Comments on: Oldschool, Contradiction, and a Strawman

By: Jim Harrington

Jim Harrington — Tue, 04 Apr 2006 17:57:05 +0000

I have only briefly looked through Alicki and Horedecki’s paper so far, so I will need to digest it more before commenting on their work. I would like to point out, however, some work towards a “quantum hard drive” from a different perspective than the Hamiltonian approach.

In Chapter 5 of my thesis, I wrote about joint work with Charlene Ahn on constructing a purely local scheme for maintaining quantum memory in a two-dimensional array. We considered placing classical controllers that regularly perform measurements on nearby qubits (and which are allowed to fail at some given rate).

Any such finite system will eventually degrade (since there is a nonzero probability of a fatally large cluster of errors occuring), but we showed that provided the individual qubit and controller error rates are below certain values, then the relaxation time for the system grows exponentially with the size of the system. Hence, quantum memory can in principle be locally maintained in two dimensions for very long times with only moderate overhead.

In connection with John Preskill’s comments, we tried to demonstrate that two-dimensional quantum memory is analagous to one-dimensional classical memory.

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By: Greg Kuperberg

Greg Kuperberg — Fri, 31 Mar 2006 05:31:18 +0000

Is it unfair of me to have a knee-jerk reaction against papers that claim to prove that quantum computation is impossible? It is yet conceivable someone will think of some completely new reason that they are impossible. Even so, papers like this one seem just plain inadequate.

Is it a matter of style? There was a Bugs Bunny cartoon with Christopher Columbus arguing back and forth with the King of Spain: “Flat!” “Round!” “Flat!” “Round!” I suppose that debates with outright mutual contradictions are useful in politics, or even psychology. I am not used to them in mathematics or CS theory. Are they more accepted in theoretical physics?

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By: John Preskill

John Preskill — Thu, 30 Mar 2006 20:49:15 +0000

To summarize: The difference between the two-dimensional and four-dimensional Kitaev models is much like the difference between the one-dimensional and two-dimensional Ising models. The 1D Ising model, or the 2D Kitaev model, gets disordered at any finite temperature, because the energy cost of creating a point defect is a constant independent of the system size. But to disorder the 2D Ising model, or the 4D Kitaev model, a string loop with length comparable to the linear size of the system must be created by thermal fluctuations, which is exponentially unlikely at sufficiently low nonzero temperature.

Quantum information *is* harder to stabilize than classical information, but it’s not impossible. In this setting, the price of quantum memory is a higher required spatial dimensionality

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By: John Preskill

John Preskill — Thu, 30 Mar 2006 20:29:43 +0000

I think that a counterexample to the claim of Alicki and Horodecki is provided by the four-dimensional version of Kitaev’s toric code (discussed in http://arxiv.org/abs/quant-ph/0110143), which protects quantum information in a manner very analogous to how a two-dimensional ferromagnet protects classical information.

As in Kitaev’s two-dimensional model, the (time-independent) Hamiltonian is a sum of commuting terms, but in this case the qubits live on 2-cells (i.e. plaquettes) in a four-dimensional cubic lattice. There is a term associated wtih each edge (the tensor product of six X’s acting on all of the 2-cells that meet at that edge) and a term associated with each cube (the tensor product of the six Z’s acting on all of the 2-cells contained in that cube). If the lattice covers a 4-torus, there are 6 encoded qubits, one associated with each of the cohomologically nontrivial 2-surfaces that wrap around the torus.

Consider how Z errors are controlled in this model (the control of X errors works the same way, with the lattice and the dual lattice interchanged). If Z errors occur on a connected “droplet” of 2-cells, each of the edges on the boundary of the droplet is excited. So the energy is proportional to the length of the boundary of the droplet.

Now we need to consider the competition between the energy and the entropy of droplets. But (just as in the corresponding analysis done many years ago for the two-dimensional Ising model by Peierls), the right way to think about it is to consider the dynamics of the loops of “string” that bound the droplets. The number of string loops of a specified length grows exponentially with length. Therefore, if the temperature is low enough, long loops of string are exponentially rare, due to the Boltzmann suppression.

For the memory to fail, a thermal fluctuation must induce a loop to grow so large that it sweeps around the lattice. The time scale for such a thermal fluctuation grows exponentially with the size of the lattice.

Admittedly, this memory requires four spatial dimensions, so it is not realistic. But spatial dimensionality did not seem be considered in the Alicki-Horodecki argument.

By the way, in http://arxiv.org/abs/quant-ph/0110143 we considered how to realize this idea by simulating the thermal bath using quantum gates and refreshable ancillas. Charlene Ahn and I have done a much more detailed analysis of this (reported at QIP 2005, see http://www.theory.caltech.edu/~preskill/talks/QIP05_preskill.pdf, but unfortunately not yet written up for publication). But that is not what I am talking about here. Rather, I just want to consider the system described by the four-dimensional version of Kitaev’s Hamilonian in thermal equilibrium.
This is actually easier to analyze.

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By: R.R.Tucci

R.R.Tucci — Thu, 30 Mar 2006 18:50:23 +0000

Okay, let’s get those gentlemen from Poland to respond. To provoke a response, I have composed an especially vitriolic Haiku. As Bertie Wooster would say, I mean it to sting.

To dear Alicki
and also Horodecki
your paper icky

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