sflammia

October 18, 2011

Quantum Information in Quantum Many-Body Physics — Live Blogging Day 1

For the rest of the week I’m at the very interesting conference in Montreal on quantum information and quantum many-body physics. I’ll be live blogging the event. Enjoy!

Quantum Information in Quantum Many-Body Physics

Alioscia Hamma, Entanglement in Physical States, arXiv:1109.4391. Joint work with P. Zanardi and S. Santhra.

Some motivation: studying the foundations of statistical mechanics using quantum information. Want to investigate the role of entanglement in stat mech. Goal of this talk is to revise some of the recent work by Winter et al. which studies stat mech using Haar-random states and instead consider some subset of just the “physical” states, e.g. states which are subject to a local Hamiltonian evolution. Thermalization in a closed quantum system is a vexing philosophical problem. Let’s focus on entanglement; now if we look locally at the system we see a mixed state even if the global state is pure. Can we explain thermodynamics via this entanglement entropy? For a Haar-random pure state, the reduced density operator is almost maximally mixed. Now let’s define a global Hamiltonian with a weak coupling to a bath. Then the reduced state is close to the Gibbs state. However, just by a simple counting argument (Poulin, Quarry, Somma, Verstraete 2011) most physical states (i.e. ones subject to evolution with respect to a local Hamiltonian) are far from typical Haar- random states. What Alioscia et al. want to do is start from completely separable pure states and evolve with respect to a local Hamiltonian for a polynomial amount of time. Now study the entanglement in such a system. We start with a probability measure $p$ on the set of subsets of qudits. Now sample a subset and evolve for an infinitesimal amount of time. Now take a new iid sample and evolve again. Look at the reduced density operator and compute the purity, $mathrm{Tr}(rho^2)$ . Now let’s compute the mean and variance of the purity. To do this, use a superoperator formalism. Use a so-called random edge model. If you put the system on a lattice, you find that the average purity decays with an area law scaling. Next, consider a linear chain. A chain of length L cut into two pieces A and B. Instead of choosing edges from a random edge model, use a random local unitary on each site chosen in a random order (it turns out this order doesn’t matter much). Now repeatedly apply these random local unitaries to each site on the chain. The purity decays exponentially in the number of time steps. The average Rényi 2-entropy is nearly $k log d - log 2$ where $k$ is the number time steps and $d$ is the local site dimension. If we look at the $ktoinfty$ limit we get the maximally mixed state. They were not able to compute the gap of this superoperator yet, so we don’t know the rate of convergence. When the number of time steps is constant, we have an area law. Working on 2D and 3D lattices now. An open questions is how to take an energy constraint into account so that one might be able to recover a Gibbs state. Question period: David Poulin suggests keeping the unitary according to a Metropolis-Hastings rule.

Sergey Bravyi, Memory Time of 3D spin Hamiltonians with topological order, Phys. Rev. Lett. 107 150504 (2011); arXiv:1105.4159, joint work with Jeongwan Haah.

Want to store quantum information in a quantum many-body system for a long time. In a self-correcting memory, we want the natural physics of the system to do error correction for us. An example of a classical self-correcting memory: 2D ferromagnetic Ising model. There is a macroscopic energy barrier which prevents transitions between the ground states. It’s no good as a quantum memory because different ground states can be distinguished locally. (We need topological order.) Begin with an encoding of the information, i.e. an initialization of the system. To model the evolution, we use Metropolis dynamics. If a spin flip would decrease the energy, then flip it; otherwise, flip it with probability $exp(-beta Delta E)$ . If we are below the critical temperature, then the lifetime of the information is long, otherwise it is exponentially short. Now go to the quantum case. We will consider 2D and 3D quantum codes. We consider only those with Hamiltonians where the local operators are supported on a unit cell (locality). All the stabilizers must commute and must be a tensor product of Pauli operators. The ground state is frustration-free. Initial state of the quantum memory is some ground state. Now we consider dynamics. Restrict to Markovian dynamics, i.e. a master equation in Lindblad form. We will restrict the jump operators to be local with bounded operator norm, and the total number of jumps should be only of order $n$ (the number of spins). We require that the fixed point of the master equation is the Gibbs state and also detailed balance. Define the Liouville inner product $langle X , Y rangle_beta = mathrm{Tr}( exp(-beta H) X^dagger Y )$ . Detailed balance amounts to $langle X , mathcal{L}(Y) rangle_beta = langle mathcal{L}(X) , Y rangle_beta$ . As for decoding, we measure the syndrome and correct the errors. We measure the eigenvalue of each of the stabilizer operators. Whenever we record a -1 eigenvalue, we call that a defect. The decoder’s goal is trying to annihilate the defects in the way which is most likely to return the memory to its original state. Let’s ignore the computational complexity of the decoding algorithm for the moment. Then a good decoder is a minimum energy decoder, a la Dennis, Kitaev, Landahl, Preskill. (Why not consider free-energy minimizer instead?) Rather than use this decoder, we use the RG decoder of Harrington (PhD thesis) and Duclos-Cianci and Poulin (2009). To do this, identify connected clusters of errors and find the minimum enclosing box which encloses the errors. Then try to erase the cluster by finding minimum energy error within the boxes. Some of the boxes will be corrected, some will not. Now coarse-grain by a factor of two and try again. Stop when there are no defects left. Now apply the product of all the recorded erasing operators. If it returns the system to the ground state, declare success. Declare failure if the size increases to the linear size of the lattice and there are still some defects left. Now let’s define the energy barrier so that a high energy barrier gives high storage fidelity. A stabilizer Hamiltonian has energy barrier $B$ iff any error path of local operators mapping a ground state to an orthogonal ground state has energy cost at least $B$ . Lemma 1, the trace norm distance between the output of the minimum energy decoder and the original ground state is upper bounded by $lVert Phi_{mathrm{ec}}(rho(t)) - rho(0)rVert le O(tN) sum_{mge B} {N choose m} exp(-beta m) ,.$ How large does the energy barrier need to be to get self-correcting properties? Clearly a constant $B$ is insufficient. What about $B sim log N$ ? Plug this into the pervious bound and try to pick the system size (for a fixed temperature) that minimizes the RHS. Choose the system size to scale like $N_beta sim exp{beta/2}$ and you get that the storage infidelity decreases like $exp{beta^2}$ . This is a “marginal” self-correcting memory. There is a size of the system which is “too big” beyond which you lose the self-correction properties, but for the right size you get good properties. Contrast this with systems that have logical string-like operators: applying a string-like operator to the vacuum creates a pair of defects at the end points of a string. Then the energy barrier is $O(1)$ and the system can’t be self correcting. The memory time is $O(exp(beta))$ . All 2D stabilizer Hamiltonians have these string-like logical operators. In 3D, all previous attempts had string- like excitations as well. Recent results by Haah arXiv:1101.1962 provably have no string-like logical operators in 3D. Suppose that we consider a stabilizer Hamiltonian that has topological quantum order (TQO) but does not have logical string-like operators. What can we say about its energy barrier? What can we say about its memory time? We need to define “TQO” and “string-like”. For simplicity, consider only translationally invariant Hamiltonians. Definition: A finite cluster of defects C is called neutral iff C can be created from the vacuum by acting on a finite number of qubits. Otherwise C is called charged. For example, a single semi-infinite string would be a charged cluster. Definition: TQO iff 1) orthogonal ground states are indistinguishable on scale $ll L$ ; 2) Any neutral cluster of defects C can be created from the vacuum by acting on qubits inside the smallest enclosing box $b(C)$ and its constant neighborhood. Now define string-like. Consider a Pauli operator P. Suppose it creates defects only inside two boxes A1 and A2. Then a logical string segment P is called trivial iff both anchors are topologically neutral. Trivial string segments can be localized by multiplying them with stabilizers. These aren’t actually strings. The aspect ration $alpha$ is the ratio of the distance between the anchors divided by the size of the anchor regions. The no-strings rule: there exists a constant $alpha$ such that any logical string segment with aspect ratio greater than $alpha$ is trivial. Main result, theorem 1, Any sequence of local Pauli errors mapping a ground state to an orthogonal ground state must cross a logarithmic energy barrier. The constant implicit in front of the logarithmic scaling depends only the aspect ratio (no-strings constant) and the local site dimension. (The original Haah code has $alpha = 15$ .) This bound is tight and gives the $exp(beta^2)$ scaling for the minimum energy decoder. The next result gives a similar result for the more practical RG decoder. Sketch of the proof of theorem 1. No- strings rule implies that local errors can move charged clusters of defects only a little bit. Dynamics of neutral clusters is irrelevant as long as such clusters remain isolated. The no- strings rule is scale invariant, so use RG approach to analyze the energy barrier. Define a notion of “sparse” and “dense” syndrome. Lemma: dense syndromes are expensive in terms of energy cost. Any particular error path will have some sparse and some dense moments as you follow the path. Now do some RG flow. Keep only the dense points in the path. This sparsifies some of the dense ones at the next level of the RG hierarchy. Repeat. Use the no- strings rule to help localize errors. In conclusion: All stabilizer Hamiltonians in 2D local geometries have string-like logical operators. The energy barrier does not grow with the lattice size. Some 3D Hamiltonians have no string-like logical operators. Their energy barrier grows logarithmically with the size of the lattice. For small temperature $T$ , the memory time grows exponentially with $1/T^2$ . The optimal lattice size grows exponentially with $1/T$ .

Tzu-Chieh Wei, Measurement-based Quantum Computation with Thermal States and Always-on Interactions, Phys. Rev. Lett. 107 060501 (2011); arXiv:1102.5153. Joint work with Li, Browne, Kwek & Raussendorf.

Can we do MBQC with a gapped Hamiltonian where $Deltabeta ll 1$ where $Delta$ is the energy gap? Consider the toy model of the tri-valent AKLT state on a honeycomb lattice. Start with a bunch of spin-1/2 singlets on the edges and project onto the space with zero angular momentum at each vertex. If we consider the toy model of a single unit cell, then we see some features: the model is gapped (of course) and the Hamiltonian evolution is periodic (of course). So, if we perform operations at regular intervals, perhaps we can get MBQC to work if $beta Delta ll 1$ . We need to reduce the dimensionality of the central spin from 4 states to 2 states. We can use projection filtering to get this down to a GHZ state along a randomly chosen quantization axis (either of $x,y,z$ ). Distill a cluster state by measuring the POVM on the center particles. Then measure the bond particles. We can extend to 3D as well. This deterministically distills a 3D cluster state. As for fault-tolerance, you can use the Raussendorf-Harrington-Goyal scheme.

Anne E. B. Nielsen, Infinite-dimensional matrix product states, arXiv:1109.5470, joint work with Cirac and Sierra.

Recall the definition of an iMPS, we take the $Dtoinfty$ limit of a standard matrix product state. Let’s consider iMPS constructed from conformal fields. Motivation: want to identify families of states useful for investigating particular problems in quantum many-body systems. Why $Dtoinfty$ limit, and why conformal fields? Want to describe critical systems. Entanglement entropy is not limited by the area law. Power law decay of spin-spin correlation fucnctions. Long-range interactions are possible. Mathematical tools from CFT are useful. Sometimes it is possible to derive analytical results. Let’s restrict to a special model, $mathrm{SU}(2)_k$ WZW model. There are some chiral primal fields $phi_{j,m}(z)$ , and spin values range in $j in 0,1/2,ldots,k/2$ and $m$ is the z-component. There is a closed form of the wavefunction for $k=1, j=1/2$ and where the $z_i$ are uniform on the unit circle in the complex plane. Can compute the Rényi 2-entropy, the two-point spin correlation function. We can also derive a parent Hamiltonian for this model. We use two properties: the null field and the Ward identity. This Hamiltonian is non-local but 2-body. We can recover the Haldane-Shastry model in the uniform case. They computed the two- and four-point correlation functions.

David Sénéchal, The variational cluster method.

Ultimate goal: solve the Hubbard model, i.e. the simplest model of interacting electrons that you can write down. In particular, we are interested in the Green’s functions, single particle properties and the susceptibilities (2-particle properties). We base the cluster method on a tiling of the lattice (i.e. the superlattice). We solve for local information first in each block and then try to restore the connections between the blocks. We can rewrite this using a Fourier transform and parameterize in terms of two wave vectors: a discrete cluster wave vector (telling you which cluster you’re in) and a reduced wave vector (continuous). Now we decompose the Hamiltonian into a cluster term plus a perturbation which has inter-cluster hopping terms. Then we want to treat this at lowest order in perturbation theory. The approximate Green’s function at this level of approximation is very simple: take the cluster Green function and subtract the perturbation to get the total Green’s function. The lattice self-energy is approximately the cluster self-energy. Note that this method breaks translation invariance. But this can be “restored” by completing the Fourier transform (i.e. Fourier transform over the lattice sites). Then we can “periodize” the Green’s function. (This seems to be the right way to do it, rather than doing this to the self-energy, at least at half-filling where we have an exact solution for 1D.) Cluster perturbation theory (CPT) is exact when $U=0$ and when $t=0$ , and also gives exact short-range correlations. It allows a continuum of wavevectors. However, it doesn’t allow long-range order of broken symmetry. The approximation is controlled by the size of the cluster, and finite-size effects are important. There is no self-consistency condition (unlike dynamical mean-field theory). One idea is to try to capture a broken symmetry in CPT is to add a Weiss field. To set the relative weight of this term you would need some kind of principle, such as energy minimization. Instead, we use an idea due to Potthoff (M. Potthoff, Eur. Phys. J. B 32 429 (2003)). We define a functional of the Green’s function with the property that varying with respect to G gives the self-energy, and add some terms so that we get the Dyson equation. The value at the stationary point is the grand potential. There are now several approximation schemes: we could approximate the Euler equation (the equation for the stationary solution); we could approximate the functional (Hartree-Fock, etc.); we could restrict the variational space but keep the functional exact. We focus on the third method (following Potthoff). Potthoff suggested using the self-energy instead of the Green’s function and used the Legendre transform. This functional is universal in the sense that its functional form only depends on the interaction part. We can introduce a reference system which differs from the original Hamiltonian by one-body terms only. Suppose that we can solve this new Hamiltonian exactly. Then at the physical self-energy we can compute the value of the grand potential. Thus, we can find the dependence of the grand potential on the Green’s function. Showed some examples with Néel antiferromagnetism, superconductivity and a dynamical example studying the Mott transition.

Guillaume Duclos-Cianci, Local equivalence of topological order: Kitaev’s code and color codes, arXiv:1103.4606, joint work with H. Bombin and D. Poulin.

Focus on the toric code to be concrete. The main result is the following theorem: All 2D topological stabilizer codes are equivalent, meaning, there exists a local unitary mapping to a certain number of copies of Kitaev’s toric code. First review the toric code and the fact that the excitations are mutually bosonic with semionic exchange statistics and the notion of topological charge (i.e. equivalence classes of excitation configurations.) Now consider the decoding problem: given a configuration of defects, how can we decide which equivalence class we started in? Now define color codes. Gave a lot of details of the mapping taking the color code to two copies of the toric code, but I would really need some pictures to give an adequate description. Showed some results on the threshold for decoding using the mapping.

Philippe Corboz, Recent progress in the simulation of strongly correlated systems in two dimensions with tensor network algorithms, several papers, joint work with Bauer, Troyer, Mila, Vidal, White, Läuchli & Penc.

Want to use tensor networks to study strongly correlated systems. Typically, we use quantum Monte Carlo (QMC), but this fails for fermionic or frustrated systems because of the sign problem, so it is important to look for alternatives. To model fermions with a tensor network, we need to take the exchange statistics into account, and this can be done. Consider the t-J model: does the tensor network approach reproduce the “striped” states observed in some cuprates? DMRG (wide ladders) say YES, while variational and fixed-node Monte Carlo say NO. What does iPEPS say? It says stripes! Another example: the $mathrm{SU}(N)$ Heisenberg models on a 2D square lattice. For $N=2$ we know there is Néel order. The iPEPS result reproduces known results from QMC in this case; a good start. There are problems fitting as you increase the bond dimension $D$ and it is difficult to extrapolate. For $N=3$ and $N=4$ the sign problem means you can’t use QMC. Focus on the case $N=4$ . Here they find a new ground state which has considerably lower energy than previous variational results. The new ground state exhibits dimerization, and the color variation across each dimer exhibits Néel order. (“Dimer-Néel order”) Here the colors just refer to the 4 different degrees of freedom. This seems to be the predicted ground state as opposed to the plaquette state predicted by linear flavor-wave theory.

September 29, 2011

Ig Nobels 2011

Yes, it’s that time of year again. The Ig Nobel Prize ceremony was held this evening at Harvard. My favorite is the winner of the Ig Nobel Peace Prize, which went to

Arturas Zuokas, the mayor of Vilnius, Lithuania, for demonstrating that the problem of illegally parked luxury cars can be solved by running them over with an armoured tank.

This could have plausibly gotten him an Ig Nobel in economics instead, since it’s all about incentives. The video is priceless:

As of right now, the list of winners is still not up at the home page for the Annals of Improbable Research. That link should start working shortly, but until then I found this article from the BBC which gives complete coverage of the event. Enjoy!
Of course, this means that we will know the winners of the “real” Nobels soon enough. Put your best guess for the Physics prize in the comments.

September 21, 2011

Theoretical Physics StackExchange

Correction added 4 Oct: The site is in public beta now.
Correction added 27 Sep: At least for now, the site is in a closed beta, restricted to the people who initially committed to using it. Once that changes, we’ll let you know here.
There is a new Q&A website for theoretical physics started by our own Joe Fitzsimons. Go check it out!
http://theoreticalphysics.stackexchange.com/
The site will remain in beta for 60 to 90 days, at which time it will become a permanent StackExchange website…if there is enough activity during the 60-90 days! So go check out the website, ask and answer vexing quantum questions, and witness the power of crowdsourcing.

September 20, 2011

Two kilopontiffs, and Pontiff++

This post is the 2000th post here at the Quantum Pontiff!
And it seems that the former proprietor of this blog just couldn’t stop blogging… so go check out his new blog, Pontiff++.

September 20, 2011

Markus Greiner named MacArthur Fellow

Markus Greiner from Harvard was just named a 2011 MacArthur Fellow. For the experimentalists, Markus needs no introduction, but there might be a few theorists out there who still don’t know his name. Markus’ work probes the behavior of ultracold atoms in optical lattices.
When I saw Markus speak at SQuInT in February, I was tremendously impressed with his work. He spoke about his invention of a quantum gas microscope, a device which is capable of getting high fidelity images of individual atoms in optical lattices. He and his group have already used this tool to study the physics of the bosonic and fermionic Bose-Hubbard model that is (presumably) a good description of the physics in these systems. The image below is worth a thousand words.

Yep, those are individual atoms, resolved to within the spacing of the lattice. The ultimate goal is to obtain individual control of each atom separately within the lattice. Even with Markus’ breakthroughs, we are still a long way from having a quantum computer in an optical lattice. But I don’t think it is a stretch to say that his work is bringing us to the cusp of having a truly useful quantum simulator, one which is not universal for quantum computing but which nonetheless helps us answer certain physics questions faster than our best available classical algorithms and hardware. Congratulations to Markus!

September 16, 2011

Q-circuit v2.0

Many readers are familiar with the LaTeX package called Q-circuit that I coauthored with Bryan Eastin. If you aren’t familiar with it, it is a set of macros that helps make typesetting quantum circuits easy, efficient and (reasonably) intuitive. The results are quite beautiful, if I do say so myself, as can be seen in the picture to the left.
In the past year Bryan and I began getting emails from Q-circuit users who were experiencing some bugs. It turns out that the issue was usually an incompatibility between Q-circuit v1.2 and Xy-pic v3.8, an update to a package that Q-circuit relies on heavily.
Thanks to the user feedback and some support from the authors of Xy-pic, we were able to stamp out the bugs. (Probably… no guarantees!) Thus I present to you the latest version of Q-circuit!
Download Q-circuit v2.0
There is more info on the Q-circuit website, where you will find the tutorial, some examples, and you can also enjoy the painfully retro green-on-black motif. (Let the haters hate… I like it.) A few additional technical details:

Nothing has been added to the new version. It is as near as possible to the old version while still functioning with Xy-pic version 3.8.x.
The old version of Q-circuit works better with Xy-pic version 3.7. (When using Xy-pic 3.7, Q-circuit 2.0 makes PDFs with slightly pixelated curves.)
The arXiv is still using Xy-pic 3.7 and they don’t know when they’ll update to 3.8.

Finally, a big thank you to my coauthor Bryan for putting in so much hard work to make Q-circuit a success!

September 14, 2011

Entangled LIGO

The quest to observe gravitational waves has been underway for several years now, but as yet there has been no signal. To try to detect gravitational waves, the LIGO collaboration basically uses huge kilometer-scale Michaelson-type interferometers, one of which is seen in the aerial photo to the left. When a gravitational wave from, say, a supernova or in-spiraling pair of black holes arrives at the detector, the wave stretches and shrinks spacetime in the transverse directions, moving the test masses at the ends of the interferometer arms and hence changing the path length of the interferometer, creating a potentially observable signal.
The problem is, the sensitivity requirements are extreme. So extreme in fact, that within a certain frequency band the limiting noise comes from vacuum fluctuations of the electromagnetic field. Improving the signal-to-noise ratio can be achieved by a “classical” strategy of increasing the circulating light power, but this strategy is limited by the thermal response of the optics and can’t be used to further increase sensitivity.
But as we all know, the quantum giveth and the quantum taketh away. Or alternatively, we can fight quantum with quantum! The idea goes back to a seminal paper by Carl Caves, who showed that using squeezed states of light could reduce the uncertainty in an interferometer.
What’s amazing is that in a new paper, the LIGO collaboration has actually succeeded for the first time in using squeezed light to increase the sensitivity of one of its gravity wave detectors. Here’s a plot of the noise at each frequency in the detector.The red line shows the reduced noise when squeezed light is used. To get this to work, the squeezed quadrature must be in phase with the amplitude (readout) quadrature of the observatory output light, and this results in path entanglement between the photons in the two beams in the arms of the interferometer. The fluctuations in the photon counts can only be explained by stronger-than-classical correlation among the photons.
It looks like quantum entanglement might play a very important role in the eventual detection of gravitational waves. Tremendously exciting stuff.

September 9, 2011

Stability of Topological Order at Zero Temperature

From today’s quant-ph arXiv listing we find the following paper:

Stability of Frustration-Free Hamiltonians, by S. Michalakis & J. Pytel

This is a substantial generalization of one of my favorite results from last year’s QIP, the two papers by Bravyi, Hastings & Michalakis and Bravyi & Hastings.
In this new paper, Michalakis and Pytel show that any local gapped frustration-free Hamiltonian which is topologically ordered is stable under quasi-local perturbations. Whoa, that’s a mouthful… let’s try to break it down a bit.
Recall that a local Hamiltonian for a system of n spins is one which is a sum of polynomially many terms, each of which acts nontrivially on at most k spins for some constant k. Although this definition only enforces algebraic locality, let’s go ahead and require geometric locality as well by assuming that the spins all live on a lattice in d dimensions and all the interactions are localized to a ball of radius 1 on that lattice.
Why should we restrict to the case of geometric locality? There are at least two reasons. First, spins on a lattice is an incredibly important special case. Second, we have very few tools for analyzing quantum Hamiltonians which are k-local on a general hypergraph. Actually, few means something closer to none. (If you know any, please mention them in the comments!) On cubic lattices, we have many powerful techniques such Lieb-Robinson bounds, which the above results make heavy use of [1].
We say a Hamiltonian is frustration-free if the ground space is composed of states which are also ground states of each term separately. Thus, these Hamiltonians are “quantum satisfiable”, as a computer scientist would say. This too is an important requirement, since it is one of the most general classes of Hamiltonians about which we have any decent understanding. There are several key features of frustration-free Hamiltonians, but perhaps chief among them is the consistency of the ground space. The ground states on a local patch of spins are always globally consistent with the ground space of the full Hamiltonian, a fact which isn’t true for frustrated models.
We further insist that the Hamiltonian is gapped, which in this context means that there is some constant γ>0 independent of the system size which lower bounds the energy of any eigenstate orthogonal to the ground space. The gap assumption is extremely important since it is again closely related to the notion of locality. The spectral gap sets an energy scale and hence also a length scale, the correlation length. For two disjoint regions of spins separated by a length L in the lattice, the connected correlation function for any pair or local operators decays exponentially in L.
The last property, topological order, can be tricky to define. One of the key insights of this paper is a new definition of a sufficient condition for topological stability that the authors call local topological order. Roughly speaking, this new condition says that ground states of the local Hamiltonian are not distinguishable by any (sufficiently) local operator, except up to small effects that vanish rapidly in a neighborhood of the support of the local operator. Thus, the ground space can be used to encode quantum information which is insensitive to local operators! Since nature presumably acts locally and hence can’t corrupt the (nonlocally encoded) quantum information, systems with topological order would seem to be great candidates for quantum memories. Indeed, this was exactly the motivation when Kitaev originally defined the toric code.
Phew, that was a lot of background. So what exactly did Michalakis and Pytel prove, and why is it important? They proved that if a Hamiltonian satisfying the above criteria is subject to a sufficiently weak but arbitrary quasi-local perturbation then two things are stable: the spectral gap and the ground state degeneracy. (Quasi-local just means that strength of the perturbation decays sufficiently fast with respect to the size of the supporting region.) A bit more precisely, the spectral gap remains bounded from below by a constant independent of the system size, and the ground state degeneracy splits by an amount which is at most exponentially small in the size of the system.
There are several reasons why these stability results are important. First of all, the new result is very general: generic frustration-free Hamiltonians are a substantial extension of frustration-free commuting Hamiltonians (where the BHM and BH papers already show similar results). It means that the results potentially apply to models of topological quantum memory based on subsystem codes, such as that proposed by Bombin, where the syndrome measurements are only two-body. Second, the splitting of the ground state degeneracy determines the dephasing (T2) time for any qubits encoded in that ground space. Hence, for a long-lived quantum memory, the smaller the splitting the better. These stability results promise that even imperfectly engineered Hamiltonians should have an acceptably small splitting of the ground state degeneracy. Finally, a constant spectral gap means that when the temperature of the system is such that kT<<γ, thermal excitations are suppressed exponentially by a Boltzmann factor. The stability results show that the cooling requirements for the quantum memory do not increase with the system size.
Ah, but now we have opened a can of worms by mentioning temperature… The stability (or lack there of) of topological quantum phases at finite temperature is a fascinating topic which is the focus of much ongoing research, and perhaps it will be the subject of a future post. But for now, congratulations to Michalakis and Pytel on their interesting new paper.

[1] Of course, Lieb-Robinson bounds continue to hold on arbitrary graphs, it’s just that the bounds don’t seem to be very useful.

September 2, 2011

Geocentrism Revival

Seriously?

A few conservative Roman Catholics are pointing to a dozen Bible verses and the church’s original teachings as proof that Earth is the center of the universe, the view that was at the heart of the church’s clash with Galileo Galilei four centuries ago.

I can confidently speak for all of the quantum pontiffs when I say that we reject the geocentric view of the universe. I never thought I would have to boldly stand up for these beliefs, yet here I am.

… Those promoting geocentrism argue that heliocentrism, or the centuries-old consensus among scientists that Earth revolves around the sun, is a conspiracy to squelch the church’s influence.

This sentence nearly made my head explode.
First of all, heliocentrism is of course not agreed upon as scientific fact. As readers of this blog surely know, General Relativity teaches us that there are an infinite number of valid coordinate systems in which one can describe the universe, and we needn’t choose the one with the sun or earth at the origin to get the physics right (though one or the other might be more convenient for a specific calculation.)
Second, you’ve gotta love the form of argument which I affectionately call “argument by conspiracy theory”, in which any evidence against your position is waved away as the work of a secret organization with interests aligned against you. Oh, what’s that? You don’t have any evidence for the existence of this secret society? Well, that simply proves how cunning they are and merely strengthens the argument by conspiracy theory!

“Heliocentrism becomes dangerous if it is being propped up as the true system when, in fact, it is a false system,” said Robert Sungenis, leader of a budding movement to get scientists to reconsider. “False information leads to false ideas, and false ideas lead to illicit and immoral actions — thus the state of the world today.… Prior to Galileo, the church was in full command of the world, and governments and academia were subservient to her.”

So in case you were wondering: yes, this guy is serious. In fact, he is also happy to charge you $50 to attend his conference, or sell you one of several books on the topic, as well as some snazzy merchandise like coffee mugs and t-shirts that say “Galileo was wrong” on the front. (Hint: they don’t say “Einstein was right” on the back.)
To Mr. Sungenis and his acolytes: I implore you. Please just stop. It’s embarrassing for both of us. And if you’re worried about your bottom line, then consider going into climate change denial instead, which I hear is quite lucrative.

August 18, 2011

The first SciRate flame war

Maybe it’s not a war, but it is at least a skirmish.
The first shot was fired by a pseudonymous user named gray, who apparently has never scited any papers before and just arrived to bash an author of this paper for using a recommendation engine to… cue the dramatic music… recommend his own paper!
In an effort to stem this and future carnage, I’m taking to the quantum pontiff bully pulpit. This is probably better suited for the SciRate blog, but Dave didn’t give me the keys to that one.
Since it wasn’t obvious to everyone: SciRate is not a place for trolls to incite flame wars. Use the comments section of this post if you want to do that. (Kidding.) Comments on SciRate should have reasonable scientific merit, such as (at minimum) recommending a paper that was overlooked in the references, or (better) posting questions, clarifications, additional insights, etc. As an example, look at some of the excellent substantive comments left by prolific scirater Matt Hastings, or this discussion.
Nor is SciRate the place for insipid dull self-promotional comments and/or gibberish.
Now to make things fun, let’s have a debate in the comments section about the relative merits of introducing comment moderation on SciRate. Who is for it, who is against it, and what are the pros and cons? And who volunteers to do the moderating?
As for “gray” or any other troll out there: if you want to atone for your sins, my quantum confessional booth is always open.