How Quantinuum researchers used quantum tensor networks to measure the properties of quantum particles at a phase transition
Quantum tensor networks demonstrate potential exponential resource reduction in both time and memory for calculation of critical state properties in digital quantum computers
When thinking about changes in phases of matter, the first images that come to mind might be ice melting or water boiling. The critical point in these processes is located at the boundary between the two phases – the transition from solid to liquid or from liquid to gas.
Phase transitions like these get right to the heart of how large material systems behave and are at the frontier of research in condensed matter physics for their ability to provide insights into emergent phenomena like magnetism and topological order. In classical systems, phase transitions are generally driven by thermal fluctuations and occur at finite temperature. On the contrary, quantum systems can exhibit phase transitions even at zero temperatures; the residual fluctuations that control such phase transitions at zero temperature are due to entanglement and are entirely quantum in origin.
Quantinuum researchers recently used the H1-1 quantum computer to computationally model a group of highly correlated quantum particles at a quantum critical point — on the border of a transition between a paramagnetic state (a state of magnetism characterized by a weak attraction) to a ferromagnetic one (characterized by a strong attraction).
Simulating such a transition on a classical computer is possible using tensor network methods, though it is difficult. However, generalizations of such physics to more complicated systems can pose serious problems to classical tensor network techniques, even when deployed on the most powerful supercomputers. On a quantum computer, on the other hand, such generalizations will likely only require modest increases in the number and quality of available qubits.
In a technical paper submitted to the arXiv, Probing critical states of matter on a digital quantum computer, the Quantinuum team demonstrated how the powerful components and high fidelity of the H-Series digital quantum computers could be harnessed to tackle a 128-site condensed matter physics problem, combining a quantum tensor network method with qubit reuse to make highly productive use of the 20-qubit H1-1 quantum computer.
Reza Haghshenas, Senior Advanced Physicist, and the lead author the paper said, “This is the kind of problem that appeals to condensed-matter physicists working with quantum computers, who are looking forward to revealing exotic aspects of strongly correlated systems that are still unknown to the classical realm. Digital quantum computers have the potential to become a versatile tool for working scientists, particularly in fields like condensed matter and particle physics, and may open entirely new directions in fundamental research.”
The role of quantum tensor networks
Tensor networks are mathematical frameworks whose structure enables them to represent and manipulate quantum states in an efficient manner. Originally associated with the mathematics of quantum mechanics, tensor network methods now crop up in many places, from machine learning to natural language processing, or indeed any model with a large number of interacting, high-dimensional mathematical objects.
The Quantinuum team described using a tensor network method--the multi-scale entanglement renormalization ansatz (MERA)--to produce accurate estimates for the decay of ferromagnetic correlations and the ground state energy of the system. MERA is particularly well-suited to studying scale invariant quantum states, such as ground states at continuous quantum phase transitions, where each layer in the mathematical model captures entanglement at different scales of distance.
“By calculating the critical state properties with MERA on a digital quantum computer like the H-Series, we have shown that research teams can program the connectivity and system interactions into the problem,” said Dave Hayes, Lead of the U.S. quantum theory team at Quantinuum and one of the paper’s authors. “So, it can, in principle, go out and simulate any system that you can dream of.”
Simulating a highly entangled quantum spin model
In this experiment, the researchers wanted to accurately calculate the ground state of the quantum system in its critical state. This quantum system is composed of many tiny quantum magnets interacting with one another and pointing in different directions, known as a quantum spin model. In the paramagnetic phase, tiny, individual magnets in the material are randomly oriented, and only correlated with each other over small length-scales. In the ferromagnetic phase, these individual atomic magnetic moments align spontaneously over macroscopic length scales due to strong magnetic interactions.
In the computational model, the quantum magnets were initially arranged in one dimension, along a line. To describe the critical point in this quantum magnetism problem, particles in the line needed to be entangled with one another in a complex way, making this as a very challenging problem for a classical computer to solve in high dimensional and non-equilibrium systems.
“That's as hard as it gets for these systems,” Dave explained. “So that's where we want to look for quantum advantage – because we want the problem to be as hard as possible on the classical computer, and then have a quantum computer solve it.”
To improve the results, the team used two error mitigation techniques, symmetry-based error heralding, which is made possible by the MERA structure, and zero-noise extrapolation, a method originally developed by researchers at IBM. The first involved enforcing local symmetry in the model so that errors affecting the symmetry of the state could be detected. The second strategy, zero-noise extrapolation, involves adding noise to the qubits to measure the impact it has, and then using those results to extrapolate the results that would be expected under conditions with less noise than was present in the experiment.
Future applications
The Quantinuum team describes this sort of problem as a stepping-stone, which allows the researchers to explore quantum tensor network methods on today’s devices and compare them either to simulations or analytical results produced using classical computers. It is a chance to learn how to tackle a problem really well before quantum computers scale up in the future and begin to offer solutions that are not possible to achieve on classical computers.
“Potentially, our biggest applications over the next couple of years will include studying solid-state systems, physics systems, many-body systems, and modeling them,” said Jenni Strabley, Senior Director of Offering Management at Quantinuum.
The team now looks forward to future work, exploring more complex MERA generalizations to compute the states of 2D and 3D many-body and condensed matter systems on a digital quantum computer – quantum states that are much more difficult to calculate classically.
The H-Series allows researchers to simulate a much broader range of systems than analog devices as well as to incorporate quantum error mitigation strategies, as demonstrated in the experiment. Plus, Quantinuum’s System Model H2 quantum computer, which was launched earlier this year, should scale this type of simulation beyond what is possible using classical computers.
About Quantinuum
Quantinuum, the world’s largest integrated quantum company, pioneers powerful quantum computers and advanced software solutions. Quantinuum’s technology drives breakthroughs in materials discovery, cybersecurity, and next-gen quantum AI. With over 500 employees, including 370+ scientists and engineers, Quantinuum leads the quantum computing revolution across continents.
At the heart of quantum computing’s promise lies the ability to solve problems that are fundamentally out of reach for classical computers. One of the most powerful ways to unlock that promise is through a novel approach we call Generative Quantum AI, or GenQAI. A key element of this approach is the Generative Quantum Eigensolver (GQE).
GenQAI is based on a simple but powerful idea: combine the unique capabilities of quantum hardware with the flexibility and intelligence of AI. By using quantum systems to generate data, and then using AI to learn from and guide the generation of more data, we can create a powerful feedback loop that enables breakthroughs in diverse fields.
Unlike classical systems, our quantum processing unit (QPU) produces data that is extremely difficult, if not impossible, to generate classically. That gives us a unique edge: we’re not just feeding an AI more text from the internet; we’re giving it new and valuable data that can’t be obtained anywhere else.
The Search for Ground State Energy
One of the most compelling challenges in quantum chemistry and materials science is computing the properties of a molecule’s ground state. For any given molecule or material, the ground state is its lowest energy configuration. Understanding this state is essential for understanding molecular behavior and designing new drugs or materials.
The problem is that accurately computing this state for anything but the simplest systems is incredibly complicated. You cannot even do it by brute force—testing every possible state and measuring its energy—because the number of quantum states grows as a double-exponential, making this an ineffective solution. This illustrates the need for an intelligent way to search for the ground state energy and other molecular properties.
That’s where GQE comes in. GQE is a methodology that uses data from our quantum computers to train a transformer. The transformer then proposes promising trial quantum circuits; ones likely to prepare states with low energy. You can think of it as an AI-guided search engine for ground states. The novelty is in how our transformer is trained from scratch using data generated on our hardware.
Here's how it works:
- We start with a batch of trial quantum circuits, which are run on our QPU.
- Each circuit prepares a quantum state, and we measure the energy of that state with respect to the Hamiltonian for each one.
- Those measurements are then fed back into a transformer model (the same architecture behind models like GPT-2) to improve its outputs.
- The transformer generates a new distribution of circuits, biased toward ones that are more likely to find lower energy states.
- We sample a new batch from the distribution, run them on the QPU, and repeat.
- The system learns over time, narrowing in on the true ground state.
To test our system, we tackled a benchmark problem: finding the ground state energy of the hydrogen molecule (H₂). This is a problem with a known solution, which allows us to verify that our setup works as intended. As a result, our GQE system successfully found the ground state to within chemical accuracy.
To our knowledge, we’re the first to solve this problem using a combination of a QPU and a transformer, marking the beginning of a new era in computational chemistry.
The Future of Quantum Chemistry
The idea of using a generative model guided by quantum measurements can be extended to a whole class of problems—from combinatorial optimization to materials discovery, and potentially, even drug design.
By combining the power of quantum computing and AI we can unlock their unified full power. Our quantum processors can generate rich data that was previously unobtainable. Then, an AI can learn from that data. Together, they can tackle problems neither could solve alone.
This is just the beginning. We’re already looking at applying GQE to more complex molecules—ones that can’t currently be solved with existing methods, and we’re exploring how this methodology could be extended to real-world use cases. This opens many new doors in chemistry, and we are excited to see what comes next.
Last year, we joined forces with RIKEN, Japan's largest comprehensive research institution, to install our hardware at RIKEN’s campus in Wako, Saitama. This deployment is part of RIKEN’s project to build a quantum-HPC hybrid platform consisting of high-performance computing systems, such as the supercomputer Fugaku and Quantinuum Systems.
Today, a paper published in Physical Review Research marks the first of many breakthroughs coming from this international supercomputing partnership. The team from RIKEN and Quantinuum joined up with researchers from Keio University to show that quantum information can be delocalized (scrambled) using a quantum circuit modeled after periodically driven systems.
"Scrambling" of quantum information happens in many quantum systems, from those found in complex materials to black holes. Understanding information scrambling will help researchers better understand things like thermalization and chaos, both of which have wide reaching implications.
To visualize scrambling, imagine a set of particles (say bits in a memory), where one particle holds specific information that you want to know. As time marches on, the quantum information will spread out across the other bits, making it harder and harder to recover the original information from local (few-bit) measurements.
While many classical techniques exist for studying complex scrambling dynamics, quantum computing has been known as a promising tool for these types of studies, due to its inherently quantum nature and ease with implementing quantum elements like entanglement. The joint team proved that to be true with their latest result, which shows that not only can scrambling states be generated on a quantum computer, but that they behave as expected and are ripe for further study.
Thanks to this new understanding, we now know that the preparation, verification, and application of a scrambling state, a key quantum information state, can be consistently realized using currently available quantum computers. Read the paper here, and read more about our partnership with RIKEN here.
In our increasingly connected, data-driven world, cybersecurity threats are more frequent and sophisticated than ever. To safeguard modern life, government and business leaders are turning to quantum randomness.
What is quantum randomness, and why should you care?
The term to know: quantum random number generators (QRNGs).
QRNGs exploit quantum mechanics to generate truly random numbers, providing the highest level of cryptographic security. This supports, among many things:
- Protection of personal data
- Secure financial transactions
- Safeguarding of sensitive communications
- Prevention of unauthorized access to medical records
Quantum technologies, including QRNGs, could protect up to $1 trillion in digital assets annually, according to a recent report by the World Economic Forum and Accenture.
Which industries will see the most value from quantum randomness?
The World Economic Forum report identifies five industry groups where QRNGs offer high business value and clear commercialization potential within the next few years. Those include:
- Financial services
- Information and communication technology
- Chemicals and advanced materials
- Energy and utilities
- Pharmaceuticals and healthcare
In line with these trends, recent research by The Quantum Insider projects the quantum security market will grow from approximately $0.7 billion today to $10 billion by 2030.
When will quantum randomness reach commercialization?
Quantum randomness is already being deployed commercially:
- Early adopters use our Quantum Origin in data centers and smart devices.
- Amid rising cybersecurity threats, demand is growing in regulated industries and critical infrastructure.
Recognizing the value of QRNGs, the financial services sector is accelerating its path to commercialization.
- Last year, HSBC conducted a pilot combining Quantum Origin and post-quantum cryptography to future-proof gold tokens against “store now, decrypt-later” (SNDL) threats.
- And, just last week, JPMorganChase made headlines by using our quantum computer for the first successful demonstration of certified randomness.
On the basis of the latter achievement, we aim to broaden our cybersecurity portfolio with the addition of a certified randomness product in 2025.
How is quantum randomness being regulated?
The National Institute of Standards and Technology (NIST) defines the cryptographic regulations used in the U.S. and other countries.
- NIST’s SP 800-90B framework assesses the quality of random number generators.
- The framework is part of the FIPS 140 standard, which governs cryptographic systems operations.
- Organizations must comply with FIPS 140 for their cryptographic products to be used in regulated environments.
This week, we announced Quantum Origin received NIST SP 800-90B Entropy Source validation, marking the first software QRNG approved for use in regulated industries.
What does NIST validation mean for our customers?
This means Quantum Origin is now available for high-security cryptographic systems and integrates seamlessly with NIST-approved solutions without requiring recertification.
- Unlike hardware QRNGs, Quantum Origin requires no network connectivity, making it ideal for air-gapped systems.
- For federal agencies, it complements our "U.S. Made" designation, easing deployment in critical infrastructure.
- It adds further value for customers building hardware security modules, firewalls, PKIs, and IoT devices.
The NIST validation, combined with our peer-reviewed papers, further establishes Quantum Origin as the leading QRNG on the market.
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It is paramount for governments, commercial enterprises, and critical infrastructure to stay ahead of evolving cybersecurity threats to maintain societal and economic security.
Quantinuum delivers the highest quality quantum randomness, enabling our customers to confront the most advanced cybersecurity challenges present today.