research-article

Open access

Tower: data structures in Quantum superposition

Authors:

Michael CarbinAuthors Info & Claims

Proceedings of the ACM on Programming Languages, Volume 6, Issue OOPSLA2

Article No.: 134, Pages 259 - 288

https://doi.org/10.1145/3563297

Published: 31 October 2022 Publication History

Abstract

Emerging quantum algorithms for problems such as element distinctness, subset sum, and closest pair demonstrate computational advantages by relying on abstract data structures. Practically realizing such an algorithm as a program for a quantum computer requires an efficient implementation of the data structure whose operations correspond to unitary operators that manipulate quantum superpositions of data.

To correctly operate in superposition, an implementation must satisfy three properties --- reversibility, history independence, and bounded-time execution. Standard implementations, such as the representation of an abstract set as a hash table, fail these properties, calling for tools to develop specialized implementations.

In this work, we present Core Tower, the first language for quantum programming with random-access memory. Core Tower enables the developer to implement data structures as pointer-based, linked data. It features a reversible semantics enabling every valid program to be translated to a unitary quantum circuit.

We present Boson, the first memory allocator that supports reversible, history-independent, and constant-time dynamic memory allocation in quantum superposition. We also present Tower, a language for quantum programming with recursively defined data structures. Tower features a type system that bounds all recursion using classical parameters as is necessary for a program to execute on a quantum computer.

Using Tower, we implement Ground, the first quantum library of data structures, including lists, stacks, queues, strings, and sets. We provide the first executable implementation of sets that satisfies all three mandated properties of reversibility, history independence, and bounded-time execution.

Supplementary Material

Auxiliary Archive (oopslab22main-p217-p-archive.zip)

Appendices of paper.

Download
399.63 KB

References

[1]

Scott Aaronson. 2002. Quantum Lower Bound for Recursive Fourier Sampling. https://doi.org/10.48550/ARXIV.QUANT-PH/0209060 arxiv:0209060.

[2]

Scott Aaronson, Nai-Hui Chia, Han-Hsuan Lin, Chunhao Wang, and Ruizhe Zhang. 2019. On the Quantum Complexity of Closest Pair and Related Problems. arxiv:1911.01973.

[3]

Héctor Abraham. 2019. Qiskit: An Open-source Framework for Quantum Computing.

[4]

Daniel S. Abrams and Seth Lloyd. 1997. Simulation of Many-Body Fermi Systems on a Universal Quantum Computer. Phys. Rev. Lett., 79 (1997), Sep, https://doi.org/10.1103/PhysRevLett.79.2586

[5]

T. Altenkirch and J. Grattage. 2005. A Functional Quantum Programming Language. In IEEE Symposium on Logic in Computer Science. https://doi.org/10.1109/LICS.2005.1

Digital Library

[6]

Andris Ambainis. 2003. Quantum walk algorithm for element distinctness. arxiv:0311001.

[7]

Srinivasan Arunachalam, Vlad Gheorghiu, Tomas Jochym-O’Connor, Michele Mosca, and Priyaa Varshinee Srinivasan. 2015. On the robustness of bucket brigade quantum RAM. New Journal of Physics, 17, 12 (2015), Dec, https://doi.org/10.1088/1367-2630/17/12/123010

[8]

Holger Bock Axelsen and Robert Glück. 2013. Reversible Representation and Manipulation of Constructor Terms in the Heap. In Reversible Computation, Gerhard W. Dueck and D. Michael Miller (Eds.).

[9]

Sumeet Bajaj, Anrin Chakraborti, and Radu Sion. 2016. Practical Foundations of History Independence. IEEE Transactions on Information Forensics and Security, 11, 2 (2016), https://doi.org/10.1109/TIFS.2015.2491309

Digital Library

[10]

Adriano Barenco, André Berthiaume, David Deutsch, Artur Ekert, Richard Jozsa, and Chiara Macchiavello. 1997. Stabilization of Quantum Computations by Symmetrization. SIAM J. Comput., 26, 5 (1997), https://doi.org/10.1137/S0097539796302452

Digital Library

[11]

J. S. Bell. 1964. On the Einstein Podolsky Rosen paradox. Physics, 1 (1964), Nov, https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195

[12]

C. H. Bennett. 1973. Logical Reversibility of Computation. IBM Journal of Research and Development, 17, 6 (1973), https://doi.org/10.1147/rd.176.0525

Digital Library

[13]

Charles H. Bennett. 1989. Time/Space Trade-Offs for Reversible Computation. SIAM J. Comput., 18, 4 (1989), Aug, https://doi.org/10.1137/0218053

Digital Library

[14]

Charles H. Bennett and Gilles Brassard. 2014. Quantum cryptography: Public key distribution and coin tossing. Theoretical Computer Science, 560 (2014), https://doi.org/10.1016/j.tcs.2014.05.025

Digital Library

[15]

Daniel J. Bernstein, Stacey Jeffery, Tanja Lange, and Alexander Meurer. 2013. Quantum Algorithms for the Subset-Sum Problem. In Post-Quantum Cryptography.

[16]

Dominic W. Berry, Mária Kieferová, Artur Scherer, Yuval R. Sanders, Guang Hao Low, Nathan Wiebe, Craig Gidney, and Ryan Babbush. 2017. Improved techniques for preparing eigenstates of fermionic Hamiltonians. npj Quantum Information, 4 (2017), https://doi.org/10.1038/s41534-018-0071-5

[17]

Benjamin Bichsel, Maximilian Baader, Timon Gehr, and Martin Vechev. 2020. Silq: A High-Level Quantum Language with Safe Uncomputation and Intuitive Semantics. In ACM SIGPLAN Conference on Programming Language Design and Implementation. https://doi.org/10.1145/3385412.3386007

Digital Library

[18]

Aaron Bohannon, J. Nathan Foster, Benjamin C. Pierce, Alexandre Pilkiewicz, and Alan Schmitt. 2008. Boomerang: Resourceful Lenses for String Data. In ACM SIGPLAN Symposium on Principles of Programming Languages. https://doi.org/10.1145/1328438.1328487

Digital Library

[19]

Kyle E. C. Booth, Bryan O' Gorman, Jeffrey Marshall, Stuart Hadfield, and Eleanor Rieffel. 2021. Quantum-accelerated constraint programming. Quantum, https://doi.org/10.22331/q-2021-09-28-550

[20]

William Bowman, Roshan James, and Amr Sabry. 2011. Dagger Traced Symmetric Monoidal Categories and Reversible Programming. In Workshop on Reversible Computation.

[21]

Harry Buhrman, Bruno Loff, Subhasree Patro, and Florian Speelman. 2021. Limits of quantum speed-ups for computational geometry and other problems: Fine-grained complexity via quantum walks. arxiv:2106.02005.

[22]

Harry Buhrman, Bruno Loff, Subhasree Patro, and Florian Speelman. 2022. Memory Compression with Quantum Random-Access Gates. https://doi.org/10.48550/ARXIV.2203.05599 arxiv:2203.05599.

[23]

Shantanav Chakraborty, András Gilyén, and Stacey Jeffery. 2019. The Power of Block-Encoded Matrix Powers: Improved Regression Techniques via Faster Hamiltonian Simulation. https://doi.org/10.4230/LIPICS.ICALP.2019.33 arxiv:1804.01973.

[24]

Lijie Chen and Ramis Movassagh. 2021. Quantum Merkle Trees. https://doi.org/10.48550/ARXIV.2112.14317 arxiv:2112.14317.

[25]

K-W Cheng and C-C Tseng. 2002. Quantum full adder and subtractor. Electron. Lett., 38 (2002), https://doi.org/10.1049/el:20020949

[26]

Mitchell Chiew, Kooper de Lacy, Chao-Hua Yu, Sam Marsh, and Jingbo Wang. 2019. Graph comparison via nonlinear quantum search. Quantum Information Processing, 18 (2019), 08, https://doi.org/10.1007/s11128-019-2407-2

Digital Library

[27]

Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su. 2018. Toward the first quantum simulation with quantum speedup. Proceedings of the National Academy of Sciences, 115, 38 (2018), https://doi.org/10.1073/pnas.1801723115

[28]

D. Coppersmith. 1994. An approximate Fourier transform useful in quantum factoring.

[29]

Yongshan Ding, Xin-Chuan Wu, Adam Holmes, Ash Wiseth, Diana Franklin, Margaret Martonosi, and Frederic T. Chong. 2020. SQUARE: Strategic Quantum Ancilla Reuse for Modular Quantum Programs via Cost-Effective Uncomputation. In International Symposium on Computer Architecture. https://doi.org/10.1109/ISCA45697.2020.00054

Digital Library

[30]

Thomas G. Draper. 2000. Addition on a Quantum Computer. arxiv:quant-ph/0008033.

[31]

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. 2014. A Quantum Approximate Optimization Algorithm. https://doi.org/10.48550/ARXIV.1411.4028 arxiv:1411.4028.

[32]

Edward Fredkin and Tommaso Toffoli. 1982. Conservative logic. International Journal of Theoretical Physics, 21 (1982), https://doi.org/10.1007/BF01857727

[33]

Craig Gidney. 2022. Quantum Dictionaries without QRAM. https://doi.org/10.48550/ARXIV.2204.13835 arxiv:2204.13835.

[34]

Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. 2008. Quantum Random Access Memory. Physical Review Letters, 100, 16 (2008), Apr, https://doi.org/10.1103/physrevlett.100.160501

[35]

Oded Goldreich and Rafail Ostrovsky. 1996. Software Protection and Simulation on Oblivious RAMs. J. ACM, 43, 3 (1996), May, https://doi.org/10.1145/233551.233553

Digital Library

[36]

Alexander S. Green, Peter LeFanu Lumsdaine, Neil J. Ross, Peter Selinger, and Benoît Valiron. 2013. Quipper: A Scalable Quantum Programming Language. In ACM SIGPLAN Conference on Programming Language Design and Implementation. https://doi.org/10.1145/2491956.2462177

Digital Library

[37]

Lov K. Grover. 1996. A Fast Quantum Mechanical Algorithm for Database Search. In ACM Symposium on Theory of Computing. https://doi.org/10.1145/237814.237866

Digital Library

[38]

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. 2009. Quantum Algorithm for Linear Systems of Equations. Physical Review Letters, 103, 15 (2009), Oct, https://doi.org/10.1103/physrevlett.103.150502

[39]

Jason D. Hartline, Edwin S. Hong, Alexander E. Mohr, William R. Pentney, and Emily C. Rocke. 2002. Characterizing History Independent Data Structures. In Algorithms and Computation, Prosenjit Bose and Pat Morin (Eds.).

[40]

Tue Haulund. 2017. Design and Implementation of a Reversible Object-Oriented Programming Language. https://doi.org/10.48550/ARXIV.1707.07845 arxiv:1707.07845.

[41]

Md Saiful Islam, Muhammad Mahbubur Rahman, Zerina Begum, and Mohd Z Hafiz. 2009. Low cost quantum realization of reversible multiplier circuit. Information technology journal, 8, 2 (2009), https://doi.org/10.3923/itj.2009.208.213

[42]

Stacey Jeffery, Robin Kothari, and Frederic Magniez. 2013. Nested Quantum Walks with Quantum Data Structures. In ACM-SIAM Symposium on Discrete Algorithms. https://doi.org/10.1137/1.9781611973105.106

[43]

Ivan Kassal, James D. Whitfield, Alejandro Perdomo-Ortiz, Man-Hong Yung, and Alán Aspuru-Guzik. 2011. Simulating Chemistry Using Quantum Computers. Annual Review of Physical Chemistry, 62, 1 (2011), https://doi.org/10.1146/annurev-physchem-032210-103512

[44]

Iordanis Kerenidis and Anupam Prakash. 2016. Quantum Recommendation Systems. https://doi.org/10.48550/ARXIV.1603.08675 arxiv:1603.08675.

[45]

E Knill. 1996. Conventions for quantum pseudocode.

[46]

Gushu Li, Li Zhou, Nengkun Yu, Yufei Ding, Mingsheng Ying, and Yuan Xie. 2020. Projection-Based Runtime Assertions for Testing and Debugging Quantum Programs. In ACM Conference on Object-Oriented Programming, Systems, Languages, and Applications. https://doi.org/10.1145/3428218

Digital Library

[47]

Richard L. Liboff. 1980. Introductory quantum mechanics. Addison-Wesley.

[48]

Christopher Lutz. 1986. Janus: a time-reversible language. Letter to R. Landauer.

[49]

Olivia Di Matteo, Vlad Gheorghiu, and Michele Mosca. 2020. Fault-Tolerant Resource Estimation of Quantum Random-Access Memories. IEEE Transactions on Quantum Engineering, 1 (2020), https://doi.org/10.1109/tqe.2020.2965803

[50]

Donald R. Morrison. 1968. PATRICIA–Practical Algorithm To Retrieve Information Coded in Alphanumeric. J. ACM, 15, 4 (1968), Oct, https://doi.org/10.1145/321479.321481

Digital Library

[51]

Moni Naor and Vanessa Teague. 2001. Anti-persistence: History independent data structures. In ACM Symposium on Theory of Computing. https://doi.org/10.1145/380752.380844

Digital Library

[52]

Michael A. Nielsen and Isaac L. Chuang. 2010. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press.

Digital Library

[53]

Alexandru Paler, Oumarou Oumarou, and Robert Basmadjian. 2020. Parallelizing the queries in a bucket-brigade quantum random access memory. Physical Review A, 102, 3 (2020), Sep, https://doi.org/10.1103/physreva.102.032608

[54]

Anouk Paradis, Benjamin Bichsel, Samuel Steffen, and Martin Vechev. 2021. Unqomp: Synthesizing Uncomputation in Quantum Circuits. In ACM SIGPLAN Conference on Programming Language Design and Implementation. https://doi.org/10.1145/3453483.3454040

Digital Library

[55]

Jennifer Paykin, Robert Rand, and Steve Zdancewic. 2017. QWIRE: A Core Language for Quantum Circuits. In ACM SIGPLAN Symposium on Principles of Programming Languages. https://doi.org/10.1145/3009837.3009894

Digital Library

[56]

Romain Péchoux, Simon Perdrix, Mathys Rennela, and Vladimir Zamdzhiev. 2020. Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory. In Lecture Notes in Computer Science. Springer International Publishing. https://doi.org/10.1007/978-3-030-45231-5_29

Digital Library

[57]

Peter Selinger and Benoît Valiron. 2005. A Lambda Calculus for Quantum Computation with Classical Control. Typed Lambda Calculi and Applications, https://doi.org/10.1017/S0960129506005238

Digital Library

[58]

Run-Hua Shi. 2021. Quantum Bloom Filter and Its Applications. IEEE Transactions on Quantum Engineering, https://doi.org/10.1109/TQE.2021.3054623

[59]

Peter W. Shor. 1997. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM J. Comput., 26, 5 (1997), Oct, https://doi.org/10.1137/S0097539795293172

Digital Library

[60]

Krysta Svore, Martin Roetteler, Alan Geller, Matthias Troyer, John Azariah, Christopher Granade, Bettina Heim, Vadym Kliuchnikov, Mariia Mykhailova, and Andres Paz. 2018. Q#: Enabling Scalable Quantum Computing and Development with a High-level DSL. In Real World Domain Specific Languages Workshop. https://doi.org/10.1145/3183895.3183901

Digital Library

[61]

Michael Kirkedal Thomsen and Holger Bock Axelsen. 2015. Interpretation and Programming of the Reversible Functional Language RFUN. In Symposium on the Implementation and Application of Functional Programming Languages. https://doi.org/10.1145/2897336.2897345

Digital Library

[62]

Carlin Vieri. 1995. Pendulum–a reversible computer architecture.

[63]

Xiao Shaun Wang, Kartik Nayak, Chang Liu, T-H. Hubert Chan, Elaine Shi, Emil Stefanov, and Yan Huang. 2014. Oblivious Data Structures. In ACM SIGSAC Conference on Computer and Communications Security. https://doi.org/10.1145/2660267.2660314

Digital Library

[64]

Dave Wecker, Krysta M. Svore, and Krysta M. Svore. 2014. LIQUi|>: A Software Design Architecture and Domain-Specific Language for Quantum Computing. February.

[65]

Andrew C. Yao. 1982. Protocols for Secure Computations. In Symposium on Foundations of Computer Science. https://doi.org/10.5555/1382436.1382751

Digital Library

[66]

Tetsuo Yokoyama and Robert Glück. 2007. A Reversible Programming Language and Its Invertible Self-Interpreter. In ACM SIGPLAN Symposium on Partial Evaluation and Semantics-Based Program Manipulation. https://doi.org/10.1145/1244381.1244404

Digital Library

[67]

Charles Yuan and Michael Carbin. 2022. Tower: Data Structures in Quantum Superposition. https://doi.org/10.5281/zenodo.6819031

Digital Library

[68]

Samee Zahur and David Evans. 2013. Circuit Structures for Improving Efficiency of Security and Privacy Tools. In IEEE Symposium on Security and Privacy. https://doi.org/10.1109/SP.2013.40

Digital Library

Cited By

Puram VGeorge KThomas J(2024)Quantum Representation for Deterministic Push-Down AutomataElectronics10.3390/electronics1322453113:22(4531)Online publication date: 18-Nov-2024
https://doi.org/10.3390/electronics13224531
Venev HGehr TDimitrov DVechev M(2024)Modular Synthesis of Efficient Quantum UncomputationProceedings of the ACM on Programming Languages10.1145/36897858:OOPSLA2(2097-2124)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689785
Kang CLee JOh H(2024)Statistical Testing of Quantum Programs via Fixed-Point Amplitude AmplificationProceedings of the ACM on Programming Languages10.1145/36897168:OOPSLA2(140-164)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689716
Show More Cited By

Index Terms

Tower: data structures in Quantum superposition

Recommendations

Twist: sound reasoning for purity and entanglement in Quantum programs

Quantum programming languages enable developers to implement algorithms for quantum computers that promise computational breakthroughs in classically intractable tasks. Programming quantum computers requires awareness of entanglement, the phenomenon in ...
Ket Quantum Programming
Quantum programming languages (QPL) fill the gap between quantum mechanics and classical programming constructions, simplifying the development of quantum applications. However, most QPL addresses the inherent quantum programming problem, neglecting ...
Quantum Programming With Mixed States

In this paper we offer a programming approach to quantum computation using mixed states. Mixed-state quantum systems generalise standard (pure) quantum systems by allowing the state of the system to be a probabilistic distribution of pure states. We ...

Comments

Information & Contributors

Information

Published In

cover image Proceedings of the ACM on Programming Languages

Proceedings of the ACM on Programming Languages Volume 6, Issue OOPSLA2

October 2022

1932 pages

EISSN:2475-1421

DOI:10.1145/3554307

Editor:
Philip Wadler
University of Edinburgh, UK

Issue’s Table of Contents

Copyright © 2022 Owner/Author.

This work is licensed under a Creative Commons Attribution 4.0 International License.

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 31 October 2022

Published in PACMPL Volume 6, Issue OOPSLA2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Badges

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

7
Total Citations
View Citations
766
Total Downloads

Downloads (Last 12 months)412
Downloads (Last 6 weeks)55

Reflects downloads up to 01 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Puram VGeorge KThomas J(2024)Quantum Representation for Deterministic Push-Down AutomataElectronics10.3390/electronics1322453113:22(4531)Online publication date: 18-Nov-2024
https://doi.org/10.3390/electronics13224531
Venev HGehr TDimitrov DVechev M(2024)Modular Synthesis of Efficient Quantum UncomputationProceedings of the ACM on Programming Languages10.1145/36897858:OOPSLA2(2097-2124)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689785
Kang CLee JOh H(2024)Statistical Testing of Quantum Programs via Fixed-Point Amplitude AmplificationProceedings of the ACM on Programming Languages10.1145/36897168:OOPSLA2(140-164)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689716
Yuan CCarbin M(2024)The T-Complexity Costs of Error Correction for Control Flow in Quantum ComputationProceedings of the ACM on Programming Languages10.1145/36563978:PLDI(492-517)Online publication date: 20-Jun-2024
https://dl.acm.org/doi/10.1145/3656397
Yuan CVillanyi ACarbin M(2024)Quantum Control Machine: The Limits of Control Flow in Quantum ProgrammingProceedings of the ACM on Programming Languages10.1145/36498118:OOPSLA1(1-28)Online publication date: 29-Apr-2024
https://dl.acm.org/doi/10.1145/3649811
Ren JSui YCheng XFeng YZhao J(2024)Dynamic Transitive Closure-based Static Analysis through the Lens of Quantum SearchACM Transactions on Software Engineering and Methodology10.1145/364438933:5(1-29)Online publication date: 4-Jun-2024
https://dl.acm.org/doi/10.1145/3644389
Zhou LBarthe GStrub PLiu JYing M(2023)CoqQ: Foundational Verification of Quantum ProgramsProceedings of the ACM on Programming Languages10.1145/35712227:POPL(833-865)Online publication date: 11-Jan-2023
https://dl.acm.org/doi/10.1145/3571222

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

Figures

Tables

Media

View Issue’s Table of Contents