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Calculate multiple energy levels with a tensor network method



Solving the equations of quantum mechanics is of paramount importance to understanding how materials work at a fundamental level. At its core, the central equation of quantum mechanics – known as the Schrödinger equation – can be formulated as an eigenvalue problem in linear algebra. Solving the eigenvalues of these equations leads to the structure of the problem at the energy level. The lowest energies are the most important for modeling a system. By finding the lowest energies, systems can be accurately modeled before they are even created in the laboratory.

A method that efficiently computes the smallest absolute energy solutions is based on a set of linear algebra techniques known as tensor networks. The best-known network method, called Density Matrix Renormalization Group (DMRG) was first proposed nearly 30 years ago. Tensor networks have thus been used to simulate a wide variety of physical models and they have made important predictions.

This invention is a key extension of the fundamental concepts of GRDM. Rather than calculating the eigenvalues sequentially (one after the other), this invention makes it possible to calculate a large number of these eigenvalues simultaneously. This solution is very reliable and fast to calculate. Moreover, it performs better than all other methods, including those that work sequentially. More particularly, the proposed method is highly applicable to the design and characterization of superconducting circuit technologies – a leading candidate for building a quantum computer. The research team has even already demonstrated that the observed properties of these circuits can be accurately modeled by this method.


This technology simultaneously calculates more than one of the lowest eigenvalues. It has applications in calculating the properties of superconducting qubits and more.

Tensor networks have long been known to efficiently find the extreme eigenvalues at the ends of the full spectrum, especially for systems with local area-law interactions.

Finding the energy of the ground state with DMRG is often insufficient to completely characterize a given system. This is a common problem in many-body systems. Finding excited energy levels beyond the ground state is also necessary to better understand quantum systems in general.

This invention is an algorithm that finds more than one energy level at the same time. More importantly, this method avoids the accuracy issues that other DMRG techniques suffer from, which makes this invention very reliable in comparison.

One application is in superconducting circuits, which consist of a network or mesh of many superconducting junctions. With this invention, the representation of the mesh problem in a network of tensors makes it possible to determine the lowest energy levels of the circuit. These values can be used to find relevant experimental quantities – such as qubit coherence times – and find a high degree of accuracy with known experimental observations.

Figure 1 and Figure 2 (see PDF).



  • Linear scaling based on system size.

  • Resolution of energy levels according to the chosen precision of the tensor network algorithm.

  • Can solve any number of excitations simultaneously.



  • Previously unsolvable superconducting circuit models can be solved using this method.

  • Fast and efficient: the method makes it possible to determine hundreds of excitations for large-scale superconducting circuits.

  • Avoid common mistakes: resolve degeneracies and don't omit energy levels.

  • Model independent: this method does not depend on any model and can, in principle, work with any physical system.

  • Several interested potential partners and institutions: potential industrial partners such as IBM, Microsoft, Google, Northrup Grumman and academic institutions such as MIT Lincoln Laboratory, ETH Zurich, Berkeley and Yale.


  • Design and characterization of superconducting circuits.

  • Potential application to other interacting systems (example: in the field of quantum materials).



Technology Readiness Level (TRL): 6.5 - this includes a fully functional Tensor Network Library (DMRjulia).

  • Simulations of previously unsolvable models have already been carried out, notably the fluxonium circuit.

  • Extensions of the Hubbard models are in progress.

  • The research team forms collaborations to make maximum use of this method.



Patent application filed.




Project Director: François Nadeau

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