Reversible computing: Difference between revisions

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The [[exclusive or]] (XOR) gate is irreversible because its two inputs cannot be unambiguously reconstructed from its single output, or alternatively, because information erasure is not reversible. However, a reversible version of the XOR gate—the [[controlled NOT gate]] (CNOT)—can be defined by preserving one of the inputs as a 2nd output. The three-input variant of the CNOT gate is called the [[Toffoli gate]]. It preserves two of its inputs ''a,b'' and replaces the third ''c'' by <math>c\oplus (a\cdot b)</math>. With <math>c=0</math>, this gives the AND function, and with <math>a\cdot b=1</math> this gives the NOT function. Because AND and NOT together is a [[functional completeness|functionally complete]] set, the Toffoli gate is universal and can implement any [[Boolean function]] (if given enough initialized [[ancilla bit]]s).

Surveys of reversible circuits, their construction and optimization, as well as recent research challenges, are available.<ref>Rolf Drechsler, Robert Wille. From Truth Tables to Programming Languages: Progress in the Design of Reversible Circuits. International Symposium on Multiple-Valued Logic, 2011. http://www.informatik.uni-bremen.de/agra/doc/konf/11_ismvl_reversible_circuit_design_tutorial.pdf</ref><ref>{{cite journal |last1=Saeedi |first1=Mehdi |last2=Markov |first2=Igor L. |title=Synthesis and optimization of reversible circuits—a survey |journal=ACM Computing Surveys |date=1 February 2013 |volume=45 |issue=2 |pages=1–34 |doi=10.1145/2431211.2431220 |arxiv=1110.2574 |s2cid=6302811 }}</ref><ref>Rolf Drechsler and Robert Wille. Reversible Circuits: Recent Accomplishments and Future Challenges for an Emerging Technology. International Symposium on VLSI Design and Test, 2012. http://www.informatik.uni-bremen.de/agra/doc/konf/2012_vdat_reversible_circuits_accompl_chall.pdf</ref><ref>{{cite journal |last1=Cohen |first1=Eyal |last2=Dolev |first2=Shlomi |last3=Rosenblit |first3=Michael |title=All-optical design for inherently energy-conserving reversible gates and circuits |journal=Nature Communications |date=26 April 2016 |volume=7 |issue=1 |pages=11424 |doi=10.1038/ncomms11424 |pmid=27113510 |pmc=4853429 |bibcode=2016NatCo...711424C }}</ref><ref>{{Cite journal|last1 =Ang|first1 = Y. S.|last2 = Yang|first2 = S. A.|last3 = Zhang|first3 = C.|last4 = Ma|first4 = Z. S.|last5 = Ang|first5 = L. K.|date = 2017|title = Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate|journal = Physical Review B|volume = 96|issue = 24|pages = 245410|doi = 10.1103/PhysRevB.96.245410|arxiv = 1711.05906|bibcode = 2017PhRvB..96x5410A| s2cid=51933139 }}</ref>

Similarly, in the [[Turing machine]] model of computation, a reversible Turing machine is one whose transition function is invertible, so that each machine state has at most one predecessor.

[[:fr:Yves Lecerf|Yves Lecerf]] proposed a reversible Turing machine in a 1963 paper,<ref>Lecerf (Y.): [http://vadeker.net/corpus/reversible/lecerf.pdf Logique Mathématique : Machines de Turing réversibles.] Comptes rendus des séances de l'académie des sciences, 257: 2597–2600, 1963.</ref> but apparently unaware of Landauer's principle, did not pursue the subject further, devoting most of the rest of his career to ethnolinguistics. In 1973 [[Charles H. Bennett (physicist)|Charles H. Bennett]], at IBM Research, showed that a universal Turing machine could be made both logically and thermodynamically reversible,<ref>C. H. Bennett, "[http://www.dna.caltech.edu/courses/cs191/paperscs191/bennett1973.pdf Logical reversibility of computation]", IBM Journal of Research and Development, vol. 17, no. 6, pp. 525–532, 1973</ref> and therefore able in principle to perform an arbitrarily large number of computation steps per unit of physical energy dissipated, if operated sufficiently slowly. Thermodynamically reversible computers could perform useful computations at useful speed, while dissipating considerably less than ''[[kT (energy)|kT]]'' of energy per logical step. In 1982 [[Edward Fredkin]] and [[Tommaso Toffoli]] proposed the [[Billiard ball computer]], a mechanism using classical hard spheres to do reversible computations at finite speed with zero dissipation, but requiring perfect initial alignment of the balls' trajectories, and Bennett's review<ref>{{cite journal |last1=Bennett |first1=Charles H. |title=The thermodynamics of computation—a review |journal=International Journal of Theoretical Physics |date=December 1982 |volume=21 |issue=12 |pages=905–940 |doi=10.1007/BF02084158 |bibcode=1982IJTP...21..905B |s2cid=17471991 }}</ref> compared these "Brownian" and "ballistic" paradigms for reversible computation. Aside from the motivation of energy-efficient computation, reversible logic gates offered practical improvements of [[bit manipulation|bit-manipulation]] transforms in cryptography and computer graphics. Since the 1980s, reversible circuits have attracted interest as components of [[quantum algorithm]]s, and more recently in photonic and nano-computing technologies where some switching devices offer no [[signal gain]].

=== Reversible Turing Machines (RTMs) ===

Surveys of reversible circuits, their construction and optimization, as well as recent research challenges, are available.<ref>Rolf Drechsler, Robert Wille. From Truth Tables to Programming Languages: Progress in the Design of Reversible Circuits. International Symposium on Multiple-Valued Logic, 2011. http://www.informatik.uni-bremen.de/agra/doc/konf/11_ismvl_reversible_circuit_design_tutorial.pdf</ref><ref>{{cite journal |last1=Saeedi |first1=Mehdi |last2=Markov |first2=Igor L. |title=Synthesis and optimization of reversible circuits—a survey |journal=ACM Computing Surveys |date=1 February 2013 |volume=45 |issue=2 |pages=1–34 |doi=10.1145/2431211.2431220 |arxiv=1110.2574 |s2cid=6302811 }}</ref><ref>Rolf Drechsler and Robert Wille. Reversible Circuits: Recent Accomplishments and Future Challenges for an Emerging Technology. International Symposium on VLSI Design and Test, 2012. http://www.informatik.uni-bremen.de/agra/doc/konf/2012_vdat_reversible_circuits_accompl_chall.pdf</ref><ref>{{cite journal |last1=Cohen |first1=Eyal |last2=Dolev |first2=Shlomi |last3=Rosenblit |first3=Michael |title=All-optical design for inherently energy-conserving reversible gates and circuits |journal=Nature Communications |date=26 April 2016 |volume=7 |issue=1 |pages=11424 |doi=10.1038/ncomms11424 |pmid=27113510 |pmc=4853429 |bibcode=2016NatCo...711424C }}</ref><ref>{{Cite journal|last1 =Ang|first1 = Y. S.|last2 = Yang|first2 = S. A.|last3 = Zhang|first3 = C.|last4 = Ma|first4 = Z. S.|last5 = Ang|first5 = L. K.|date = 2017|title = Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate|journal = Physical Review B|volume = 96|issue = 24|pages = 245410|doi = 10.1103/PhysRevB.96.245410|arxiv = 1711.05906|bibcode = 2017PhRvB..96x5410A| s2cid=51933139 }}</ref>

The Reversible Turing Machine (RTM) is a foundational model in reversible computing. An RTM is defined as a Turing machine whose transition function is invertible, ensuring that each machine configuration (state and tape content) has at most one predecessor configuration. This guarantees backward determinism, allowing the computation history to be traced uniquely <ref>{{cite web |url=https://scispace.com/pdf/what-do-reversible-programs-compute-uwj26erp4f.pdf |title=What do reversible programs compute? |website=SciSpace |access-date=April 26, 2025}}</ref>.

Formal definitions of RTMs have evolved over the last decades. While early definitions focused on invertible transition functions, more general formulations allow for bounded head movement and cell modification per step. This generalization ensures that the set of RTMs is closed under composition (executing one RTM after another results in another RTM) and inversion (the inverse of an RTM is also an RTM), forming a group structure for reversible computations <ref>{{cite web |url=https://arxiv.org/pdf/1603.08715 |title=The group of reversible Turing machines |website=arXiv |access-date=April 26, 2025}}</ref>. This contrasts with some classical TM definitions where composition might not yield a machine of the same class <ref>{{cite web |url=https://arxiv.org/pdf/1603.08715 |title=The group of reversible Turing machines |website=arXiv |access-date=April 26, 2025}}</ref>. The dynamics of an RTM can be described by a global transition function that maps configurations based on a local rule <ref>{{cite web |url=https://arxiv.org/pdf/2404.07288 |title=TOPOLOGICAL ENTROPY OF TURING COMPLETE DYNAMICS |website=arXiv |access-date=April 26, 2025}}</ref>.

[[:fr:Yves Lecerf|Yves Lecerf]] proposed a reversible Turing machine in a 1963 paper,<ref>Lecerf (Y.): [http://vadeker.net/corpus/reversible/lecerf.pdf Logique Mathématique : Machines de Turing réversibles.] Comptes rendus des séances de l'académie des sciences, 257: 2597–2600, 1963.</ref> but apparently unaware of Landauer's principle, did not pursue the subject further, devoting most of the rest of his career to ethnolinguistics.

A landmark result by [[Charles H. Bennett]] in 1973 demonstrated that any standard Turing machine can be simulated by a reversible one <ref>C. H. Bennett, "[http://www.dna.caltech.edu/courses/cs191/paperscs191/bennett1973.pdf Logical reversibility of computation]", IBM Journal of Research and Development, vol. 17, no. 6, pp. 525–532, 1973</ref>. Bennett's construction involves augmenting the TM with an auxiliary "history tape". The simulation proceeds in three stages <ref>{{cite web |url=https://arxiv.org/html/2405.20842v1 |title=Compositional Reversible Computation |website=arXiv |access-date=April 26, 2025}}</ref>:

# '''Compute:''' The original TM's computation is simulated, and a record of every transition rule applied is written onto the history tape.

# '''Copy Output:''' The final result on the work tape is copied to a separate, initially blank output tape. This copy operation itself must be done reversibly (e.g., using CNOT gates).

# '''Uncompute:''' The simulation runs in reverse, using the history tape to undo each step of the forward computation. This process erases the work tape and the history tape, returning them to their initial blank state, leaving only the original input (preserved on its tape) and the final output on the output tape.

This construction proves that RTMs are computationally equivalent to standard TMs in terms of the functions they can compute, establishing that reversibility does not limit computational power in this regard <ref>{{cite web |url=https://arxiv.org/html/2405.20842v1 |title=Compositional Reversible Computation |website=arXiv |access-date=April 26, 2025}}</ref>. However, this standard simulation technique comes at a cost. The history tape can grow linearly with the computation time, leading to a potentially large space overhead, often expressed as <code>S'(n) = O(S(n)T(n))</code> where S and T are the space and time of the original computation <ref>C. H. Bennett, "[http://www.dna.caltech.edu/courses/cs191/paperscs191/bennett1973.pdf Logical reversibility of computation]", IBM Journal of Research and Development, vol. 17, no. 6, pp. 525–532, 1973</ref>. Furthermore, history-based approaches face challenges with local compositionality; combining two independently reversibilized computations using this method is not straightforward <ref>{{cite web |url=https://arxiv.org/html/2405.20842v1 |title=Compositional Reversible Computation |website=arXiv |access-date=April 26, 2025}}</ref>. This indicates that while theoretically powerful, Bennett's original construction is not necessarily the most practical or efficient way to achieve reversible computation, motivating the search for methods that avoid accumulating large amounts of "garbage" history <ref>{{cite web |url=https://arxiv.org/html/2405.20842v1 |title=Compositional Reversible Computation |website=arXiv |access-date=April 26, 2025}}</ref>.

RTMs compute precisely the set of injective (one-to-one) computable functions <ref>{{cite web |url=https://scispace.com/pdf/what-do-reversible-programs-compute-uwj26erp4f.pdf |title=What do reversible programs compute? |website=SciSpace |access-date=April 26, 2025}}</ref>. They are not strictly universal in the classical sense because they cannot directly compute non-injective functions (which inherently lose information). However, they possess a form of universality termed "RTM-universality" and are capable of self-interpretation <ref>{{cite web |url=https://scispace.com/pdf/what-do-reversible-programs-compute-uwj26erp4f.pdf |title=What do reversible programs compute? |website=SciSpace |access-date=April 26, 2025}}</ref>.

==Commercialization==