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2.5. Summary and further reading

This introductory chapter covered some of the basic principles of quantum computation and, in doing so, hopefully, made a convincing argument as to why we should expect the programs running on quantum hardware to become more complex in the future, with the intertwining of classical and quantum computations – processes we refer to as hybrid quantum-classical programs. Prior to that, we also presented quantum compilation, an emerging discipline that is introducing many new problems and ideas to the established corpus of work on compiler research.

If this quantum taster has intrigued you or you would like to learn the basics from people who actually know what they are talking about, nothing beats the reference book for quantum information and quantum computing by Nielsen and Chuang Nielsen, 2016Michael A. Nielsen and Isaac L. Chuang. 2016. Quantum Computation and Quantum Information (10th Anniversary edition). Cambridge University Press. A fascinating alternative perspective on quantum theory has also been developed within the programme of categorical quantum mechanics, for which the illustrious “Dodo book” Coecke, 2017Bob Coecke and Aleks Kissinger. 2017. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press. doi: 10.1017/9781316219317 would be the go-to introductory material¹.

At the risk of turning this thesis into absolutely shameless Oxford self-promotion, guess what else was a product of this university’s world-class research? The quantum circuit itself! These diagrams came from theoretical physicists (no surprise here) interested in capturing thought experiments in quantum information theory Deutsch, 1989David Deutsch. 1989. Quantum Computational Networks. In Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. Royal Society, 73--90.

The idea caught on, and soon, software tools were created to facilitate building such diagrams. The Quantum Computation Language (QCL) was one of the first Ömer, 2000Bernhard Ömer. 2000. Quantum Programming in QCL. (January 2000). Retrieved from http://tph.tuwien.ac.at/ oemer/doc/quprog.pdf. Quantum software² has since proliferated, especially as the possibility of actually performing these thought experiments on quantum hardware became more tangible. The result were software packages for quantum computing, designed for the automatic transformation and optimisation of quantum computations for execution on real hardware Javadi., 2024Ali Javadi-Abhari, Matthew Treinish, Kevin Krsulich, Christopher J. Wood, Jake Lishman, Julien Gacon, Simon Martiel, Paul D. Nation, Lev S. Bishop, Andrew W. Cross, Blake R. Johnson and Jay M. Gambetta. 2024. Quantum computing with Qiskit. arXiv: 2405.08810 [quant-ph] Cirq D., 2024 Cirq Developers. 2024. Cirq Steiger, 2018Damian S. Steiger, Thomas Häner and Matthias Troyer. 2018. ProjectQ: an open source software framework for quantum computing. Quantum 2 (January 2018, 49). doi: 10.22331/q-2018-01-31-49 Sivara., 2020Seyon Sivarajah, Silas Dilkes, Alexander Cowtan, Will Simmons, Alec Edgington and Ross Duncan. 2020. t|ket⟩: a retargetable compiler for NISQ devices. Quantum Science and Technology 6, 1 (November 2020, 014003). doi: 10.1088/2058-9565/ab8e92​ – we called them quantum compilers.

A recent development for quantum compilers focuses on scalability and first-class support for hybrid quantum-classical computations. Quantum circuits that include some form of classical control have been variously called “dynamic circuits” (e.g. 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), “adaptive circuits” (e.g. Smith, 2024Kevin C. Smith, Abid Khan, Bryan K. Clark, S.M. Girvin and Tzu-Chieh Wei. 2024. Constant-Depth Preparation of Matrix Product States with Adaptive Quantum Circuits. PRX Quantum 5, 3 (Septempter 2024, 030344). doi: 10.1103/prxquantum.5.030344), “circuits with measurements and feedforward” (e.g. 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), and “circuits assisted by local operations and classical communication” (e.g. Piroli, 2021Lorenzo Piroli, Georgios Styliaris and J. Ignacio Cirac. 2021. Quantum Circuits Assisted by Local Operations and Classical Communication: Transformations and Phases of Matter. Physical Review Letters 127, 22 (November 2021, 220503). doi: 10.1103/physrevlett.127.220503).

Besides supporting advances 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, hybrid classical-quantum computations are central to many quantum computing applications. As put recently by Alam and Clark Alam, 2024Faisal Alam and Bryan K. Clark. 2024. Learning dynamic quantum circuits for efficient state preparation. arXiv: 2410.09030 [quant-ph]​:

“[…] dynamic quantum circuits are a crucial milestone on the roadmap to fault-tolerant quantum computers.”

We have covered a small subset of applications of hybrid quantum-classical computations. Quantum teleportation is undoubtedly one of the oldest Bennett, 1993Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres and William K. Wootters. 1993. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters 70, 13 (March 1993, 1895--1899). doi: 10.1103/physrevlett.70.1895. The block-encoding technique that we discussed in section 2.3 is the foundation of several algorithms, including the Quantum Singular Value Decomposition (QSVT) Gilyén, 2019András Gilyén, Yuan Su, Guang Hao Low and Nathan Wiebe. 2019. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, June 2019. ACM, 193--204. doi: 10.1145/3313276.3316366 and the Linear Combination of Unitaries (LCU) Chakra., 2024Shantanav Chakraborty. 2024. Implementing any Linear Combination of Unitaries on Intermediate-term Quantum Computers. Quantum 8 (October 2024, 1496). doi: 10.22331/q-2024-10-10-1496 Sze, 2025Michelle Wynne Sze, Yao Tang, Silas Dilkes, David Muñoz Ramo, Ross Duncan and Nathan Fitzpatrick. 2025. Hamiltonian dynamics simulation using linear combination of unitaries on an ion trap quantum computer. arXiv: 2501.18515 [quant-ph]. Measurement-based quantum computing (MBQC) was introduced in Rausse., 2001Robert Raussendorf and Hans J. Briegel. 2001. A One-Way Quantum Computer. Physical Review Letters 86, 22 (May 2001, 5188--5191). doi: 10.1103/PhysRevLett.86.5188 and is forming the base for some photonic quantum computing architectures Bartol., 2023Sara Bartolucci, Patrick Birchall, Hector Bombín, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Terry Rudolph and Chris Sparrow. 2023. Fusion-based quantum computation. Nature Communications 14, 1 (February 2023). doi: 10.1038/s41467-023-36493-1 Bouras., 2021J. Eli Bourassa, Rafael N. Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q. Baragiola, Saikat Guha, Guillaume Dauphinais, Krishna K. Sabapathy, Nicolas C. Menicucci and Ish Dhand. 2021. Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer. Quantum 5 (February 2021, 392). doi: 10.22331/q-2021-02-04-392. 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.

On the other hand, repeat until success schemes Paetzn., 2014Adam Paetznick and Krysta M. Svore. 2014. Repeat-until-success: non-deterministic decomposition of single-qubit unitaries. Quantum Information & Computation 14, 15–16 (November 2014, 1277–1301) are widespread in state preparation routines and will play a key role in fault-tolerant (FT) quantum computing. Arguably, the most well-known scheme for FT is magic state distillation Bravyi, 2005Sergey Bravyi and Alexei Kitaev. 2005. Universal quantum computation with ideal Clifford gates and noisy ancillas. Physical Review A 71, 2 (February 2005, 022316). doi: 10.1103/PhysRevA.71.022316, a procedure expected to be a core building block of many FT architectures. State preparation is generally a 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 Fowler, 2012Austin G. Fowler, Matteo Mariantoni, John M. Martinis and Andrew N. Cleland. 2012. Surface codes: Towards practical large-scale quantum computation. Physical Review A 86, 3 (Septempter 2012, 032324). doi: 10.1103/physreva.86.032324.

Finally, quantum error-correcting (QEC) codes themselves must be implemented using hybrid programs. The quantum error correction (QEC) literature is vast and can get very technical very quickly, but diving into it promises bountiful rewards. The field is one of quantum information’s fastest-evolving areas of research. These work-in-progress lecture notes Gottes., 2024Daniel Gottesman. 2024. Surviving as a Quantum Computer in a Classical World. (February 2024). Retrieved on 08/01/2025 (lecture notes) from https://www.cs.umd.edu/class/spring2024/cmsc858G/QECCbook-2024-ch1-15.pdf by a coryphaeus of the field make for excellent introductory material.

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