You might want to look at https://arxiv.org/pdf/1301.1995.pdf - for one thing it shows you can get larger depth for other noise models.
...(continued)Thank you for your profound work.
I have a few questions reading through the first few pages.(1) The notation for CP maps is a bit confusing; The dot product is defined to be the adjoint map on page 3, i.e., $A\cdot \rho := A \rho A^\dagger$. Does $E\cdot \rho$ mean $E(\rho)$ for $E\in CP(\math
...(continued)Based on TableS2 and Fig S5, the physical noise strength "p" that characterizes your device is approximately p=1%. Based on Fig1c, your simulations suggest that a noise strength of p=1% results in a postselected logical error rate of L~=10%. This value of L~=10% is 300 times worse than the L~=0.03%
...(continued)Hi Tianfeng. The doubled density operator (DDO) is different from PDM in several ways: (i) The PDM assigns one Hibert space for each event, the DDO assigns two local spaces for each event; (ii) The PDM is Hermitian but not positive semidefinite in general, the DDO is in general not Hermitian; (iii)
...(continued)Hi Zhian,
I'm a bit confused that your work seems to be the same idea as pseudo-density operator (PDM). Of course your math tricks are a little different but they have the same form. Maybe I miss someting... Could you tell me what is the iessentially difference between double density matrix and PD
Thank you for your thought-provoking work. May I take this opportunity to bring to your attention my preprint arxiv:2010.01167? There I make similar arguments in section V.D, although in my preprint I do not work out the technical details as thoroughly as you do.
...(continued)> Originally, quantum computers emerged as a promising platform to
solve certain computational problems that would otherwise be unfeasible to solve on classical computers.This is true, but let's not forget that quantum computers were also originally studied for the potential energy consumption
The following papers also consider fast-forwarding and reversal of Hamiltonian evolution in quantum systems, and may be of interest:
https://arxiv.org/abs/2205.01131
https://arxiv.org/abs/2205.01122
https://arxiv.org/abs/1903.10568
Dear Craig and Arthur, thanks for your answers, they are very helpful!
...(continued)Dear Ryan, thanks for the interest in our work!
Our comparison is between our query complexity upper bound and the best upper bound given in arXiv:2111.08152. For kappa=10^3 we have an effective constant upper bound of ~800 versus ~6023 given in arXiv:2111.08152 (Dominic told us the smaller numbe
I only regret that I have but one Scite to give.
...(continued)Hi Marius,
I agree with Craig -- hopefully the following different perspective also helps.
Sorry for the confusion, when I said logarithmic difference, I meant logarithmic in the dimension of the Hilbert space, i.e. linear in the number of qubits being used in the computation.
Let $n$ be the numb
...(continued)Interesting work! I want to note, however, that the bound you are comparing to from arXiv:2111.08152 (Costa et al.) was not meant to accurately give the constant factor except as an upper bound. When we've tested it numerically, we find that the actual constant factor is about five orders of magnitu
...(continued)Marius, you're thinking of the overhead in terms of gates instead of in terms of the area*time. Yes, error correction adds gate count overhead that grows linearly (or worse) with the number of qubits times the duration. But in terms of area*time, which is what actually determines the cost of buildin
...(continued)Dear Arthur, thanks for answering my question!
Yes, you understood it right. :)However, the statement about the logarithmic overhead is far from obvious to me.
I imagine an argument that goes like this: Let us say we have access to K parallel classical processes, independently of the input
...(continued)Hi, thank you for your comment.
Yes, the opportunity cost arguments are more general than just for QRAM, they also apply to any error corrected quantum computer (and more generally, any quantum computer with active gates). So, it many cases it would make more sense to compare an $n$ qubit quantum
...(continued)It seems to me that your arguments concerning error correction of circuit-based QRAM generalize to all quantum circuits. In particular, your Figure 1 seems to be a generic quantum circuit, not necessarily a QRAM.
I don't know much about quantum error correction, so I am wondering what does this
...(continued)Craig, yes, I've been thinking about it concretely since seeing the paper and posting my initial comment. However, in relation to your comment on the consequences, I should say that none of that depends on QRAM (or QROM): the polynomial code allows for a finite number of transversal Toffoli gates at
...(continued)Joe, you should really do a numerical estimate of how well this address encoding idea would work, with a specific chosen code. I'd be interested to see the result. My guess is you'll find the constant factors just don't work out. But if it does work out, it's a big deal.
Suppose your technique wo
...(continued)Hi Bibek,
Thanks for your helpful comments & sorry for the very slow reply. Let me reply to your two questions or points in reverse order.(1) first of all, we think very highly of your experimental demonstration of Grover's and it's an important milestone in demonstrating how it can be done and
...(continued)Joe: Yes, thanks for clarifying—you're right that the specific mechanism I described isn't actually an issue. On second thought, my statement was really just equivalent to saying that QRAM can easily propagate an error into (a superposition of) high-weight errors. I agree that with a proper code tha
...(continued)Sam: Quantum polynomial codes allow for only a bounded number of transversal Toffolis which depend on the distance. If you choose the code distance large enough you can make it enough to support enough Toffolis to be able to implement the routing circuit. Each time you apply a Toffoli, it reduces th
...(continued)Hi Connor, that's a really interesting observation -- I would love to incorporate it (with attribution of course) into a revision of our paper, if you would allow it. The output register need not be explicitly encoded (for QRACM you can use an arrangement of X gates, e.g.) but your argument would st
...(continued)Hi Joe, Sam,
I wanted to comment that this idea of encoding the address and only performing corrections between queries may have some additional subtleties—it’s not obvious to me that it’s quite so straightforward.In particular, it seems to me that with this approach you should also encode th
...(continued)I should add that I have not seen this approach in the literature, but it strikes me as a rather obvious one. As I think you realised in your second comment, you can make a Fredkin gate out of a Toffoli and two cnots (which are also transversal in polynomial codes), so you can implement the switchin
Making the code public would help putting these impressive numbers into context :-)
Yes, that’s what I mean. It gives you the best of both worlds. Similar to gate level efficiency but for any finite QRAM you have an error correction threshold.
On further thought, I see what you're saying: transversal gates are closed under composition, and QRAM circuits can be built from Toffoli + X + ancilla, therefore the two cases I distinguished above are the same case.
...(continued)Thank you for your comment. In our analysis of "circuit" QRAM, we assume error correction on the bucket-brigade, but we also consider "gate" QRAM, and different approaches to implement the bucket-brigade without fault tolerance on each node (specifically in Section VII).
We'll look into quantum p
...(continued)One quick comment on the treatment of bucket-brigade QRAM in the current paper and more generally in the literature: There seems to be an inbuilt assumption that full fault-tolerance must occur within the memory (there is a reference to the distance of surface code required and to magic state distil
...(continued)You can check the distance with a program to see that it is 2. If you construct a parity check matrix for the code that James is talking about given the stabilizers at the top of page 9 you find that the distance is 2.
For example, you can use the [compute_code_distance function in the python pack
The reason of code distance 3 of Fig. 4 is explained in the latest version of the manuscript entitled as "A genus-two code" arXiv:2211.12695 .
Just wanted drop a line saying that I love how the term "non-Abelion" was introduced in the abstract without definition and yet was immediately obvious what it was. Hallmark of an excellent (and apt!) name!
...(continued)Hi Koen, thanks for the question.
I would say the main purpose of the Hamiltonian is that it defines the eigenstates which we prepare. Since the Hamiltonian is a commuting projector Hamiltonian, one route would be to directly measure its terms and then correct for the outcomes. However, the proto
...(continued)Impressive results!
What I'm confused about: how does the specific Hamiltonian come into this simulation? Do I understand correctly that the applied gate sequence works well because we have a good analytical understanding of the simulated system (so that we can use a known gate sequence for state
Very interesting!
Very interesting!
Thanks!
I've added these results to [Metriq](https://metriq.info/Submission/561) in case anyone wants to see how it compares to other platforms
Awesome! Great work!
...(continued)Hi Victory, thanks for your question.
The two works have different objectives. The Google paper you mention creates an **Abelian topological order (TO)**, namely the Z_2 toric code. On top of this, they create and control lattice defects in this state. These are one-dimensional defects in the two
...(continued)Can someone explain to me, a non-expert in this field how this paper differs from https://scirate.com/arxiv/2210.10255 except that it's done on a different hardware platform? The abstract for this paper says
> In this work, we present the first unambiguous realization of non-Abelian TO and demonst
...(continued)Hi Jason,
Thank you for your comment.
I just realized that the type-II fusion used in the proposed scheme is a variant that measures $\\{ZZ, YX\\}$, which is shown in Fig. 10(a). It can be verified easily that this type of fusion is sufficient for the scheme in Fig. 4. Moreover, since $Z_1$ an
...(continued)Hi all,
I agree with Seok-Hyung that the Type II fusion scheme given in Figure 3 is actually measuring $X_1 Z_2$ and $Z_1 X_2$, and that failure leads to measurement of $X$ on one qubit and $Z$ on the other. This is problematic for the boosted fusion protocol presented here, because a $Z$ measur
...(continued)Very nice result, though I feel the way the statement of the main result in the abstract should be more precise. It seems the result applies to $f$ which has integer points as the extreme points of its solution set, or, better said, any $f: \mathbb{R}^n \rightarrow \mathbb{R} $ whose set of minimize
...(continued)Thank you, Craig. We intend to explore deeper circuits (hopefully in the ballpark of what you've suggested) in the future. Pushing the circuits to the brink of failure at the chosen error rate would, no doubt, be a useful experiment.
Also, our goal here was to establish the value of the DS-ZNE idea
...(continued)This paper should really consider circuit depths beyond 20 or 30. Table I should include depths 100, 1000, 10000, 100000 and 1000000. Push it until it breaks.
For scale, note that you want ~100 non-Clifford gates for the circuit to not be classically simulable. This is why you'd at least want to
...(continued)Hi,
First, congratulations on the publication of your paper! Your paper seems to be very appealing to researchers studying MBQC and photonic quantum computing like me.
I have one question regarding your work. In page 6, it is written that a type-II fusion failure leads to $Z$ measurements on both
"Another interesting feature is the intermediate connectivity between qubits, which
can be made variable by shuttling atoms during the computation" Is any neutral atom computing vendor seriously considering this _during computation_?