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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.

The 55th Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics () from June 3-7, 2024, will feature presentations from ҹɫֱ��’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 ҹɫֱ�� is advancing quantum computing:

‍Speaker: Patty Lee, Chief Scientist for Hardware Technology Development
Date and Time: June 3, 2:30 pm – 3:45 pm
Location: Room 202AB

Speaker: Robert Delaney, Advanced Physicist
Date and Time: June 4, 10:45 am – 12:45 pm
Location: Room 203B

Speaker: Julia Cline, Advanced Physicist
Date and Time: June 4, 2:00 pm – 4:00 pm
Location: Room 203A

(Poster Session I)
Presenter: Ivaylo Madjarov, Numerical Physicist
Date and Time: June 4, 4:00 pm – 6:00 pm
Location: Hall BC

Speaker: Chris Gilbreth, Lead Physicist
Date and Time: June 7, 11:30 am – 11:42 am
Location: Room 201BC

‍

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 , 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, ҹɫֱ��

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 ҹɫֱ��, with support from PAQ Ile de France.

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

May 14^th

2:20pm – 2:35pm | Hall Z – 3rd floor

The Trapped-Ion Quantum Processors at ҹɫֱ��: ҹɫֱ�� 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 ҹɫֱ��’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 ҹɫֱ��: 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. ҹɫֱ�� 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, ҹɫֱ��

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 ҹɫֱ�� 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, ҹɫֱ��

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, ҹɫֱ��

We are surrounded by optimization problems – for example, what’s the most efficient route for getting all your chores done on a Sunday? What’s the best way to pack a suitcase? Modern businesses can’t escape the importance of optimization problems, they’re critical in everything from charting shipping routes to setting prices.��

To solve such real-world examples, experts build mathematical models and explore computer algorithms capable of finding the optimal path through a problem. In many cases, as they scale, problems become intractable to even the most powerful classical supercomputers. Research suggests that for some problems, quantum algorithms offer some new promise. Our researchers have explored a quantum approach to a widely applicable optimization problem called “Max-Cut”, where one cuts a graph to snip as many vertices as possible.

Finding exact solutions to the Max-Cut problem in a reasonable amount of time would have practical applications in a wide range of situations, including supply chain management, machine scheduling, image recognition, quality control, fraud detection, patient diagnostics, and electric circuit design. For a generic graph, this problem is really hard: a computer scientist would call it “NP hard”. There is no known classical algorithm to solve Max-Cut for a generic graph whose runtime is polynomial in the number of vertices L, and it is strongly believed that no such classical algorithm exists. Many other useful optimization problems have a similar problem: they may simply be too expensive to solve exactly with classical computers. Back in the real world, this explains why many aspects of daily life run sub-optimally. Consider the experience of multiple drivers delivering a succession of small goods from the same vendor, often packaged in clearly oversized boxes. The costs of this sort of inefficiency accrue in terms of time, money, and environmental impact, locally and at the full scale of the global economy.

Our team has been working on applying a quantum solution to the Max-Cut problem based on the adiabatic theorem of quantum mechanics. Using the adiabatic theorem to solve an optimization problem involves encoding the problem into the qubits (setting up the Hamiltonian), then letting the system slowly evolve some parameter, carefully keeping it in the ground state the whole time. This method is an all-purpose solver for classically hard optimization problems, but it comes at a large computational cost: the “slow” evolution means applying lots of expensive gates to perform the many time steps needed.��

Our team figured out that instead of taking many expensive steps they could instead take a limited amount without destroying the convergence, as long as the optimization problem has a classical Hamiltonian. They call this “Floquet adiabatic evolution” and find that this approach reduces the required number of gates by several orders of magnitude.

Contrary to variational quantum algorithms such as Quantum Approximate Optimization Algorithm (QAOA), these low circuit depths can be achieved without classical optimization of parameters (whose sensitivity to noise and scaling behavior is not well understood).

Extrapolating their numerical simulation results, the team estimated that there may be a quantum speedup for this problem with a 2-qubit gate infidelity around 10^-5 and roughly 2000 qubits. Our H1 system already boasts a world-class 2-qubit gate infidelity of 8.8 × 10^-4, and we are well on our way towards even better fidelity with more qubits. You can see our roadmap here, and read the paper .

In the meantime, the paper proposes that this method could be used as a quantum computing benchmark for application-oriented problems, making a valuable contribution to the , of which ҹɫֱ�� is a founding member.