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All of these results are for Shor's algorithms. More specifically, the results are for various derivatives of Shor's original algorithms. These derivatives are specialized for problems that are relevant in cryptography. They provide various constant factor improvements with respect to the number of

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How about using shor's algorithm?

Regarding your first point, I have had exactly the same thoughts. It seems that the "blocking strategy" cannot be applied to the Raussendorf et. al robustness measure and I would be interested if there is a different way of extending low-dimensional solutions. Simulating a quantum circuit with $n$ m

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Opp.... I wrote a reply yesterday but as a separate comment (see below). My main question is essentially the same as Patrick's above: these LPs are typically only tractable up to 5 qubits so you need some suboptimal method to perform larger simulations.

The last point confuses me somewhat: if I understand correctly your algorithm only supports potentially non-classical input states followed by Pauli measurements. How can this be used to simulate an arbitrary quantum circuit? With an MBQC approach, the initial state would have to scale with the size

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Thanks Robert for confirming those technical points.

I hope it is OK if I ask a few more questions here.

Regarding the classical simulation algorithm and the consequences of the non-multiplicativity of your robustness monotone. I am wondering what you are able to say about the negativity of *k

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Campbell writes: ``For odd dimension, we have that if ρ and σ both have positive Wigner functions, then ρ⊗σ will also have a positive Wigner function. I would expect any qubit counterpart of the Veitch el al result to also have this property."

=> We confirm that our phase point operators and Wign

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I agree that the construction lacks this 'nice' property under composition, but I would argue that this was expected... and is a feature rather than a bug. One of the main lessons I take from Angela Karanjai's paper https://scirate.com/arxiv/1802.07744 is that the phase space on which you define an

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I thought it worth flagging up that there remain significant differences between the qubit Wigner functions introduced here and those encountered in odd dimension. My impression from reading the abstract was that all these differences had been removed, but this is not the case. There are lots of

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Podolsky is like banana: One knows how to spell it, but one does not know how to stop.

Thanks! We'll add a citation. :)

FYI, there is a minor (but amusing) typo in the first paragraph: "Einstein-Pododolsky-Rosen" should be just "Einstein-Podolsky-Rosen."

Note for those working in quantum algorithms: the `random access quantum memory` (RAQM) implemented in this manuscript deals with addressing data in qubits, where the qubit index is selected by classical control. This should not be confused the `quantum random access memory` (QRAM), where a superpos

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There is a more recent article by Appleby, following up on the one that is Ref. 1 in your paper, which may be pertinent: arXiv:1602.09002.

Hi,

I have looked at the paper that you refer to: leakage, that is, the qubits are not in |0> or |1> but in some other state (say, the excited state |2>) are indeed very concrete issues that one has to deal with, as it makes getting error information unreliable ('silent stabilizer'). It then part

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Hi Mikhail,

In the paper you linked, the author addresses the possibility that the qubits may "leak." Qubit is often encoded in two lowest energy eigenstates of some physical system(e.g., transmon, ion, photon, quantum dot, etc.) but of course, the actual system can have higher energy states. If

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See the recent article https://arxiv.org/pdf/1702.07688.pdf. The author makes a professional analysis of errors and error correction (of which I am not capable) and his conclusions are worth considering

Thanks for explanation. The rotating frame will not help if all the energy differences are not exactly equal (equidistant energy spectrum). This very specific requirement will never be satisfied in any real system, where the time evolution will be chaotic. This is the general case.

See the recent

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If someone is interested, the earlier version is still available [here][1]

The pith stays the same but the new version strongly emphasises that we are faced with two alternatives for

> the relation between a physical theory T and the system of logic supporting inferences on T-propositions

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Sorry for not being clear: x would be an N-bit string and \sum_x \Psi(x) |x> is the state of the quantum computer after having applied some sequence of gates. In making my comment I thought about a physics analogy with x being the position of a particle, of course one can discrete this space and rep

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Wow, thanks! Those are some impressive typo-spotting skills! :)

> This might imply that multi-agent paradoxes are linked to the notion of contextuality

That is exactly the point of [Sebastian Fortin and Olimpia Lombardi][1] (but with the scope limited to the original F-R argument):

> the contradiction resulting from the F-R argument is inferred by making c

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I loved the remark that "Heisenberg in his later years expresses his appreciation of the mathematical side by claiming that it was his own work in the first place."

Subtle typo on page 3: "but new very little about matrices". And another on page 16: "change slowly in phases space". Also, "propert

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At section 5.1, second paragraph, it is mentioned that Optimal fitting flat must contain the mean of the point set. Is there a simple proof for this fact?

Barbara, I don't quite understand: what Is \Psi(x) and what is x, as applied to a quantum system with N qubits? Also, since you have mentioned **energy**, are oscillations of the quantum amplitudes in time with frequencies delta E/h taken into account?

Dear Mikhail,

the reason that quantum computing theorists have a different perspective has two aspects. First, one has to be careful in stating what we need to control. If we move from quantum computing to classical stochastic computing (from a Schrödinger equation to a diffusion equation), we re

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Hi Mikhail,

You are basically saying that the exact overlap between the ideal and non-ideal state decays exponentially in N. I agree with you on this but you are merely attacking a straw man here. Exponentially small overlap does not imply that you cannot do fault-tolerant quantum computation. No

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Dear Barbara,
I agree, and this is also my point: " **it cannot be said that the theory of quantum error correction and fault-tolerance guarantees that robust quantum computers will ever be built**.
As well as that **this is given by "the physics"**.

In particular the **quantum physics** which

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Dear Elizabeth, I fully agree with what you are saying.

Except that faulty gates are not the only source of errors. There are also unwanted interactions within the system of qubits, as well as between this system and the environment, and also because the initial state cannot be exactly |000...>,

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> It's not a complete analogy to the authentication game that Sandu has
> the parties play

Here's a closer classical analogy: Alice prepares an ensemble of classical bits. Each bit is random (determined by a coin toss) and, for some fixed partitioning of the bits into pairs, Alice writes down i

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Victory Omole, The Nobel prize was given "for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems."

Here "Individual quantum systems*" means individual atoms and individual photons, definitely NOT many-body quantum systems, as you apparently

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My comment was not so much in reply to Dyakonov's paper (i.e. a corroboration or refutation of his points) but rather a general contribution to the discussion.

I think we sometimes tend to respond to critics by rewriting their criticism into something more reasonable. Are correlations in the power spectrum of noise something that may be influenced by our gates and may await more experimental data and more theoretical insight? Yes, probably. Is this what

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Naturally, it cannot be said that the theory of quantum error correction and fault-tolerance guarantees that robust quantum computers will ever be built. One particular challenge is the ability to turn gates off and on: in almost all implementations this works by meeting resonance conditions which a

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Even without studying quantum information theory, there is an immediate way to see that quantum computation is more robust than classical analog computation. This argument is based on linearity. Consider a sequence of unitary operators $U_1 ,...,U_T$. For concreteness, each $U_i$ acts on two qub

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>My previous experience tells me that it is never possible to control a many-body quantum (or even classical) system on a microscopic level.

If your previous experience tells you this, then i encourage you to gain new experience because experimentalists have been controlling many-body quantum sys

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Dear Steve,

Thank you very much for your invitation and your friendly attitude! However it seems to me that we live in parallel worlds: the world of quantum information theory, which has a rather poor experimental support, and the world of physics, where the relationship between theory and experi

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Michel, you should come to a quantum information conference!

You may feel like your critiques have fallen on deaf ears, but that is not true. If we don't pay explicit attention it is because, as Aram and others have pointed out above, we feel that we have already addressed your concerns. I know

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OK, I have not studied these papers yet.
Let us return to this subject in 10 years when hopefully someone will manage to factor 15 by Shor (full Shor's algorithm, please, no cheating with the "compiled version".

This is awesome!

One more thing. The fidelity between the actual and ideal states will be exponentially small but this is not a barrier to FTQC. The same occurs with classical memory.

The classical analogue of the fidelity expression you quote is: "What are the odds that all the spins in your hard drive are poi

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That is some evidence it can be efficiently scaled up. You would have said the same thing if those experiments were not done at all; but the number and quality of qubits experimentalists encode fault tolerantly seem to be increasing with time and effort and there is no evidence it's going to stop.

This objection is a bit of a moving target. First there is the proposal that there is some reason that quantum computers cannot work _in principle_. Then when people respond by saying that this reason is (a) vague and unspecified and (b) contrary to accepted principles of locality in physics, the

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Of course there will be some undesired extra terms in the Hamiltonian, which we can call V. However, V is not a completely arbitrary $2^n$-dimensional matrix. We can expand it as
$$ V = V_0 + V_1 + V_2 + \cdots + V_n .$$
where $V_j$ contains only tensor products of $j$ non-identity Pauli matrice

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So far there is no evidence that this can be efficiently scaled up

The good news is that some basic principles of quantum error-correction have already been demonstrated on existing quantum computing hardware, e.g. see https://arxiv.org/pdf/1806.02359.pdf and references therein.

I would like to see an experimental realization of these ideas with a large enough number of qubits, say 10 -20

In quantum error correction we deal with this problem by actively performing error-detecting measurement and correcting the error. The measurement will be faulty in reality, but this can be dealt with by repeating the measurement; see Section IV of https://arxiv.org/abs/quant-ph/0110143 for example.

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No doubt that the Schroedinger equation describes the evolution of the wavefunction, hence of its 2^N amplitudes. The problem is that in reality the Hamiltonian, and hence the evolution of the quantum amplitudes, can never be exactly what you would like it to be: H=H_0 +V. Also, the initial conditio

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The wavevector is determined exactly as the solution to the equation
$$ \frac{d}{dt}|\psi(t)\rangle = -iH(t) |\psi(t)\rangle.$$
While the state has $2^n$ dimensions, those do not show up as experimentalist-tunable parameters in this equation. How many tunable parameters are there? Well, those s

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