SciRate Comments

Comment on 1003.1150 by joe.renes

Paper Id: 1003.1150
Paper Title: Approximate Quantum Error Correction via Complementary Observables
Authors: Joseph M. Renes

joe.renes said: Another brief overview provided here.

Comment on 1003.0703 by joe.renes

Paper Id: 1003.0703
Paper Title: Duality of privacy amplification against quantum adversaries and data compression with quantum side information
Authors: Joseph M. Renes

joe.renes said: I wrote up a brief overview here.

Comment on 1002.3226 by Kris Krogh

Paper Id: 1002.3226
Paper Title: Making nonlocal reality compatible with relativity
Authors: H. Nikolic

Kris Krogh said: Rezn,

If, as you say, this paper is laughable, it should not take much of your time to point to some obvious flaw. Are you asking us lesser intellects to simply accept your word that this is so -- without your saying why, or who you are?

Comment on 1002.3226 by Rezn

Paper Id: 1002.3226
Paper Title: Making nonlocal reality compatible with relativity
Authors: H. Nikolic

Rezn said: Kris,

If you don't see what is wrong with this paper then I don't want to argue with you, this would be a waste of my time.

Comment on 1002.3226 by Kris Krogh

Paper Id: 1002.3226
Paper Title: Making nonlocal reality compatible with relativity
Authors: H. Nikolic

Kris Krogh said: Rezn,

I don't see anything wrong with this paper. Can you show there is? It's easy to post denigrating comments with no substance, while hiding behind anonymity. Do you think that contributes to scientific discourse?

Comment on 1002.3427 by aram

Paper Id: 1002.3427
Paper Title: Discretization of quantum pure states and local random unitary channel
Authors: Dong Pyo Chi, Kabgyun Jeong

aram said: The claims in this paper are wrong. An epsilon-randomizing map requires >=(1-eps)*d unitaries to work even for a single input, since anything within trace distance epsilon of the maximally mixed state has to have rank at least (1-eps)*d.

Comment on 1002.3226 by Rezn

Paper Id: 1002.3226
Paper Title: Making nonlocal reality compatible with relativity
Authors: H. Nikolic

Rezn said: This is the most entertaining and laughable paper I've read in many years ! Unfortunately for the author, I'm afraid that was not the purpose...

Comment on 1002.0846 by ari.mizel

Paper Id: 1002.0846
Paper Title: Fixed-gap adiabatic quantum computation
Authors: Ari Mizel

ari.mizel said: I'm uncomfortable saying that we've got something equivalent to a 2D lattice with polylog(L) range interactions. I think that the only way to resolve this is for us to work out an example together via skype or whatever.

Comment on 1002.0846 by matt.hastings

Paper Id: 1002.0846
Paper Title: Fixed-gap adiabatic quantum computation
Authors: Ari Mizel

matt.hastings said: Thanks for the detailed reply Ari. As I say, I really think this is great if it works. Regarding the correlation decay theorems, for any Hamiltonian H, and any operators A and B, we need to know 4 things to bound the correlation function of A and B. For definiteness, let me assume that the Hamiltonian acts on qubits on a two-dimensional lattice; I understand that you say that the lattice changes, but in fact this should just mean that the interactions change with L: as L increases, we increase the range of interactions beyond nearest neighbor.

We write the Hamiltonian H as a sum of term of sets of diameter at most R, where R is the interaction range. We need to know:
1)How does R depends on L. I believe that you are saying that R can scale as polylog(L). That is, the interactions can be embedded in this two-dimensional lattice with a range that is a polylog of L.

2)A bound, J, on the norm of these terms in the Hamiltonian. I believe that this is O(1) for your system.

3)The support of A and B. I believe that they both include at most polylog(L) sites, and that they are a distance L away from each other.

4)The spectral gap. The claim is that this is bounded below by an L-independent constant.

Given (1) and (2), we can bound the Lieb-Robinson velocity by polylog(L). Then, we can bound the correlation of A and B by ||A|| ||B|| polylog(L) exp(-L/polylog(L)), which goes to zero for large L.

So, one of those 4 claims I made about the Hamiltonian above is inconsistent with having correlation between the spins. I'd be very interested to see where I misunderstood your construction. Perhaps you require longer range interactions? If you prefer to talk over email, let me know.

If you prefer to discuss over email

Comment on 1002.0846 by ari.mizel

Paper Id: 1002.0846
Paper Title: Fixed-gap adiabatic quantum computation
Authors: Ari Mizel

ari.mizel said: I've never used Scirate before, and my experience is that it's very hard to communicate physics clearly via short text messages. But, I'm going to give it one try, and I'm happy to talk to interested folks in person or on the phone. (By the way, thanks for the comments and criticisms Matt and the mysterious anonymous RS. I welcome more comments from everybody, and even if I don't manage to respond on Scirate, I'll certainly take them into account as I try to improve the paper.)

1) On error detection/correction in GSQC:

The appendix of the paper describes a direct recipe for mapping any gate model circuit to a GSQC Hamiltonian and ground state. It discusses single qubit gates and two qubit gates. The recipe requires no creativity; you give me the circuit, I write down the Hamiltonian and ground state immediately.

There are two subtleties.

a) For fixed-gap GSQC, one should be sure that the gate model circuit includes a teleportation step after each gate. Just pretend that your gate model circuit is paranoid about leakage or something like that.

b) A subtlety arises whenever the gate model circuit includes measurements. The gate model measurement is mapped into a GSQC projection Hamiltonian; rather than simulating a measurement, GSQC performs a projection. The Bell measurement associated with teleportation becomes a Bell-state projection as described in the main text. Single qubit measurements become single qubit projections (as described in the error correction part of the appendix).

That's it. If you give me any gate model circuit, I just apply the recipe to turn it into a GSQC Hamiltonian and ground state.

The paper maintains that any GSQC Hamiltonian constructed using the recipe will have a gap that depends upon the value of Lambda in the projection Hamiltonians but not upon the number of qubits or number of gates. In order to maintain a finite gap, one needs a finite Lambda. But a finite Lambda leads to imperfect projections, and the GSQC ground state ends up simulating a _faulty_ gate model calculation.

This is when error correction comes in. To address these faulty gates, we can apply our regular GSQC recipe to translate a gate model circuit that _includes_ error correction. Including error correction won't effect the gap if we use the same Lambda, but now the errors in the ground state get cleaned up by the error correction in the circuit.

Perhaps the phrase "a GSQC version of quantum error detecting codes" was misleading. The phrase "GSQC version" was just meant to emphasize that no active measuring and pulsing is going on like in gate model error detection/correction. All we are doing is translating into GSQC, using the usual recipe, a gate model circuit that includes error detection/correction.

As an aside, it turns out that you can just use error detection rather than error correction in GSQC, as mentioned in the paper, but this is just an aside. If you'd prefer to use error correction, go ahead, but I suspect that you'll get a lower threshold. Pick your favorite code, from the surface code to the Steane code, use all the associated gate model fault tolerant circuits and simply translate them to GSQC using the recipe.

2) On correlations:

I believe that GSQC requires local operators alone whenever gate model error correction can be done with local gates alone (which is usually the case. See, for instance, D. Gottesman, "Fault-Tolerant Quantum Computation with Local Gates," J. Modern Optics 47, 333-345 (2000).) After all, we just translate our gate model circuit into GSQC using our recipe. Let me suggest the following resolution to your question, Matt, and please tell me if it sounds right to you.

To perform quantum error correction, you must encode quantum information into logical qubits. Under the encoding, the quantum information that used to be stored in physical operators is stored instead in logical operators that involve poly(log(L)) physical operators. This is true in the gate model, and therefore it's also true in GSQC.

Is there is any theorem limiting the correlations between a collection of poly(log(L)) physical operators near site 1 and a collection of poly(log(L)) physical operators near site L in the presence of local interactions between the physical operators? Before you answer, note that we use more levels of concatenated error correction as the number of steps L in the gate model calculation increases. Since the interactions in the "lattice" of GSQC qubits is determined by the gate model circuit, and the gate model circuit depends on L, the lattice changes with L. Bounding such correlations between logical operators as a function of L seems very tricky given that the number of physical operators in the logical operators, the definition of the logical operators in terms of the physical operators, and the lattice of interactions (which is determined by the gate model circuit) between logical operators all depend on L. The theorems in Hastings and Koma, Commun. Math. Phys. 265, 781 - 804 (2006), for instance, don't seem to apply to this case.

Does this sound reasonable?