Tensor networks for Quantum computing

Tensors are mathematical objects that generalize scalars, vectors, and matrices to higher dimensions. They are used extensively in fields such as physics, engineering, and computer science, particularly in areas involving geometry, mechanics, and machine learning.

A tensor can be thought of as a multi-dimensional array of numbers that transforms according to certain rules under a change of coordinates. This property makes them very useful for expressing physical laws in a form that is independent of the choice of coordinate system.

In the context of machine learning, tensors are often used to represent data and operations in neural networks, where they can handle multiple dimensions of data and facilitate complex mathematical computations efficiently.

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The beginning of the series.

To my surprise Quantum Computer hardware by October 24, 2023 is built on 1180 qubits! Not to my surprise - of course on cold neutral atoms (mentioned that I worked with them in my PhD research). Again, surprisingly not by IBM or Google but some startup in the US, Atom Computing, Berkeley CA.

Another motivation to learn Quantum Mechanics and Quantum computing and become a Quantum Computer engineer (majority positions are for quantum engineers).

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Tensor Networks.

Now about Tensor Networks. It turns out that qubits can be simulated on classical devices, on GPUs. Not only simulated but Quantum Algorithms can even inspire some classical algorithms! So it is very interesting to see the connection between Tensor Networks and Quantum Computing.

Additionally, let me show how Tensor Networks can be used:

• Matrix Product States, factorization of wavefunction
• QEC: quantum error correction codes to make quantum computers less prone to errors
• Machine Learning: decomposition
• Quantum Field Theory and Condensed matter
• Quantum chemistry: design new molecules
• Graph theory
• Ofc Neuroscience (Wikipedia)
• Image processing
• For quantum spacetime simulation, tabletop experiments and playing with synthetic curvature

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A Promising Tool for Noise Characterization in Quantum Computers

Quantum computers are entering the utility era thanks to quantum error mitigation techniques that help reduce the effects of noise. However, characterizing quantum processes for tens to hundreds of qubits is challenging. A tensor-network-based quantum process tomography method has been developed to address this issue. This method allows for the accurate characterization of realistic correlated noise models affecting individual layers of quantum circuits. Combined with a noise-aware tensor network error mitigation protocol, it can provide accurate estimations even on deep circuit instances, making it a valuable tool for practical error characterization and mitigation in quantum computing.

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What is the Importance of Noise Characterization in Quantum Computers?

Quantum computers are currently entering what is termed as the utility era. This advancement has been made possible due to quantum error mitigation, a collection of techniques for mitigating and eventually eliminating the effects of noise in quantum computers without relying on quantum error correction. Despite recent progress, quantum error correction is currently out of reach for useful quantum computing.

Many error mitigation techniques require ideally perfect knowledge of the noise channels on the device, that is, of the actual physical processes implemented on the real machine instead of the ideal unitary gates or ideal sharp measurements. Characterizing quantum processes for tens to hundreds of qubits is, however, not a trivial task. Standard state and process tomography requires an exponential amount of resources as a function of the number of qubits.

Different techniques have been proposed to overcome this key issue and other inherent difficulties in tomographic methods such as the enforcement of physical constraints. Examples include twirling methods that tailor the investigated processes to specific simpler forms, classical shadow methods to reconstruct quantum processes, tensor network methods to characterize non-Markovian evolution processes, compressed sensing techniques for low-rank quantum processes, and methods that ensure meaningful tomography by appropriately restricting the reconstructed process to physical subspaces.

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How Can Tensor Networks Help in Noise Characterization?

In a recent study, a tensor-network-based quantum process tomography method was developed and applied to the problem of noise characterization in near-term devices. This method finds an efficient tensor network representation of the quantum process under scrutiny. The method was benchmarked with ideal circuits, i.e., unitary transformations up to 10 qubits and a noisy operation on five qubits subject to single-qubit errors.

However, some applications such as probabilistic error cancellation (PEC) or tensor network error mitigation (TEM) require the knowledge of the performance of individual gates or layers of gates. Therefore, the focus was on the characterization of each individual layer of a given circuit. Moreover, the process can be split into an ideal and a noisy part and characterize only the latter. These two modifications have the advantage of alleviating the numerical requirements of the method.

Realistic types of correlated noise, including the noisy model observed on IBM devices, were considered and systems of up to 20 qubits in size were investigated.

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What are the Results of the Tensor Network-Based Noise Learning Procedure?

The tensor-network-based noise learning procedure was studied by running several numerical experiments for the characterization of various correlated noise channels with brickwork-like structure and realistic noise parameters, which are of great relevance in near-term quantum computing.

The necessary amount of experimental settings and measurement shots to obtain an accurate reconstruction was discussed. It was found that collecting statistics on just a limited number of random experiments with informationally complete states and measurements provides sufficient data to accomplish the task. In particular, it was observed that linearly many experimental samples in the number of qubits suffice to ensure very good reconstructions.

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How Can This Method Be Used for Practical Error Characterization and Mitigation?

The tensor-network-based noise characterization protocol can be combined with a recently proposed noise-aware tensor network error mitigation protocol for correcting outcomes in noisy circuits, resulting in accurate estimations even on deep circuit instances. This positions the tensor-network-based noise characterization protocol as a valuable tool for practical error characterization and mitigation in the near-term quantum computing era.

In conclusion, the tensor network-based noise characterization method provides a promising alternative for noise characterization in near-term quantum computers. It allows for accurate characterization of realistic correlated noise models affecting individual layers of quantum circuits and its performance on systems composed of up to 20 qubits. This method, combined with a noise-aware tensor network error mitigation protocol, can result in accurate estimations even on deep circuit instances, making it a valuable tool for practical error characterization and mitigation in the near-term quantum computing era.

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