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It's clear from foundational research that qudits can provide an enhanced control of the Hilbert space over qubits, and I've encountered references that highlight improved robustness and noise tolerance in quantum protocols such as QKD when using higher-dimensional systems. Furthermore, the number of controlled-sign gates needed to implement a Toffoli gate can be reduced when using qudits (as illustrated with qutrits, in this paper). This could hint at a more efficient circuit design or even improved error correction capabilities with qudits. Yet, it's also understood that active error correction might not be a primary focus for NISQ devices due to the limited number of available qubits.

However, while the theoretical advantages are insightful, I'm curious about the potential implications and possible benefits when integrating qudits into NISQ devices. Additionally, given that institutions like Fermilab are actively researching or even building qudit-based hardware (as detailed in this Snowmass paper), there seems to be at least some practical interest in their potential in the NISQ era.

Decoherence and Noise

Are there any indications or studies suggesting that qudits might exhibit inherent advantages in terms of reduced decoherence or susceptibility to certain types of noise on NISQ devices, beyond the specific context of QKD?

Quantum Simulation

One of the promising applications of NISQ devices lies in their potential for quantum simulation. With qudits' richer state space, is there any evidence to suggest they might facilitate improved or more efficient quantum simulations of complex systems?

Hardware Considerations

I'm aware that creating and manipulating qudits tends to be more challenging than qubits. But is there any existing or emerging hardware architecture where qudits might have an experimental edge, or perhaps a unique synergy, when implemented on NISQ devices?

While a previous discussion on Physics Stack Exchange touched upon some benefits of qudits, I'm particularly interested in their integration and potential benefits within the NISQ regime, given the practical challenges and opportunities it presents.

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    $\begingroup$ I've always thought that qudits of dimension $\gt 2$ to be of no significant theoretical benefit than that of qubits - that is in one of Aaronson's gotchya questions in his exam that he gave ChatGPT. The Physics SE question you link to is informative though. Nonetheless have you seen this question? $\endgroup$ Sep 21, 2023 at 16:53
  • $\begingroup$ I had not seen that question, thanks for the pointer. I suppose what sparked the question was a thought along the lines of; given the complexities associated with implementing qudits and their not-so-obvious (potential?) advantages, it's interesting that they're a subject of somewhat active exploration in both theory and hardware, especially in the context of NISQ devices. $\endgroup$
    – banercat
    Sep 21, 2023 at 18:23
  • $\begingroup$ Hello! I waited until the end of the bounty period to write my answer. So that the bounty doesn't completely go to waste, I wrote an answer even though it is probably not the answer that you were hoping to see. However, I did spend about 4.5 hours on it (wow, how did time fly!), and I'd be very willing to discuss any part of it with you if you are open to discussion! $\endgroup$ Sep 30, 2023 at 18:56

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Preamble

Versions of this question arise over and over again, probably because most of us are accustomed to thinking that "more is better" rather than "less is more", for example you already mentioned finding this thread from 2014 and Mark already pointed out this question from 2022. Your question is essentially the same: "are there advantages of using qudits rather than qubits?" except that your question comes from a more well-informed starting point (thanks in part to your question being asked later, and also being able to benefit from seeing the answers to the 2014 question and 9 years worth of other information that's been available since then, including the emergence of the Snowmass whitepaper), and focuses on NISQ (a term that was first used in 2018 and therefore didn't exist in 2014).

On QCSE alone, the question has been asked more times:

But I will try to focus more on NISQ and the other newer aspects of your question, such as the Snowmass paper.

Robustness against noise and NISQ devices

"the number of controlled-sign gates needed to implement a Toffoli gate can be reduced when using qudits (as illustrated with qutrits, in this paper). This could hint at a more efficient circuit design or even improved error correction capabilities with qudits. it's also understood that active error correction might not be a primary focus for NISQ devices due to the limited number of available qubits."

The reason why error correction is not practical for NISQ devices is because you would need millions of qubits for any meaningful application of error correction to a meaningful problem. Preskill's paper that introduced the term NISQ discussed quantum computers with 50 to a few hundred qubits, which is orders of magnitude smaller than what is needed for meaningful error correction on a meaninful problem. You might wonder if "maybe by using qudits the number of qubits/qudits required for practically meaningful error correction will go down enough for it to be relevant for NISQ?" I wouldn't count on it. The paper you mentioned in the above quote, talks about needing 3 two-qubit gates instead of 5 for implementing the Toffoli gate. That is not the several orders of magnitude of reduction in gate count needed for needing meaningfully less error correction, and similarly, no qudit-based error correcting schemes use several orders of magnitude fewer resources nor are they several orders of magnitude more robust to noise: otherwise everyone would be working on qudit-based quantum computing much more! This brings me to the next section of this answer:

Status of making qudit-based NISQ hardware

"Additionally, given that institutions like Fermilab are actively researching or even building qudit-based hardware (as detailed in this Snowmass paper), there seems to be at least some practical interest in their potential in the NISQ era."

Fermilab is an enormous institution that employs more than 2000 people at any time, and I have a pet peeve against failures to separate institutions from researchers. I see people in the quantum computing community say things like "Google did this" and "Microsoft said that" when in fact the people running Google and Microsoft do not have an undergrad-level understanding of how quantum computing works. Research groups do not ask their dean (or director of research) for permission every time they publish a paper, or work on something, so the "institution" doesn't know and can't endorse all research that goes on in the institution (even if they could monitor publications that use their name in their affiliation, the people in charge cannot have enough scientific expertise in every single sub-field to know what research is nonsense and what is high-quality).

There's dozens of problems that I see with the Snowmass 2021 (white)paper. It would take hours to describe all of these in detail, but hopefully it's enough to say that none of the ~40 co-authors are experts (by any stretch of the imagination!) in quantum hardware, which might be considered ironic since the title of the paper is "Quantum hardware for HEP algorithms and sensing." I only recognize two names in that entire list, and they're both two of the only people not from FermiLab or nearby institutions (these are Norm Tubman, a quantum chemist, and Davide Venturelli who specializes in AQC, which is not the topic of the paper), I do not recognize any other names in that author list as people who I would trust on the topic of quantum computing without doing more detailed "due diligence". The two corresponding authors have grad-student levels of citations on Google Scholar (or less, considering that their field is high-energy physics), and next-to-zero experience in quantum computing.

The paper has only 17 citations on Google Scholar, mainly from people who are co-authors of the paper. I wouldn't use this paper as evidence that experts are seriously considering making large-scale NISQ hardware using qudits. Remember that the "I" in NISQ is "intermediate", and that Preskill's original paper talks about quantum computers with 50 to a few hundred qubits. Therefore, there is a huge, huge, gap between some students and postdocs putting together a whitepaper about some experiments that they're considering to do with a handful of qudits, versus the construction of actual NISQ devices that are built as the result of tens of millions of dollars of investment (cf. Fermilab's entire annual budget being only a few hundred million dollars). The institutions that are really putting tens-to-hundreds of millions of dollars into making NISQ devices (Google, IBM, D-Wave, etc.) are not even thinking about using qudits.

Quantum simulations

"One of the promising applications of NISQ devices lies in their potential for quantum simulation. With qudits' richer state space, is there any evidence to suggest they might facilitate improved or more efficient quantum simulations of complex systems?"

First of all, I want to comment on the word choice here. Instead of saying "one of the most promising" you have said "one of the promising". The word "promising" means "showing signs of future success" or "likely to succeed or to yield good results" or "shows signs that it is going to be successful or enjoyable" based on the first three results on Google (the Oxford, Merriam-Webster, and Cambridge dictionaries in that order).

Quantum simulation is my field, and I want to make it very clear that for quantum computers to outperform what we can already do on classical computers for simulating quantum systems, there is no "sign of future success" and it is not "likely to succeed" anymore than it is "likely to fail" unless we go far, far beyond the "NISQ" regime. We will need millions of qubits to do on a full-fledged fault-tolerant quantum computer what we could basically do in 2018 with modest resources on a classical computer for the same problem. That is again several orders of magnitude beyond the number of qubits in the largest quantum computers that will ever be considered "NISQ" devices. I do appreciate that you cited a Nature paper from 2022 about quantum simulations (in fact one of my former students from 2015 is a co-author on it!), but it says "Hybrid digital–analogue devices that exist today already promise substantial flexibility in near-term applications" and promising "flexibility" does not mean promising meaningful applications.

Summary

The question: "are there advantages of using qudits over qubits?" has been asked every year by many people since the term "qudit" started being used in the 90s and likely far earlier than that when people were studying the power of quantum information in the 1980s.

When we learned about qutrits and qudits in "Introduction to Quantum Information Processing" (a 2008 course taught by Michele Mosca), we were told that these were not likely to be used in practical quantum computer implementations, and as you can see from the 2014, 2018, 2020 and 2022 Stack Exchange threads linked in this answer, there's a handful of edge cases in which qudits might provide some advantage (e.g. implementing Toffoli gates or certain aspects of QKD, but in both cases the disadvantages from an implementation perspective probably outweigh any advantage, otherwise more people would be working on qudit-implementations). This is despite trillions of dollars and tens of thousands of very smart full-time researchers being involved in quantum computing research over at least the last 40 years.

Quantum error correction is very likely to be a necessary procedure for a quantum computer to be able to do something like factoring numbers with fewer resources than a classical computer, and it turns problems that require 1000s of un-corrected qubits into problems that require 1,000,000s of error-corrected qubits. NISQ machines by "definition" do not have millions of qubits, and the small handful of advantages of using qudits for very specific applications (e.g. QKD and Toffoli gates) that we have managed to discover despite so much research going into this field, are not going to bridge the chasm from NISQ to million-qubit fault-tolerant machines, without the emergence of a major breakthrough.

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    $\begingroup$ +1; you're never one to shy away from offering your honest assessment. $\endgroup$ Sep 30, 2023 at 20:39

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