How 黑料社 researchers used quantum tensor networks to measure the properties of quantum particles at a phase transition
April 9, 2023
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 鈥 the transition from solid to liquid or from liquid to gas.聽
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.聽聽
黑料社 researchers recently used the H1-1 quantum computer to computationally model a group of highly correlated quantum particles at a quantum critical point 鈥 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.聽 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, , the 黑料社 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, 鈥淭his 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.鈥
The role of quantum tensor networks
Abstract representation of the 128-site MERA used in this work
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.聽
The 黑料社 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.聽
鈥淏y 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,鈥 said Dave Hayes, Lead of the U.S. quantum theory team at 黑料社 and one of the paper鈥檚 authors. 鈥淪o, it can, in principle, go out and simulate any system that you can dream of.鈥
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.聽
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.聽
鈥淭hat's as hard as it gets for these systems,鈥 Dave explained. 鈥淪o that's where we want to look for quantum advantage 鈥 because we want the problem to be as hard as possible on the classical computer, and then have a quantum computer solve it.鈥
To improve the results, the team used two error mitigation techniques, symmetry-based error heralding, which is made possible by the MERA structure, and , 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 黑料社 team describes this sort of problem as a stepping-stone, which allows the researchers to explore quantum tensor network methods on today鈥檚 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.聽
鈥淧otentially, our biggest applications over the next couple of years will include studying solid-state systems, physics systems, many-body systems, and modeling them,鈥 said Jenni Strabley, Senior Director of Offering Management at 黑料社.
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 鈥 quantum states that are much more difficult to calculate classically.聽
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, 黑料社鈥檚 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 黑料社
黑料社,聽the world鈥檚 largest integrated quantum company, pioneers powerful quantum computers and advanced software solutions. 黑料社鈥檚 technology drives breakthroughs in materials discovery, cybersecurity, and next-gen quantum AI. With over 500 employees, including 370+ scientists and engineers, 黑料社 leads the quantum computing revolution across continents.聽
Blog
October 30, 2025
Scalable Quantum Error Detection
Typically, Quantum Error Detection (QED) is viewed as a short-term solution鈥攁 non-scalable, stop-gap until full fault tolerance is achieved at scale.
That鈥檚 just changed, thanks to a serendipitous discovery made by our team. Now, QED can be used in a much wider context than previously thought. Our team made this discovery while studying the contact process, which describes things like how diseases spread or how water permeates porous materials. In particular, our team was studying the quantum contact process (QCP), a problem they had tackled before, which helps physicists understand things like phase transitions. In the process (pun intended), they came across what senior advanced physicist, Eli Chertkov, described as 鈥渁 surprising result.鈥
While examining the problem, the team realized that they could convert detected errors due to noisy hardware into random resets, a key part of the QCP, thus avoiding the exponentially costly overhead of post-selection normally expected in QED.
To understand this better, the team developed a new protocol in which the encoded, or logical, quantum circuit adapts to the noise generated by the quantum computer. They quickly realized that this method could be used to explore other classes of random circuits similar to the ones they were already studying.
The team put it all together on System Model H2 to run a complex simulation, and were surprised to find that they were able to achieve near break-even results, where the logically encoded circuit performed as well as its physical analog, thanks to their clever application of QED. 聽Ultimately, this new protocol will allow QED codes to be used in a scalable way, saving considerable computational resources compared to full quantum error correction (QEC).
Researchers at the crossroads of quantum information, quantum simulation, and many-body physics will take interest in this protocol and use it as a springboard for inventing new use cases for QED.
Stay tuned for more, our team always has new tricks up their sleeves.
When will quantum computers outperform classical ones?
This question has hovered over the field for decades, shaping billion-dollar investments and driving scientific debate.
The question has more meaning in context, as the answer depends on the problem at hand. We already have estimates of the quantum computing resources needed for Shor鈥檚 algorithm, which has a superpolynomial advantage for integer factoring over the best-known classical methods, threatening cryptographic protocols. Quantum simulation allows one to glean insights into exotic materials and chemical processes that classical machines struggle to capture, especially when strong correlations are present. But even within these examples, estimates change surprisingly often, carving years off expected timelines. And outside these famous cases, the map to quantum advantage is surprisingly hazy.
Researchers at 黑料社 have taken a fresh step toward drawing this map. In a new theoretical framework, Harry Buhrman, Niklas Galke, and Konstantinos Meichanetzidis introduce the concept of 鈥渜ueasy instances鈥 (quantum easy) 鈥 problem instances that are comparatively easy for quantum computers but appear difficult for classical ones.
From Problem Classes to Problem Instances
Traditionally, computer scientists classify problems according to their worst-case difficulty. Consider the problem of Boolean satisfiability, or SAT, where one is given a set of variables (each can be assigned a 0 or a 1) and a set of constraints and must decide whether there exists a variable assignment that satisfies all the constraints. SAT is a canonical NP-complete problem, and so in the worst case, both classical and quantum algorithms are expected to perform badly, which means that the runtime scales exponentially with the number of variables. On the other hand, factoring is believed to be easier for quantum computers than for classical ones. But real-world computing doesn鈥檛 deal only in worst cases. Some instances of SAT are trivial; others are nightmares. The same is true for optimization problems in finance, chemistry, or logistics. What if quantum computers have an advantage not across all instances, but only for specific 鈥減ockets鈥 of hard instances? This could be very valuable, but worst-case analysis is oblivious to this and declares that there is no quantum advantage.
To make that idea precise, the researchers turned to a tool from theoretical computer science: Kolmogorov complexity. This is a way of measuring how 鈥渞egular鈥 a string of bits is, based on the length of the shortest program that generates it. A simple string like 0000000000 can be described by a tiny program (鈥減rint ten zeros鈥), while the description of a program that generates a random string exhibiting no pattern is as long as the string itself. From there, the notion of instance complexity was developed: instead of asking 鈥渉ow hard is it to describe this string?鈥, we ask 鈥渉ow hard is it to solve this particular problem instance (represented by a string)?鈥 For a given SAT formula, for example, its polynomial-time instance complexity is the size of the smallest program that runs in polynomial time and decides whether the formula is satisfiable. This smallest program must be consistently answering all other instances, and it is also allowed to declare 鈥淚 don鈥檛 know鈥.
In their new work, the team extends this idea into the quantum realm by defining polynomial-time quantum instance complexity as the size of the shortest quantum program that solves a given instance and runs on polynomial time. This makes it possible to directly compare quantum and classical effort, in terms of program description length, on the very same problem instance. If the quantum description is significantly shorter than the classical one, that problem instance is one the researchers call 鈥渜耻别补蝉测鈥: quantum-easy and classically hard. These queasy instances are the precise places where quantum computers offer a provable advantage 鈥 and one that may be overlooked under a worst-case analysis.
Why 鈥淨ueasy鈥?
The playful name captures the imbalance between classical and quantum effort. A queasy instance is one that makes classical algorithms struggle, i.e. their shortest descriptions of efficient programs that decide them are long and unwieldy, while a quantum computer can handle the same instance with a much simpler, faster, and shorter program. In other words, these instances make classical computers 鈥渜ueasy,鈥 while quantum ones solve them efficiently and finding them quantum-easy. The key point of these definitions lies in demonstrating that they yield reasonable results for well-known optimisation problems.
By carefully analysing a mapping from the problem of integer factoring to SAT (which is possible because factoring is inside NP and SAT is NP-complete) the researchers prove that there exist infinitely many queasy SAT instances. SAT is one of the most central and well-studied problems in computer science that finds numerous applications in the real-world. The significant realisation that this theoretical framework highlights is that SAT is not expected to yield a blanket quantum advantage, but within it lie islands of queasiness 鈥 special cases where quantum algorithms decisively win.
Algorithmic Utility
Finding a queasy instance is exciting in itself, but there is more to this story. Surprisingly, within the new framework it is demonstrated that when a quantum algorithm solves a queasy instance, it does much more than solve that single case. Because the program that solves it is so compact, the same program can provably solve an exponentially large set of other instances, as well. Interestingly, the size of this set depends exponentially on the queasiness of the instance!
Think of it like discovering a special shortcut through a maze. Once you鈥檝e found the trick, it doesn鈥檛 just solve that one path, but reveals a pattern that helps you solve many other similarly built mazes, too (even if not optimally). This property is called algorithmic utility, and it means that queasy instances are not isolated curiosities. Each one can open a doorway to a whole corridor with other doors, behind which quantum advantage might lie.
A North Star for the Field
Queasy instances are more than a mathematical curiosity; this is a new framework that provides a language for quantum advantage. Even though the quantities defined in the paper are theoretical, involving Turing machines and viewing programs as abstract bitstrings, they can be approximated in practice by taking an experimental and engineering approach. This work serves as a foundation for pursuing quantum advantage by targeting problem instances and proving that in principle this can be a fruitful endeavour.
The researchers see a parallel with the rise of machine learning. The idea of neural networks existed for decades along with small scale analogue and digital implementations, but only when GPUs enabled large-scale trial and error did they explode into practical use. Quantum computing, they suggest, is on the cusp of its own heuristic era. 鈥净耻谤颈蝉迟颈肠蝉鈥 will be prominent in finding queasy instances, which have the right structure so that classical methods struggle but quantum algorithms can exploit, to eventually arrive at solutions to typical real-world problems. After all, quantum computing is well-suited for small-data big-compute problems, and our framework employs the concepts to quantify that; instance complexity captures both their size and the amount of compute required to solve them.
Most importantly, queasy instances shift the conversation. Instead of asking the broad question of when quantum computers will surpass classical ones, we can now rigorously ask where they do. The queasy framework provides a language and a compass for navigating the rugged and jagged computational landscape, pointing researchers, engineers, and industries toward quantum advantage.
From September 16th 鈥 18th, (QWC) brought together visionaries, policymakers, researchers, investors, and students from across the globe to discuss the future of quantum computing in Tysons, Virginia.
黑料社 is forging the path to universal, fully fault-tolerant quantum computing with our integrated full-stack. With our quantum experts were on site, we showcased the latest on 黑料社 Systems, the world鈥檚 highest-performing, commercially available quantum computers, our new software stack featuring the key additions of Guppy and Selene, our path to error correction, and more.
Highlights from QWC
Dr. Patty Lee Named the Industry Pioneer in Quantum
The Quantum Leadership Awards celebrate visionaries transforming quantum science into global impact. This year at QWC, Dr. Patty Lee, our Chief Scientist for Hardware Technology Development, was named the Industry Pioneer in Quantum! This honor celebrates her more than two decades of leadership in quantum computing and her pivotal role advancing the world鈥檚 leading trapped-ion systems. .
Keynote with 黑料社's CEO,聽Dr. Rajeeb聽Hazra
At QWC 2024, 黑料社鈥檚 President & CEO, Dr. Rajeeb 鈥淩aj鈥 Hazra, took the stage to showcase our commitment to advancing quantum technologies through the unveiling of our roadmap to universal, fully fault-tolerant quantum computing by the end of this decade. This year at QWC 2025, Raj shared the progress we鈥檝e made over the last year in advancing quantum computing on both commercial and technical fronts and exciting insights on what鈥檚 to come from 黑料社. .
Panel Session:聽Policy Priorities for Responsible Quantum and AI
As part of the Track Sessions on Government & Security, 黑料社鈥檚 Director of Government Relations, Ryan McKenney, discussed 鈥淧olicy Priorities for Responsible Quantum and AI鈥 with Jim Cook from Actions to Impact Strategies and Paul Stimers from Quantum Industry Coalition.
Fireside Chat:聽Establishing a Pro-Innovation Regulatory Framework
During the Track Session on Industry Advancement, 黑料社鈥檚 Chief Legal Officer, Kaniah Konkoly-Thege,聽and Director of Government Relations, Ryan McKenney, discussed the importance of 鈥淓stablishing a Pro-Innovation Regulatory Framework鈥.