“Computers are useless without error correction”
- Anonymous

If you stumble while walking, you can regain your balance, recover, and keep walking. The ability to function when mistakes happen is essential for daily life, and it permeates everything we do. For example, a windshield can protect a driver even when it’s cracked, and most cars can still drive on a highway if one of the tires is punctured. In fact, most commercially operated planes can still fly with only one engine. All of these things are examples of what engineers call “fault-tolerance”, which just describes a system’s ability to tolerate faults while still functioning.

When building a computer, this is obviously essential. It is a truism that errors will occur (however rarely) in all computers, and a computer that can’t operate effectively and correctly in the presence of faults (or errors) is not very useful. In fact, it will often be wrong - because errors won’t be corrected.

In a new paper from Quantinuum’s world class quantum error correction team, we have made a hugely significant step towards one of the key issues faced in quantum error correction – that of executing fault-tolerant gates with efficient codes. 

This work explores the use of “genon braiding” – a cutting-edge concept in the study of topological phases of matter, motivated by the mathematics of category theory, and both related to and inspired by our prior groundbreaking work on non-Abelian anyons. 

The native fault tolerant properties of braided toric codes have been theoretically known for some time, and in this newly published work, our team shares how they have discovered a technique based on “genon braiding” for the construction of logical gates which could be applied to “high rate” error correcting codes – meaning codes that require fewer physical qubits per logical qubit, which can have a huge impact on scaling.

Stepping along the path to fault-tolerance

In classical computing, building in fault-tolerance is relatively easy. For starters, the hardware itself is incredibly robust and native error rates are very low. Critically, one can simply copy each bit, so errors are easy to detect and correct. 

Quantum computing is, of course, much trickier with challenges that typically don’t exist in classical computing. First off, the hardware itself is incredibly delicate. Getting a quantum computer to work requires us to control the precise quantum states of single atoms. On top of that, there’s a law of physics called the no cloning theorem, which says that you can’t copy qubits. There are also other issues that arise from the properties that make quantum computing so powerful, such as measurement collapse, that must be considered.

Some very distinguished scientists and researchers have thought about quantum error correcting including Steane, Shor, Calderbank, and Kitaev [9601029.pdf (arxiv.org), 9512032.pdf (arxiv.org), arXiv:quant-ph/9707021v1 9 Jul 1997].  They realized that you can entangle groups of physical qubits, store the relevant quantum information in the entangled state (called a “logical qubit”), and, with a lot of very clever tricks, perform computations with error correction.

There are many different ways to entangle groups of physical qubits, but only some of them allow for useful error detection and correction. This special set of entangling protocols is called a “code” (note that this word is used in a different sense than most readers might think of when they hear “code” - this isn’t “Hello World”). 

A huge amount of effort today goes into “code discovery” in companies, universities, and research labs, and a great deal of that research is quite bleeding-edge. However, discovering codes is only one piece of the puzzle: once a code is discovered, one must still figure out how to compute with it. With any specific way of entangling physical qubits into a logical qubit you need to figure out how to perform gates, how to infer faults, how to correct them, and so on. It’s not easy!

Quantinuum has one of the world’s leading teams working on error correction and has broken new ground many times in recent years, often with industrial or scientific research partners. Among many firsts, we were the first to demonstrate real-time error correction (meaning a fully-fault tolerant QEC protocol). This included many milestones: repeated real-time error correction, the ability to perform quantum "loops" (repeat-until-success protocols), and real-time decoding to determine the corrections during the computation. We were also the first to perform a logical two-qubit gate on a commercial system. In one of our most recent demonstrations, in partnership with Microsoft, we supported the use of error correcting techniques to achieve the first demonstration of highly reliable logical qubits, confirming our place at the forefront of this research – and indeed confirming that Quantinuum’s H2-1 quantum computer was the first – and at present only – device in the world capable of what Microsoft characterizes as Level 2 Resilient quantum computing. 

Introducing new, exotic error correction codes

While codes like the Steane code are well-studied and effective, our team is motivated to investigate new codes with attractive qualities. For example, some codes are “high-rate”, meaning that you get more logical qubits per physical qubit (among other things), which can have a big impact on outlooks for scaling – you might ultimately need 10x fewer physical qubits to perform advanced algorithms like Shor’s. 

Implementing high-rate codes is seductive, but as we mentioned earlier we don’t always know how to compute with them. A particular difficulty with high-rate codes is that you end up sharing physical qubits between logical qubits, so addressing individual logical qubits becomes tricky. There are other difficulties that come from sharing physical qubits between logical qubits, such as performing gates between different logical qubits (scientists call this an “inter-block” gate).

One well-studied method for computing with QEC codes is known as “braiding”. The reason it is called braiding is because you move particles, or “braid” them, around each other, which manipulates logical quantum information. In our new paper, we crack open computing with exotic codes by implementing “genon” braiding. With this, we realize a paradigm for constructing logical gates which we believe could be applied to high-rate codes (i.e. inter-block gates).

What exactly “genons” are, and how they are braided, is beautiful and complex mathematics - but the implementation is surprisingly simple. Inter-block logical gates can be realized through simple relabeling and physical operations. “Relabeling”, i.e. renaming qubit 1 to qubit 2, is very easy in Quantinuum’s QCCD architecture, meaning that this approach to gates will be less noisy, faster, and have less overhead. This is all due to our architectures’ native ability to move qubits around in space, which most other architectures can’t do. 

Using this framework, our team delivered a number of proof-of-principle experiments on the H1-1 system, demonstrating all single qubit Clifford operations using genon braiding. They then performed two kinds of two-qubit logical gates equivalent to CNOTs, proving that genon braiding works in practice and is comparable to other well-researched codes such as the Steane code.

What does this all mean? This work is a great example of co-design – tailoring codes for our specific and unique hardware capabilities. This is part of a larger effort to find fault-tolerant architectures tailored to Quantinuum's hardware. Quantinuum scientist and pioneer of this work, Simon Burton, put it quite succinctly: “Braiding genons is very powerful. Applying these techniques might prove very useful for realizing high-rate codes, translating to a huge impact on how our computers will scale.”

Doing mathematical physics with diagrams instead of traditional formalism allows researchers to tackle difficult problems in an intuitive and mathematically strict way that opens the door to new insights and solutions. The new calculus we are developing that we refer to as ZX calculus, also known as Penrose Spin Calculus, has applications in fields as diverse as quantum chemistry, condensed matter physics, and loop quantum gravity.

In a recent paper on the arXiv, Quantinuum researchers Harny Wang, Razin A. Shaikh, and Boldizsár Poór have proven the “completeness” of this ZX calculus in finite dimensions, meaning that one can now use diagrams instead of linear algebra to perform calculations in finite dimensional quantum mechanics. This is a remarkable achievement.

“Now very complicated formulas in quantum chemistry and loop quantum gravity can be derived by diagrams,” said co-author Harny Wang.

Physicists have used graphical calculus for a long time. They are used widely in quantum field theory, in the form of Feynman diagrams, or in gravitational theory, in the form of Penrose diagrams. Graphical calculation strategies are generally very well appreciated as they replace a lot of difficult and tedious ‘formal’ mathematics with a simpler, more intuitive, but no less accurate diagrammatic approach.

Our researcher’s work on ZX and ZXW calculus (a near cousin to ZX) is the latest but most innovative shift from “shut up and calculate” to “depict and rewrite”, a shift that many researchers are sure to welcome.

ZX calculus was initially developed by scientists as a tool for working on problems in quantum mechanics that require intricate calculations. ZX calculus, created by Professor Bob Coecke and Dr. Ross Duncan, both of whom are senior scientists at Quantinuum, has developed over the course of 15 years, leading to a growing global community of researchers. This most recent paper marks the transition of important parts of ZX from ‘a work in progress’ to something that is fully formed.

Both ZX and ZXW calculus are known for efficiently expressing quantum relations such as entanglement. It is hoped these new formalisms may uncover connections between some of the most challenging problems in science and quantum computing.

Distinguished physicist Carlo Rovelli has already expressed interest in using ZX and ZXW graphical calculus for his work, stating “Indeed, there are concrete steps in place to translate quantum gravity problems into quantum computing problems, and I have hope that the powerful conceptual and technical tools developed by Bob [Coecke], Harny [Wang] and their collaborators could play a major role in this.”

In addition to interest from the gravity community, ZX is being adopted in the wider quantum computing community. Dr. Peter Shor recently worked with colleagues to develop an algorithm that maps Clifford encoders to graphical representations in the ZX calculus.

A key step in many quantum algorithms is setting everything up: you need all your dominoes in place before you can do much else. This is called “state preparation”, and it’s a trickier problem than it might seem. 

Our team has developed new protocols that can help – and in a big way. Specifically, the team worked on preparing “multivariate” functions, which just means functions that are used to explore problems with more than one variable, or in more than 1 dimension. One-dimensional problems do exist (think of a path that only goes forwards or backwards – we can call the variable “x”) but in the real world it’s much more common to have problems with many dimensions, or variables (think instead of a landscape where you can go forwards, backwards, left, right, up, and down – we can call the variables “x”, ”y”, and “z”).

Our new multivariate function quantum state preparation protocols don’t rely on some commonly-used and computationally expensive subroutines - instead they expand the desired multivariate function into well-known mathematical basis functions, called Fourier and Chebyshev functions. This makes our protocols simpler and more effective than previous options. 

Generally, state preparation is a hard problem, and costs exponentially many resources to prepare an arbitrary state. By expanding the functions in a Fourier or Chebyshev series, one can truncate the series to create good approximations, which instead uses only polynomially many resources – meaning that this method has better asymptotic scaling than many other non-heuristic methods (which are often designed to work in only one dimension anyways). 

Our team used their protocol to prepare a commonly used initial state on our H2 trapped-ion quantum processor, the bivariate Gaussian. Bivariate Gaussians are used everywhere from physics to finance, underscoring the practicality of these new protocols. They also analyzed examples potentially useful for quantum chemistry and partial differential equations.

A very nice feature of this work is that it is broadly applicable, generic, and entirely modular – meaning it can be plugged in to the beginning of almost any quantum algorithm, giving our customers and users even more flexibility and power. 

The first half of 2024 will go down as the period when we shed the last vestiges of the “wait and see” culture that has dominated the quantum computing industry. Thanks to a run of recent achievements, we have helped to lead the entire quantum computing industry into a new, post-classical era.

Today we are announcing the latest of these achievements: a major qubit count enhancement to our flagship System Model H2 quantum computer from 32 to 56 qubits. We also reveal meaningful results of work with our partner JPMorgan Chase & Co. that showcases a significant lift in performance.

But to understand the full importance of today’s announcements, it is worth recapping the succession of breakthroughs that confirm that we are entering a new era of quantum computing in which classical simulation will be infeasible.

A historic run

Between January and June 2024, Quantinuum’s pioneering teams published a succession of results that accelerate our path to universal fault-tolerant quantum computing. 

Our technical teams first presented a long-sought solution to the “wiring problem”, an engineering challenge that affects all types of quantum computers. In short, most current designs will require an impossible number of wires connected to the quantum processor to scale to large qubit numbers. Our solution allows us to scale to high qubit numbers with no issues, proving that our QCCD architecture has the potential to scale.

Next, we became the first quantum computing company in the world to hit “three 9s” two qubit gate fidelity across all qubit pairs in a production device. This level of fidelity in 2-qubit gate operations was long thought to herald the point at which error corrected quantum computing could become a reality. It has accelerated and intensified our focus on quantum error correction (QEC). Our scientists and engineers are working with our customers and partners to achieve multiple breakthroughs in QEC in the coming months, many of which will be incorporated into products such as the H-Series and our chemistry simulation platform, InQuanto™.

Following that, with our long-time partner Microsoft, we hit an error correction performance threshold that many believed was still years away. The System Model H2 became the first – and only – quantum computer in the world capable of creating and computing with highly reliable logical (error corrected) qubits. In this demonstration, the H2-1 configured with 32 physical qubits supported the creation of four highly reliable logical qubits operating at “better than break-even”. In the same demonstration, we also shared that logical circuit error rates were shown to be up to 800x lower than the corresponding physical circuit error rates. No other quantum computing company is even close to matching this achievement (despite many feverish claims in the past 12 months).

Pushing to the limits of supercomputing … and beyond

The quantum computing industry is departing the era when quantum computers could be simulated by a classical computer. Today, we are making two milestone announcements. The first is that our H2-1 processor has been upgraded to 56 trapped-ion qubits, making it impossible to classically simulate, without any loss of the market-leading fidelity, all-to-all qubit connectivity, mid-circuit measurement, qubit reuse, and feed forward.

The second is that the upgrade of H2-1 from 32 to 56 qubits makes our processor capable of challenging the world’s most powerful supercomputers. This demonstration was achieved in partnership with our long-term collaborator JPMorgan Chase & Co. and researchers from Caltech and Argonne National Lab.

Our collaboration tackled a well-known algorithm, Random Circuit Sampling (RCS), and measured the quality of our results with a suite of tests including the linear cross entropy benchmark (XEB) – an approach first made famous by Google in 2019 in a bid to demonstrate “quantum supremacy”. An XEB score close to 0 says your results are noisy – and do not utilize the full potential of quantum computing. In contrast, the closer an XEB score is to 1, the more your results demonstrate the power of quantum computing. The results on H2-1 are excellent, revealing, and worth exploring in a little detail. Here is the complete data on GitHub.

Better qubits, better results

Our results show how far quantum hardware has come since Google’s initial demonstration. They originally ran circuits on 53 superconducting qubits that were deep enough to severely frustrate high-fidelity classical simulation at the time, achieving an estimated XEB score of ~0.002. While they showed that this small value was statistically inconsistent with zero, improvements in classical algorithms and hardware have steadily increased what XEB scores are achievable by classical computers, to the point that classical computers can now achieve scores similar to Google’s on their original circuits.

In contrast, we have been able to run circuits on all 56 qubits in H2-1 that are deep enough to challenge high-fidelity classical simulation while achieving an estimated XEB score of ~0.35. This >100x improvement implies the following: even for circuits large and complex enough to frustrate all known classical simulation methods, the H2 quantum computer produces results without making even a single error about 35% of the time. In contrast to past announcements associated with XEB experiments, 35% is a significant step towards the idealized 100% fidelity limit in which the computational advantage of quantum computers is clearly in sight.

This huge jump in quality is made possible by Quantinuum’s market-leading high fidelity and also our unique all-to-all connectivity. Our flexible connectivity, enabled by our QCCD architecture, enables us to implement circuits with much more complex geometries than the 2D geometries supported by superconducting-based quantum computers. This specific advantage means our quantum circuits become difficult to simulate classically with significantly fewer operations (or gates). These capabilities have an enormous impact on how our computational power scales as we add more qubits: since noisy quantum computers can only run a limited number of gates before returning unusable results, needing to run fewer gates ultimately translates into solving complex tasks with consistent and dependable accuracy.

This is a vitally important moment for companies and governments watching this space and deciding when to invest in quantum: these results underscore both the performance capabilities and the rapid rate of improvement of our processors, especially the System Model H2, as a prime candidate for achieving near-term value.

So what of the comparison between the H2-1 results and a classical supercomputer? 

A direct comparison can be made between the time it took H2-1 to perform RCS and the time it took a classical supercomputer. However, classical simulations of RCS can be made faster by building a larger supercomputer (or by distributing the workload across many existing supercomputers). A more robust comparison is to consider the amount of energy that must be expended to perform RCS on either H2-1 or on classical computing hardware, which ultimately controls the real cost of performing RCS. An analysis based on the most efficient known classical algorithm for RCS and the power consumption of leading supercomputers indicates that H2-1 can perform RCS at 56 qubits with an estimated 30,000x reduction in power consumption. These early results should be seen as very attractive for data center owners and supercomputing facilities looking to add quantum computers as “accelerators” for their users. 

Where we go next

Today’s milestone announcements are clear evidence that the H2-1 quantum processor can perform computational tasks with far greater efficiency than classical computers. They underpin the expectation that as our quantum computers scale beyond today’s 56 qubits to hundreds, thousands, and eventually millions of high-quality qubits, classical supercomputers will quickly fall behind. Quantinuum’s quantum computers are likely to become the device of choice as scrutiny continues to grow of the power consumption of classical computers applied to highly intensive workloads such as simulating molecules and material structures – tasks that are widely expected to be amenable to a speedup using quantum computers.

With this upgrade in our qubit count to 56, we will no longer be offering a commercial “fully encompassing” emulator – a mathematically exact simulation of our H2-1 quantum processor is now impossible, as it would take up the entire memory of the world’s best supercomputers. With 56 qubits, the only way to get exact results is to run on the actual hardware, a trend the leaders in this field have already embraced.

More generally, this work demonstrates that connectivity, fidelity, and speed are all interconnected when measuring the power of a quantum computer. Our competitive edge will persist in the long run; as we move to running more algorithms at the logical level, connectivity and fidelity will continue to play a crucial role in performance.

“We are entirely focused on the path to universal fault tolerant quantum computers. This objective has not changed, but what has changed in the past few months is clear evidence of the advances that have been made possible due to the work and the investment that has been made over many, many years. These results show that whilst the full benefits of fault tolerant quantum computers have not changed in nature, they may be reachable earlier than was originally expected, and crucially, that along the way, there will be tangible benefits to our customers in their day-to-day operations as quantum computers start to perform in ways that are not classically simulatable. We have an exciting few months ahead of us as we unveil some of the applications that will start to matter in this context with our partners across a number of sectors.”
– Ilyas Khan, Chief Product Officer

Stay tuned for results in error correction, physics, chemistry and more on our new 56-qubit processor.

The 55th Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics (DAMOP) from June 3-7, 2024, will feature presentations from Quantinuum’s physicists working on the world’s leading quantum computing hardware.

Our team members will cover various topics including quantum computation, scaling quantum technologies with ion-trap, and progress across various fields within atomic, molecular, and optical physics

Meet our quantum computing experts at Table 39, from June 4th – 6th to discuss what’s new and what’s next in trapped-ion quantum computing.

Join these sessions to discover how Quantinuum is advancing quantum computing:

‍Path to Scale QCCD Architecture for Trapped Ion Quantum Computers
‍Speaker: Patty Lee, Chief Scientist for Hardware Technology Development
Date and Time: June 3, 2:30 pm – 3:45 pm
Location: Room 202AB

Scalable Multispecies Ion Transport in a Grid-Based Surface-Electrode Trap
‍Speaker: Robert Delaney, Advanced Physicist
Date and Time: June 4, 10:45 am – 12:45 pm
Location: Room 203B

Benchmarking Quantinuum’s Second-Generation Quantum Processor
‍Speaker: Julia Cline, Advanced Physicist
Date and Time: June 4, 2:00 pm – 4:00 pm
Location: Room 203A

A fast and robust cooling method for trapped-ion qubits: phonon rapid adiabatic passage (Poster Session I)
Presenter: Ivaylo Madjarov, Numerical Physicist
Date and Time: June 4, 4:00 pm – 6:00 pm
Location: Hall BC

Laser Cooling Trapped-Ion Crystal Modes Beyond the Lamb-Dicke Regime
Speaker: Chris Gilbreth, Lead Physicist
Date and Time: June 7, 11:30 am – 11:42 am
Location: Room 201BC

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With the rapid evolution of Quantum Computing, users are contemplating the best way to begin to integrate Quantum capabilities into their existing HPC and AI infrastructure. Find our experts at the ISC conference, May 12^th-16^th, in Hamburg, Germany to discuss our world leading hardware, applications, and case studies. 

Exhibit Hall

Drop by Booth K50 in the exhibit hall to meet tour team and see a display of our System Model H2 chip, Powered by Honeywell. 

If you’d like to schedule a 1:1 meeting, send us an email to schedule a time to meet. We have reserved meeting room Hall 5 at ISC, but we’d be happy to set up time to meet with you at or after the event.

Presentations

Our team will be presenting on a range of topics about integrating quantum computing into existing HPC infrastructure. They’ll be speaking about our hardware features and how you can leverage quantum computing with your existing HPC cluster.

May 13^th

2:30pm – 3:00pm | Hall 4, ground level in the First-Time Exhibitor Pitch

Understanding Opportunities with Quantum Computing: Learn about our roadmap and key strategies to accelerate your current HPC clusters with the integration of quantum computing. 

Presented by Nash Palaniswamy, Chief Commercial Officer, Quantinuum

May 14^th

2:00pm – 2:30pm | GENCI Booth K40

Simulation of Transition Metal Oxide (TMO) Atomic Layer Deposition (ALD): A Study of the modelling of electronic energies used in the reactions involved for ALD of ZrO2 and of the reactivity of organometallic precursors used in ALD technology for controlling the quality of thin film deposition on different substrates. The study is a collaboration between C12 Quantum Electronics, Air Liquide and Quantinuum, with support from PAQ Ile de France.

Presented by Maud Einhorn, Technical Account Manager, and Gabriela Cimpan, Partner Manager, Quantinuum

May 14^th

2:20pm – 2:35pm | Hall Z – 3rd floor

The Trapped-Ion Quantum Processors at Quantinuum: Quantinuum has constructed two generations of QCCD (quantum charge-coupled device) quantum processors. These processors use trapped-ions for qubits and sympathetic cooling, and shuttling operations to achieve high-fidelity gating operations on individual qubits and between any pair of qubits – making them fully-connected. In this talk, Dave will discuss Quantinuum’s efforts to rigorously benchmark the performance of these machines, highlighting their strengths and weaknesses. He’ll also give a brief survey of our efforts toward near-term quantum advantage and quantum error correction. Finally, he’ll sketch out some technological developments aimed at scaling these processors and the implications for future devices.

Presented by David Hayes, Sr. R&D Manager for Theory and Architecture

May 14^th and May 15^th

12:30pm – 1:00pm | Meeting Room Hall 5

3:30pm – 4:00pm | Meeting Room Hall 5

Quantum Computing, Error Correction, and Scaling for the Future at Quantinuum: Quantum computing promises to provide significant computational savings in valuable problems such as chemistry, materials, and cybersecurity. To make this a reality, errors in quantum operations must be suppressed significantly below where they are today, and the size of quantum computing hardware must be increased. Quantinuum has recently made significant strides in scaling to larger sizes. Join the session to hear about these exciting results, our plans to scale, and a look towards the future.

Presented by Chris Langer, Fellow and Chairman of the Technical Board, Quantinuum

May 16^th 

1:00pm – 1:20pm | Hall H, Booth L01 in the HPC Solutions Forum

‍Harnessing the potential of quantum computing: As the landscape of quantum computing continues to rapidly evolve, the question of when to invest in quantum computing knowledge remains a key strategic consideration for organizations. This talk will explore the challenge of quantum readiness by surveying some of the research collaborations Quantinuum has performed with a range of industry-leading organizations. Using real-world case studies, we will highlight the diverse array of sectors poised to benefit from early quantum adoption, including pharmaceuticals, finance, logistics, and cybersecurity. This talk begins to unpack why many first mover enterprise organizations have made significant investments in quantum readiness already, rather than deferring until the technology matures further. 

Presented by Maud Einhorn, Technical Account Manager, Quantinuum

May 16^th

4:30pm – 5:00pm | Hall Y1 - 2nd floor

Workshop on Benchmarking and Scaling the Quantum Charged Coupled Device Quantum Computing architecture in the Quantum and Hybrid Quantum-Classical Computing Approaches: The QCCD architecture provides a unique approach to quantum computing where qubits are mobile and operating zones are fixed. In contrast to QC architectures where qubit and couplings between them are fixed, the QCCD architecture naturally provides all-to-all connectivity and high-fidelity operations. Additional advanced features include mid-circuit measurement, qubit reset, conditional logic, and variable angle gates. The talk will present benchmarking of our machines and recent progress towards scaling to larger systems.

Presented by Chris Langer, Fellow and Chair of the Technical Board, Quantinuum