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For a quantum computer to be useful, it must be universal, have lots of qubits, and be able to detect and correct errors. The error correction step must be done so well that in the final calculations, you only see an error in less than one in a billion (or maybe even one in a trillion) tries. Correcting errors on a quantum computer is quite tricky, and most current error correcting schemes are quite expensive for quantum computers to run.

We’ve teamed up with researchers at the University of Colorado to – bringing the era of quantum ‘fault tolerance’ closer to reality. Current approaches to error correction involve encoding the quantum information of one qubit into several entangled qubits (called a “logical” qubit). Most of the encoding schemes (called a “code”) in use today are relatively inefficient – they can only make one logical qubit out of a set of physical qubits. As we mentioned earlier, we want lots of error corrected qubits in our machines, so this is highly suboptimal – a “low encoding rate” means that you need many, many more physical qubits to realize a machine with lots of error corrected logical qubits.

Ideally, our computers will have “high-rate” codes (meaning that you get more logical qubits per physical qubit), and researchers have identified promising schemes known as “non-local qLDPC codes”. This type of code has been discussed theoretically for years, but until now had never been realized in practice. In a , the joint team has implemented a high rate non-local qLDPC code on our H2 quantum processor, with impressive results.��

The team used the code to create 4 error protected (logical) qubits, then entangled them in a “GHZ state” with better fidelity than doing the same operation on physical qubits – meaning that the error protection code improved fidelity in a difficult entangling operation. The team chose to encode a GHZ state because it is widely used as a system-level benchmark, and its better-than-physical logical preparation marks a highly mature system.

It is worth noting that this remarkable accomplishment was achieved with a very small team, half of whom do not have specialized knowledge about the underlying physics of our processors. Our hardware and software stack are now so mature that advances can be achieved by “quantum programmers” who don’t need advanced quantum hardware knowledge, and who can run their programs on a commercial machine between commercial jobs. This places us bounds ahead of the competition in terms of accessibility and reliability.

This paper marks the first time anyone has entangled 4 logical qubits with better fidelity than the physical analog. This work is in strong synergy with our recent announcement in partnership with Microsoft, where we demonstrated logical fidelities better than physical fidelities on entangled bell pairs and demonstrated multiple rounds of error correction.��These results with two different codes underscore how we are moving into the era of fault-tolerance ahead of the competition.

The code used in this paper is significantly more optimized for architectures capable of moving the qubits around, like ours. In practice, this means that we are capable of “non-local” gates and reconfigurability. A big advantage in particular is that some of the critical operations amount to a simple relabeling of the individual qubits, which is virtually error-free.

The biggest advantage, however, is in this code’s very high encoding rate. Unlike many codes in use today, this code offers a very high rate of logical qubits per physical qubit – in fact, the number of logical qubits is proportional to the number of physical qubits, which will allow our machines to scale much more quickly than more traditional codes that have a hard limit on the number of logical qubits one can get in each code block. This is yet another proof point that our machines will scale effectively and quickly.

The Artificial Intelligence (AI) systems that have recently permeated our lives have a serious problem: they are built in a way that makes them very hard - and sometimes impossible - to understand or interpret. Luckily, our team is tackling this problem, and we’ve just published that covers the issue in detail.

It turns out that the lack of explainability in machine learning (ML) models, such as ChatGPT or Claude, comes from the way that the systems are built. Their underlying architecture (a neural network) lacks coherent structure. While neural networks can be trained to effectively solve certain tasks, the way they do it is largely (or, from a practical standpoint, almost wholly) inaccessible. This absence of interpretability in modern ML is increasingly a major concern in sensitive areas where accountability is required, such as in finance and the healthcare and pharmaceutical sectors. The “interpretability problem in AI” is therefore a topic of grave worry for large swathes of the corporate and enterprise sector, regulators, lawmakers, and the general public.��

These concerns have given birth to the field of eXplainable AI, or XAI, which attempts to solve the interpretability problem through so-called ‘post-hoc’ techniques (where one takes a trained AI model and aims to give explanations for either its overall behavior or individual outputs). This approach, while still evolving, has its own issues due to the approximate nature and fundamental limitations of post-hoc techniques.�� 

The second approach to the interpretability problem is to employ new ML models that are, by design, inherently interpretable from the start. Such an interpretable AI model comes with explicit structure which is meaningful to us “from the outside”. Realizing this in the tech we use every day means completely redesigning how machines learn - creating a new paradigm in AI. As Sean Tull, one of the authors of the paper, stated: “In the best case, such intrinsically interpretable models would no longer even require XAI methods, serving instead as their own explanation, and one of a deeper kind.”

At ҹɫֱ��, we’re continuing work to develop new paradigms in AI while also working to sharpen theoretical and foundational tools that allow us all to assess the interpretability of a given model. In , we present a new theoretical framework for both defining AI models and analyzing their interpretability. With this framework, we show how advantageous it is for an AI model to have explicit and meaningful compositional structure.

The idea of composition is explored in a rigorous way using a mathematical approach called “category theory”, which is a language that describes processes and their composition. The category theory approach to interpretability can be accomplished via a graphical calculus which was also developed in part by ҹɫֱ�� scientists, and which is finding use cases in everything from gravity to quantum computing.��

A fundamental problem in the field of XAI has been that many terms have not been rigorously defined, making it difficult to study - let alone discuss - interpretability in AI. Our paper presents the first known theoretical framework for assessing the compositional interpretability of AI models.��With our team’s work, we now have a precise and mathematically defined definition of interpretability that allows us to have these critical conversations.    

After developing the framework, our team used it to analyze the full spectrum of ML approaches. We started with Transformers (the “T” in ChatGPT), which are not interpretable – pointing to a serious issue in some of the world’s most widely used ML tools. This is in contrast with (sparse) linear models and decision trees, which we found are indeed inherently interpretable, as they are usually described.  

Our team was also able to make precise how other ML models were what they call 'compositionally interpretable'. These include models already studied by our own scientists including models, causal models, and .��������

Many of the models discussed in this paper are classical, but more broadly the use of category theory and string diagrams makes these tools very well suited to analyzing quantum models for machine learning. In addition to helping the broader field accurately assess the interpretability of various ML models, the seminal work in this paper will help us to develop systems that are interpretable by design.��

This work is part of our broader AI strategy, which includes , and – in this case - using the tools of category theory and compositionality to help us better understand AI.��

“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 from ҹɫֱ��’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 .��

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 [ ].�� 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!

ҹɫֱ�� 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, . 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. In one of our most recent demonstrations, in partnership with Microsoft, we supported the use of error correcting techniques to achieve , confirming our place at the forefront of this research – and indeed confirming that ҹɫֱ��’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 , 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 ҹɫֱ��’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 ҹɫֱ��'s hardware. ҹɫֱ�� 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, .

ҹɫֱ�� 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 ҹɫֱ��, 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 .

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 . 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, ҹɫֱ��’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, , 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 .

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 ҹɫֱ��’s market-leading high fidelity and also our unique all-to-all connectivity. Our flexible connectivity, enabled by , 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. ҹɫֱ��’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.