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Introduction to the ZX-Calculus

pennylane.ai

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Re: Introduction to the ZX-Calculus

#11
post #4

Earlier quoted context omitted.

> Why is it useful in a world where no quantum computer physically exists? Turing's and Church's investigations into the various theoretical computing calculi are enormously useful, even though both were done well before the existence of anything we'd recognize as a modern general-purpose computer.

Yeah but you at least had analog forms of things that represented mechanical work through computation (looms, relays, etc...). Nothing exists at all in terms of physical worth in relation to quantum computing and nothing is even close to being _real_ besides a model of a theoretical physical computation that has to be checked by classical means. So I don't really get why there exists a language for it.

But we do have a ton of "analog quantum computers": basically any experiment in quantum materials or quantum chemistry is something we can not simulate and something that informs us about analogs of such systems. No different from using single-purpose analog electronics experiments to predict as an analog to trajectories of spacecrafts or ocean currents or bombs or planes.

In both the classical and the quantum case, building a digital computer (a computer that has the potential to scale to large system sizes, unlike analog computers) is what is difficult.

Babbage and Lovelace imagined a digital computer in 1830s. It took 120 years until there was an actual digital computer in existence. It took the creation of an actual digital computer around 1950 for people to stop saying "digital computers are impossible because of noise". See "Probabilistic Logics and the Synthesis of Reliable Organisms From Unreliable Component" by von Neumann, to see how incredibly surprised theorists were at the time.

Quantum digital computers were conceptualized in the late 1980s, the existence of quantum error correction (the equivalent to von Neumann's paper) was shown in late 1990s. Compared to how long it took for conception-to-realization in the case of classical computers (120 years), the quantum side of things is not that terrible (30 years and counting).

Re: Introduction to the ZX-Calculus

#13
There are many quantum computers around, IBM and Google's supper conducting ones, our ion traps, optical ones by e.g. Quandela, and they are all heavily used. There exist compilers for them and they heavily employ ZX-calculus for optimisation. ZX-calculus is also the main language for optical quantum computing.

Within the year we expect them to do a task that is impossible on classical hardware within a realistic timeframe.

Btw ZX-calculus as part of categorical quantum mechanics is an alternative formalism for QM, independent of quantum computing. It has been proven to be complete for Hilbert space and linear maps.

Re: Introduction to the ZX-Calculus

#14

These are basically interaction nets with different rules than those used for lambda calculus. > Interaction nets are a graphical model of computation devised by Yves Lafont in 1990 https://en.wikipedia.org/wiki/Interaction_nets

Not really, interaction nets and linear logic don't have the compact closed categorical semantics we use here. Ros Duncan's PhD is about the differences. (and that was long before the extra structures of ZX had started)

Re: Introduction to the ZX-Calculus

#15

Earlier quoted context omitted.

Yeah but you at least had analog forms of things that represented mechanical work through computation (looms, relays, etc...). Nothing exists at all in terms of physical worth in relation to quantum computing and nothing is even close to being _real_ besides a model of a theoretical physical computation that has to be checked by classical means. So I don't really get why there exists a language for it.

Let me give you one example to illustrate the general principle. Any quantum computer we build will be too noisy to do any useful computation as is. To counter this, it is necessary to employ (quantum) error-correction at the software level to reduce the noise sufficiently to do useful computation. While a lot of quantum error-correction algorithms have been developed using traditional techniques, they are still not…

That's very true indeed, and for photonic QCs ZX has become vital for all software.
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