# Why do quantum gates have to be reversible?

One possible reason I have come up with is that we are modeling quantum gates by unitary matrices. And since unitary operations are reversible we have to be able reverse the operation in the physical world as well. This is simply done by "remembering" which state has gone in and provide it as output if the reverse operation should be applied.

But what happens if we don't "remember" the input state? Are there certain operations that can be performed without the need of reversibility?

• All quantum operations are unitary - it follows directly from the Schroedinger equation. The exception is measurement, which is not a gate. Jul 27, 2016 at 23:19
• All closed quantum systems are reversible, the open ones are irreversible. It's exactly like in classical mechanics (with a few caveats). Jul 27, 2016 at 23:41
• If the only exception is a measurement, does that mean that an operation which doesn't copy the input automatically causes the collapse of the superposition?
– Heye
Jul 28, 2016 at 6:28
• @CuriousOne for classical mechanics I read here that "Given the initial conditions of a system, and a set of mathematical laws that model reality, we want to be able to tell what state the system will be in after a given time." which for me sounds like the other way around. The model we use has to be reversible to describe the physics. Also classical gates eg. AND are not reversible and that seems to be fine, the information that is lost is simply turned into heat.
– Heye
Jul 28, 2016 at 6:34
• I didn't say that a classical gate is reversible and I have no idea what your statement is supposed to mean in this context. Physics is more than computing, much more and quantum mechanics is so much more than quantum computing, it's just that some fields narrow the view when stared at for too long and some experiences in my life suggest that computation is one of them. Jul 28, 2016 at 6:45

Quantum gates have to be reversible because quantum mechanics is reversible (and even more specifically it is unitary). It's just an observed fact about the universe. (Even measurement can be modeled as a reversible unitary operation, inconvenient though that may be.)

Actually, classical computers also have to be reversible. We just happen to be able to sidestep the problem by throwing out accumulated garbage information as we go. Throwing out garbage information during quantum computations would also be possible, but because discarded garbage information counts as a measurement and measurement tends to break quantum algorithms... not so viable.

• How does this relate to reversible computing?
– user56903
Jul 28, 2016 at 8:47
• @DirkBruere Concepts from classical reversible computing are often applied to avoid creating garbage during quantum computations. That's how I see them relating. Jul 28, 2016 at 14:56
• If a computation procedure is regarded as a curve in the state space, then the full power of quantum computation can only be achieved by a curve that stays far enough from the classical state space. Keeping the QC reversible ensures to explore the full Hilbert space and therefore to achieve an efficient QC. Otherwise your curve will easily fall in/near the classical state space so that your QC may fail.
– XXDD
Jul 28, 2016 at 16:04
• @CraigGidney, what do you mean by the statement "Quantum Mechanics is reversible", does it mean the if the time is reversed in principal everything can go back in time? If that is the case, classical mechanics is also reversible I guess, which may imply that classical computations should also be reversible. what are your thoughts on this? Mar 17, 2021 at 5:39
• @dhirajsuvarna Yes, that's what I mean. Classical mechanics is also reversible; in fact both classical and quantum mechanics satisfy Liouville's theorem which is even stronger than just reversibility. That being said, there are some corner cases where it can get ambiguous due to singularities in the math. It's not really my expertise. Mar 17, 2021 at 6:24

There is conclusion in the Phd dissertation below:

"At the beginning of my research on designing reversible logic circuit, one of the first questions I asked myself was why we need reversible logic circuits? After completing initial research on the subject matter, I was convinced that the logical irreversibility of classical logic gates inadvertently causes the heat dissipation when new values are computed and old information is lost.

[...]

Concluding, there should be methods to synthesize Boolean circuits using only reversible gates."

Shah, Dipal, "Design of Regular Reversible Quantum Circuits" (2010). Dissertations and Theses. https://doi.org/10.15760/etd.129

Landauer [Landauer61] proved that binary logic circuits built using traditional irreversible gates inevitably lead to energy dissipation, regardless of the technology used to realize the gates.

Zhirnov et al. [Zhirnov03] showed that power dissipation in any future CMOS will lead to an impossible heat removal problem and thus the speeding-up of CMOS devices will be impossible at some point which will be reached before 2020.

Bennett [Bennett73] proved that for power not to be dissipated in a binary logic circuit, it is necessary that the circuit be built from the reversible gates. A gate (or circuit) is reversible if it is a one-to-one mapping between sets of input and output values. Thus all output vectors are just permutations of input vectors. (Such a circuit can be described by a binary permutation matrix [Nielsen00]).

Bennett's theorem suggests that every future (binary) technology will have to use some kind of reversible gates in order to reduce power dissipation. This is also true for multiple-valued reversible logics, which is an additional advantage because the multi-valuedness by itself demonstrates several potential advantages over binary logic.

These potentials of MV logic so far have not been taken advantage of since they bring no technological improvements when applied to existing technologies such as CMOS.

All these fundamental results of Landauer, Bennett and Zhirnov are technology-independent but practically applicable to future nano-technologies, especially to quantum technology as being the most advanced of all the nano-technologies. They are also applicable in quantum dots and DNA circuit realization technologies."

References:

[Landauer61] R. Landauer, ―Irreversibility and Heat Generation in the Computational Process‖, IBM Journal of Research and Development, 5, 1961, pp. 183-191.

[Zhirnov03] V. V. Zhirnov, R. K. Kavin, J. A. Hutchby, and G. I. Bourianoff, ―Limits to Binary Logic Switch Scaling – A Gedanken Model‖, Proc. of the IEEE, 91, no. 11, 2003, pp. 1934-1939.

[Bennett73] C. H. Bennett, ―Logical Reversibility of Computation‖, IBM Journal of Research and Development, 17, 1973, pp. 525-532.

[Nielsen00] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.