### Summary and further reading

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## 2.4. Summary and further reading

This introductory chapter covered some of the basic principles of quantum computation, and in doing so has hopefully made a convincing argument why we should expect the programs running on quantum hardware to be hybrid quantum-classical. Prior to that, we also presented quantum compilation, an emerging discipline in the field that is introducing many new problems and ideas to the established corpus of work on compilers research.

If this quantum taster has intreagued you or you would like to learn the basics from people that actually know what they are talking about, nothing beats the reference book for quantum information and quantum computing by Nielsen and Chuang. A fascinating alternative perspective on quantum theory has also been developed within the programme of categorical quantum mechanics, for which the seminal textbook XX1 would be the go-to introductory material.

As for quantum compilers, their history is very closely intertwined with the development of the quantum circuit itself. As is often the case with diagrammatic representations, the quantum circuit was a product of the imagination of theoretical physicists – in Oxford, of course – interested in capturing thought experiments in quantum information theory (Deutsch 1989, Quantum computational networks). The idea caught on, and soon software tools were created to facilitate building such diagrams (and printing them in papers2).

As the possibility of actually performing these experiments on quantum hardware became more tangible, the need to automatically transform and optimise these diagrams became apparent. The result were software packages for quantum computing that for the first time aimed to create programs to be executed on real quantum hardware (projectq, qiskit, cirq, TKET). We called them quantum compilers.

The specificities of quantum compilation, some of which we presented in this chapter, have been the subject of research for decades at this point – and tools for quantum compilation have been mainstream3 certainly since the advent of programmable, albeit small and noisy, quantum computers.

A more recent development for quantum compilers is the focus on scalability and first class support for hybrid quantum-classical computations, which followed improvements in quantum hardware Córco., 2021A. D. Córcoles, Maika Takita, Ken Inoue, Scott Lekuch, Zlatko K. Minev, Jerry M. Chow and Jay M. Gambetta. 2021. Exploiting Dynamic Quantum Circuits in a Quantum Algorithm with Superconducting Qubits. Physical Review Letters 127, 10 (August 2021, 100501). doi: 10.1103/physrevlett.127.100501 Graham, 2023T. M. Graham, L. Phuttitarn, R. Chinnarasu, Y. Song, C. Poole, K. Jooya, J. Scott, A. Scott, P. Eichler and M. Saffman. 2023. Midcircuit Measurements on a Single-Species Neutral Alkali Atom Quantum Processor. Physical Review X 13, 4 (December 2023, 041051). doi: 10.1103/physrevx.13.041051 Pino, 2021J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, M. S. Allman, C. H. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer, C. Ryan-Anderson and B. Neyenhuis. 2021. Demonstration of the trapped-ion quantum CCD computer architecture. Nature 592, 7853 (April 2021, 209--213). doi: 10.1038/s41586-021-03318-4. We have started to see in this chapter that in this new light, quantum circuits may not be the ideal representation for quantum computations. Moving away from circuits however requires us to rebuild our quantum compilers from the ground up! This topic will be explored further in the next chapter and will keep us busy for the rest of the thesis (and certainly beyond that, too).

We have covered some examples of applications of hybrid quantum classical computations. Quantum teleportation (citation?) is certainly one of the oldest instances of this. A whole host of applications for teleportation. Measurement-based quantum computing (MBQC) was introduced in xyz etc. More recently, MBQC were shown to be executable in practice and Will also did some work on this. Hybrid programs have also been shown to be useful for implementing the Quantum Fourier Transform (QFT) Bäumer, 2024Elisa Bäumer, Vinay Tripathi, Alireza Seif, Daniel Lidar and Derek S. Wang. 2024. Quantum Fourier Transform Using Dynamic Circuits. Physical Review Letters 133, 15 (October 2024, 150602). doi: 10.1103/physrevlett.133.150602 and the Quantum Phase Estimation (QPE) algorithms Córco., 2021A. D. Córcoles, Maika Takita, Ken Inoue, Scott Lekuch, Zlatko K. Minev, Jerry M. Chow and Jay M. Gambetta. 2021. Exploiting Dynamic Quantum Circuits in a Quantum Algorithm with Superconducting Qubits. Physical Review Letters 127, 10 (August 2021, 100501). doi: 10.1103/physrevlett.127.100501, two of the most fundamental computation primitives for quantum algorithms.

TODO: refs in the Younis paper for hybrid Niu, 2024Siyuan Niu, Efekan Kokcu, Anupam Mitra, Aaron Szasz, Akel Hashim, Justin Kalloor, Wibe Albert de Jong, Costin Iancu and Ed Younis. 2024. AC/DC: Automated Compilation for Dynamic Circuits. arXiv: 2412.07969 [quant-ph]. hybrid, aka “dynamic circuits”, “adaptive circuits”, “circuits with measurements and feedforward”, or circuits assisted by local operations and classical communication (LOCC).

> “dynamic quantum circuits are a crucial milestone on the roadmap to fault-tolerant quantum computers” (Faisal 2024)

Repeat until success schemes on the other hand are very common in state preparation routines and will play a key role in fault tolerant (FT) quantum computing. Arguably the most well-known such scheme for FT is magic state distillation, a procedure expected to be a core building block of many FT architectures. State preparation is in general an ubiquitous problem for FT, as the error correcting codes that are employed initiate computations starting from a “logical” zero state, which may be expensive to prepare on the qubits of the hardware.

Finally, quantum error correcting (QEC) codes themselves must be implemented using hybrid programs. The workflow will sound very familiar to whomever has followed this chapter with an ounce of attention: Error correction is achieved through successive rounds of i) measurements (known as syndromes in QEC lingo), ii) syndrome decoding (i.e. classical computations based on the measurement outcomes that infer the errors that must have occured) and, finally, iii) error correction (classically-controlled quantum operations that correct the inferred errors). The QEC literature is vast and can get very technical very quickly, but diving into it promises bountiful rewards. The field is one of the most fast evolving areas of research in the field. These work-in-progress lecture notes by a coryphaeus of the field make for excellent introductory material.

1. Pro tip: never omit to cite your supervisor’s textbooks. Here’s another one, just as worth a read and more practical and computer science-y: YY. ↩︎

2. The Quantum Computation Language (QCL, Omer, 1998) was in many ways ahead of its time, offering an advanced programming interface including loops and control flow, similar to some of the recent tools this thesis is a loud proponent of. It presumably failed to gain traction as it did not support printing diagrams for papers. ↩︎

3. Or at least, as mainstream as can be within such a niche field. ↩︎

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