studies the geometric structures and properties of quantum information spaces. Quantum information geometry can provide insights and tools for understanding and manipulating quantum systems, such as quantum states, operations, or channels, in terms of their distances, angles, curvature, or divergence. Quantum information geometry can be used to characterize and optimize the information content and processing of quantum systems, to design and implement quantum protocols or algorithms, and to explore the connections and differences between quantum and classical information theory.

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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What are the most advanced statistical methods for quantum computing R&D?

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1 Quantum Bayesian inference

2 Quantum machine learning

3 Quantum error correction

4 Quantum hypothesis testing

5 Quantum Monte Carlo methods

6 Quantum information geometry

7 Here’s what else to consider

Research and Development (R&D)

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