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1.1. A new compilation regime

We have introduced quantum compilation by drawing an analogy with the well-developed field of classical compilers. The novel directions in which quantum compilation is taking the field make for exciting new challenges. Three new quantum-specific properties of compilation form the core motivation for this work.

Large variations in architecture #

The vast differences between proposed hardware architectures are a first distinguishing characteristic of current quantum computing developments. Unlike classical computing, where silicon-based transistors have become the definitive physical foundation for all electronic chips, the search for the most scalable and reliable technology for quantum computing is ongoing – and doubtless one of the most burning questions for the nascent industry. This introduces an incredible variety of compilation problems.

Quantum hardware designs differ both in the types of quantum particles used to implement qubits and in the control systems employed to manipulate these particles. Suggestions for the former include charged ions Kielpi., 2002D. Kielpinski, C. Monroe and D. J. Wineland. 2002. Architecture for a large-scale ion-trap quantum computer. Nature 417, 6890 (June 2002, 709--711). doi: 10.1038/nature00784 T. Pel., 1995J. I. Cirac, T. Pellizzari and P. Zoller. 1995. Decoherence, Continuous Observation, and Quantum Computing: A Cavity QED Model. Physical Review Letters 75, 21 (November 1995, 3788--3791). doi: 10.1103/PhysRevLett.75.3788, neutral atoms Jaksch, 2000D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté and M. D. Lukin. 2000. Fast Quantum Gates for Neutral Atoms. Physical Review Letters 85, 10 (Septempter 2000, 2208--2211). doi: 10.1103/physrevlett.85.2208 Deutsch, 2000Ivan H. Deutsch, Gavin K. Brennen and Poul S. Jessen. 2000. Quantum Computing with Neutral Atoms in an Optical Lattice. Fortschritte der Physik 48, 9–11 (Septempter 2000, 925--943). doi: 10.1002/1521-3978(200009)48:9/11<925::aid-prop925>3.0.co;2-a, photons Knill, 2001E. Knill, R. Laflamme and G. J. Milburn. 2001. A scheme for efficient quantum computation with linear optics. Nature 409, 6816 (January 2001, 46--52). doi: 10.1038/35051009, transmons Blais, 2007Alexandre Blais, Jay Gambetta, A. Wallraff, D. I. Schuster, S. M. Girvin, M. H. Devoret and R. J. Schoelkopf. 2007. Quantum-information processing with circuit quantum electrodynamics. Physical Review A 75, 3 (March 2007, 032329). doi: 10.1103/physreva.75.032329 and even Majorana Sau, 2010Jay D. Sau, Roman M. Lutchyn, Sumanta Tewari and S. Das Sarma. 2010. Generic New Platform for Topological Quantum Computation Using Semiconductor Heterostructures. Physical Review Letters 104, 4 (January 2010, 040502). doi: 10.1103/physrevlett.104.040502 particles. The equipment that drives the desired operations on these particles is then drawn from a jolly mixture of lasers, magnetic fields, microwaves, dilution fridges, etc. Each combination results in different trade-offs: some will render a specific computation particularly easy; others promise to scale well to large systems but are very error-prone and unreliable; others still achieve high fidelities at the expense of slow operations.

From the perspective of a compiler engineer, this means we must equip quantum compilers to handle a wide variety of hardware primitives, multiple optimisation goals, and hardware-specific program constraints. Traditional compilation is ill-equipped to handle this considerable challenge.

A comparison of machine code for different architectures illustrates the difference between the quantum and classical worlds. Classical CPUs are dominated by two architectures, x86, used mainly by Intel and AMD, and ARM, used by a wide range of desktop and mobile chip manufacturers¹.

mov eax, 5        ; Load 5 into EAX
add eax, 3        ; Add 3 to EAX
mov [result], eax ; Store the result in memory

ldr r0, =5        ; Load 5 into R0
add r0, r0, #3    ; Add 3 to R0
ldr r1, =result   ; Load address of result
str r0, [r1]      ; Store the result in memory

There are noticeable differences between the two architectures, mostly around variable naming conventions, as well as explicit memory loads ldr and stores str instructions in the case of ARM, which in x86 are handled implicitly by mov. This simplistic example naturally ignores some of the more fine-grained considerations that can make translations hard in certain edge cases. A discussion of these can be found in Ford, 2021Blake W. Ford, Apan Qasem, Jelena Tesic and Ziliang Zong. 2021. Migrating Software from x86 to ARM Architecture: An Instruction Prediction Approach. In 2021 IEEE International Conference on Networking, Architecture and Storage (NAS), October 2021. IEEE, 1--6. doi: 10.1109/NAS51552.2021.9605443. However, overall, the instructions and capabilities of the two platforms are broadly equivalent, as is confirmed by the existence of emulation tools such as Apple Rosetta.

Let us contrast this with the difference between two quantum architectures. Consider, on the one hand, an architecture that can natively perform CX and H gates on qubits (e.g. superconducting qubits, ion traps, etc.) and, on the other hand, a platform based on photons and optical components.

h q[0];
cz q[0],q[1];

bs.h(5*pi/2, pi, pi, 2*pi) m[0], m[1];
bs.h m[2], m[3];
perm([2, 1, 3, 0]) m[1], m[2], m[3], m[4];
barrier m[0], m[1], m[2], m[3], m[4], m[5];
bs.h(1.910633) m[0], m[1];
bs.h(1.910633) m[2], m[3];
bs.h(1.910633) m[4], m[5];
perm([1, 0]) m[2], m[3];
bs.h m[3], m[4];
perm([3, 0, 1, 2]) m[1], m[2], m[3], m[4];

On the left is a quantum circuit expressed in the OpenQASM2 standard Cross, 2017Andrew W. Cross, Lev S. Bishop, John A. Smolin and Jay M. Gambetta. 2017. Open Quantum Assembly Language. arXiv: 1707.03429 [quant-ph]. The right-hand side is the equivalent linear optics circuit computed by Perceval Heurtel, 2023Nicolas Heurtel, Andreas Fyrillas, Grégoire Gliniasty, Raphaël Le Bihan, Sébastien Malherbe, Marceau Pailhas, Eric Bertasi, Boris Bourdoncle, Pierre-Emmanuel Emeriau, Rawad Mezher, Luka Music, Nadia Belabas, Benoît Valiron, Pascale Senellart, Shane Mansfield and Jean Senellart. 2023. Perceval: A Software Platform for Discrete Variable Photonic Quantum Computing. Quantum 7 (February 2023, 931). doi: 10.22331/q-2023-02-21-931, expressed in a custom, OpenQASM2-like, format. The conversion is by no means straightforward! Some of the challenges include encoding qubits into multiple photon modes and mapping quantum operations to an optically realisable procedure made of optical components and measurements Felice, 2023Giovanni de Felice and Bob Coecke. 2023. Quantum Linear Optics via String Diagrams. In Proceedings 19th International Conference on Quantum Physics and Logic, Wolfson College, Oxford, UK, 27 June - 1 July 2022. Open Publishing Association, 83-100. doi: 10.4204/EPTCS.394.6.

Other architectures, such as neutral atoms, may broadly support qubit-based operations but might not offer control over individual qubits and, instead require any operations to be applied in parallel to large groups of qubits Bluvst., 2022Dolev Bluvstein, Harry Levine, Giulia Semeghini, Tout T. Wang, Sepehr Ebadi, Marcin Kalinowski, Alexander Keesling, Nishad Maskara, Hannes Pichler, Markus Greiner, Vladan Vuletić and Mikhail D. Lukin. 2022. A quantum processor based on coherent transport of entangled atom arrays. Nature 604, 7906 (April 2022, 451--456). doi: 10.1038/s41586-022-04592-6. Finally, it is to be expected that error-correcting codes that individual platforms will introduce to reduce error rates at the hardware level will introduce further constraints and new instruction sets yet again.

It is noteworthy that current trends in the classical world are also pushing compilers towards more heterogenous architectures that may include GPUs, FPGAs and other accelerators. This has led to significant changes in the design of current compilers, which we will touch upon later. Nonetheless, this shift has, so far, mostly “limited” itself to new forms of parallelism and the introduction of more specialised instruction sets rather than a fundamental redesign of existing tools and computing paradigms. The breadth of technologies and trade-offs that quantum compilers must face have no equivalent in the classical world – at least for the time being.

Asymmetric computational resources #

A second exciting paradigm shift in compilation that quantum is driving forward is cross-compilation. A common assumption in compilation is that the program is executed on the same machine (or at least the same architecture) on which it was compiled. By contrast, in cross-compilation, the compiler and the compiled binary program run on different machines, possibly with different architectures. An instance of this would be using a recent ARM system-on-chip machine to create a binary program for a traditional Windows PC with an Intel CPU. This is a supported feature of most modern compilers (and made easier by the relative similarities between processor architectures, as seen above), but such tasks are by no means trivial and can be laborious to get to work well in practice².

The situation is very different for quantum computing. Quantum computational resources are so limited that native compilation, in which the program is compiled and run on the same machine, is unfeasible – and will remain so for the foreseeable future³. When we put the possibility of pure-quantum compilation aside, we are left with a cross-compilation problem that is entirely the realm of classical computer science; the output of which happens to be destined to run on a quantum computer. This is simliar to how in classical computing, GPU programs are typically compiled on CPUs before being uploaded and executed on the GPU.

Cross-compilation presents significant challenges. As quantum programs grow in size and complexity, debugging and verifying their correctness without access to the target hardware becomes increasingly difficult Rovara, 2024Damian Rovara, Lukas Burgholzer and Robert Wille. 2024. A Framework for Debugging Quantum Programs. arXiv: 2412.12269 [quant-ph], as we hit the limits of what can be simulated classically. Quantum simulation is a vibrant research area that is the subject of theses (e.g. Flanni., 2020Stuart Flannigan. 2020. The application of quantum simulation to topological and open many-body systems Azad, 2024Fariha Azad. 2024. Tensor networks for classical andquantum simulation of open and closedquantum systems. (January 2024)) in its own right.

On the flip side, using classical hardware for quantum program compilation comes with a giant opportunity for compilers: the classical computational resources available to the compiler, measured in the size of the memory and the number of operations that can be handled, are many orders of magnitude larger (and cheaper!) than what the quantum hardware that will execute the program is capable of. We can today execute tens to hundreds of billions of operations per second (GFLOPS) on desktop computers, up to the “exascale”, i.e. $10^{15}$ FLOPS, for the largest supercomputers Dongar., 2024Strohmaier, E., Simon, H., Meuer, H. Dongarra. 2024. TOP500 List. (November 2024). Retrieved on 30/12/2024 from https://top500.org/lists/top500/list/2024/11/. Quantum hardware, on the other hand, will not be executing programs with sizes beyond 1000 error-corrected gates, or 10,000 physical gates, for another three years – that is believing the most optimistic roadmaps in the industry IBM, 2024 IBM. 2024. Expanding the IBM Quantum roadmap to anticipate the future of quantum-centric supercomputing. Retrieved on 30/12/2024 from https://www.ibm.com/quantum/blog/ibm-quantum-roadmap-2025 Quanti., 2024 Quantinuum. 2024. Quantinuum Unveils Accelerated Roadmap to Achieve Universal, Fully Fault-Tolerant Quantum Computing by 2030. Retrieved on 30/12/2024 from https://www.quantinuum.com/press-releases/quantinuum-unveils-accelerated-roadmap-to-achieve-universal-fault-tolerant-quantum-computing-by-2030.

It is expected that even a few thousand quantum gates will suffice to solve problems that our largest supercomputers struggle with. Meanwhile, every gate that must be performed comes at a high cost: it may fail, introduce errors, or take a long time to complete. It therefore behoves us to use all the classical resources at our disposal to reduce quantum operations to a minimum.

Given the strict hardware limitations, all near-term architectures are expected to face, quantum compilation must evolve into cross-compilers that are able to utilise the full power of classical hardware available to them; doing so will push the boundaries of what is possible with quantum computing just a bit further – in a field where every marginal gain may unlock new applications.

The confluence of classical and quantum compilation #

Finally, quantum compilation also stands in front of some momentous engineering challenges. As we will see in section 2.2, significant research efforts have focused on the compilation and optimisation of quantum programs expressed as quantum circuits (cf. section 2.1). This formalism has its roots in quantum information theory, the field that gave birth to quantum computing and makes for an ideal framework to develop the theory and optimisation techniques. However, it does not include any of the fundaments of compiler and programming language design that make classical software as composable and scalable as it is today.

For example, there is no concept of subroutine or function calls; neither can a program execution be branching or looping based on runtime values. This makes code reuse impossible, resulting in huge program sizes and unsurmountable challenges for scaling up compilation to problems of real-world interest Ittah, 2022David Ittah, Thomas Häner, Vadym Kliuchnikov and Torsten Hoefler. 2022. QIRO: A Static Single Assignment-based Quantum Program Representation for Optimization. ACM Transactions on Quantum Computing 3, 3 (June 2022, 1--32). doi: 10.1145/3491247. The absence of code abstractions is being felt even more acutely with the emergence of hybrid quantum-classical computations, as we discuss in section 2.3.

With applications of quantum computing that cannot be expressed as quantum circuits proliferating, a move away from circuit-based representations is becoming unavoidable Hossei., 2023Lev Bishop, Yudong Cao, Andrew Cross, Niel Hossein Ajallooiean. 2023. OpenQASM 3.0 Specification. Retrieved on 15/03/2025 from https://openqasm.com/versions/3.0/intro.html QIR Al., 2021 QIR Alliance. 2021. QIR Specification v0.1. Retrieved on 31/12/24 from https://www.qir-alliance.org/. This is also an opportunity to incorporate learnings from the decades of experience that have been gathered in classical computer science. Many of the tools and software that were originally developed for classical computations are thus being adopted and adapted to the specificities of quantum. This convergence of quantum computing and classical compiler technologies is heralding new opportunities – but also pose important questions around how to represent quantum programs and optimise them.

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