The Quantum Pontiff

Day 3 of the conference.

Quantum Information in Quantum Many-Body Physics

Guifre Vidal, Criticality, impurities and real-space renormalization, joint work with Glen Evenbly.

First an overview of the multi-scale entanglement renormalization ansatz (MERA). Basic notions: isometries, disentanglers, causal cone. Bounded causal cone width means that it is possible to efficiently evaluate the expectation value of local observables. One can also course grain a local operator by applying one level of the MERA. (For ternary MERA, a two-body term gets mapped to a two-body term.) We will focus on critical systems, and then put in impurities. To enforce scale and translation invariance, we characterize the MERA by just a single pair of tensors and repeated across the whole network. MERA seems to be a good ansatz to describe critical systems where such invariance is manifest. MERA also can exhibit polynomial decay of correlations. Define the scaling superoperator. The eigenvalues and eigenvectors of this superoperator are the scaling operators and (the exponential of) the scaling dimensions of the underlying CFT. How does one choose the optimal tensors given a local Hamiltonian? He dodges the question… doesn’t want to get into such details in this talk. Showed an example for the 1D critical Ising model with transverse field. Now let’s consider an impurity at the origin. Now we lose translation invariance, but we still have scale invariance (conformal defect). The bulk scaling operators in the absence of an impurity has zero expectation. However, in the presence of an impurity the one-point functions decay like a power of the distance to the impurity. We need to add new scaling operators and new scaling dimensions. Translation invariance is broken, we can’t possibly optimize over an extensive number of tensors (because we could never reach the thermodynamic limit). Well, desperate times call for desperate measures. Introduce something called “Principle of minimal influence” in RG flow. PMI says: Given a MERA for the initial (pure) problem, we can obtain a MERA for the problem with impurity by then varying the tensors in the causal cone for the region containing the impurity. We don’t expect the principle to work in all cases, such as in systems with spontaneous symmetry breaking where the impurity restores the symmetry. We are still exploring the limits of applicability of this principle. If you split the isometries into two “half isometries”, you get that the total number of tensors you need to optimize is still . Why is this ansatz plausible? For one thing, we can show that the scaling superoperator now exhibits power low decay with respect to the distance from the impurity. The optimization goes as follows. First optimize with respect to the bulk Hamiltonian (no impurity). Next, use the bulk tensors to map the original impurity problem to an effective impurity problem. This gives a new Hamiltonian problem with exponentially decaying interaction strengths. Finally, we can reduce the optimization via “folding in half” to that of an MPS for an open boundary problem. This same form appears in Wilson’s solution of the Kondo problem. However, the impurity MERA can be used beyond the setting of free fermions, unlike Wilson’s original approach. Showed example of transverse-field Ising model with impurity. This model allows an exact solution via CFT, and the numerics agree with the exact solution for various impurity strengths. We can extend this to more than one impurity by treating each impurity individually in their non-intersecting causal cones, and then once the causal cones intersect we do one more variation. We can also have an impurity that “cuts” the chain or a boundary between two half-chains coupled by the impurity, and even boundary conditions.

André-Marie Tremblay, Success and limitations of dynamical mean-field theory, joint work with Sordi, Sénéchal, Haule, Okamoto, Kyong and Civelli.

First recall how one “builds” a metal. We take a lattice and put orbitals on the sites. If the orbitals overlap, then it takes no energy to excite an electron. However, this doesn’t always work, e.g. in NiO. Recall the Hubbard model with on-site and hopping strength (and for nearest diagonal neighbors, etc.) When , we can Fourier transform and solve the model. When , everything is localized, and there are degenerate ground states at half filling corresponding to the different electron configurations with one electron per site. We can observe antiferromagnetism in the Hubbard model. We see a Mott transition in the region where and are of the same order, and at this point in the phase diagram neither the plane wave nor the localized wavefunctions gives a good description. One can write an effective model with Heisenberg coupling. We would like to measure Green’s functions for a single particle. Why do we care about Green’s functions? We can use them to compute the value of other observables and can measure it directly via photo emission. Fermi liquid theory: means that it is not too far from a free electron problem. We want to be able to solve in the presence of an impurity. Let’s review dynamical mean-field theory (DMFT). We begin working in infinite dimensions. Then the self-energy is a function only of the frequency. We want to compute the self-energy of the impurity problem. Then use that self-energy (which depends only on frequency) for the whole lattice. Now project the lattice onto a single site and adjust the bath so that the single site density of states obtained both ways are equal. There are some self-consistency equations. We can use a number of methods: cavity method, local nature of perturbation theory in infinite dimensions, expand around the atomic limit, effective medium theory, Potthoff self-energy functional. Use the universal self-energy functional method (see the talk by Sénéchal). Vary with respect to the parameters of the cluster (including Weiss fields). In this context, we can view DMFT as a stationary point in the variational space. What about the impurity solvers? The most trivial one is just exact diagonalization. Could also do Monte Carlo using an ergodic Markov chain. Showed a bunch of plots of these numerical methods for various models. The strengths of DMFT are: can obtain one-particle properties and phase diagrams; weaknesses include, order parameters are not coupled to observables that are quadratic in creation and annihilation operators, transport (four-point correlation functions), analytic continuation for QMC solvers. Future challenge is to find the optimum environment.

Matthias Troyer, Galois conjugates of topological phases, arXiv:1106.3267.

The question is: should we explore the computational power of non-unitary topological phases? Such phases appear through Galois conjugation of unitary topological phases. The arising models are non-Hermitian, but still have a purely real spectrum. Do they make sense? First, a review of topological phases and anyons (both Abelian and non-Abelian). Review of fusion rules. These states might appear in the Pfaffian state (Moore & Read) which has Ising anyons (level ) or in the parafermion state (Read & Rezayi) which has Fibonacci anyons (level ). How can we make more exotic states? We start with an anyonic liquid on a high-genus surface. Picture two sheets which are connected by fusing a bunch of perforations through both sheets. These create doubled (non-chiral) models. We can consider two types of quantities: flux through a hole and flux through a tube. We can build a family of basis states to describe the anyon wavefunctions by building a “skeleton” through each of the tubes. The basis states are labeled by string nets of particle labels on the skeleton lattice. Now we add the Hamiltonian for the model. We can either add flux through a tube or a hole, and each of these costs some different amount of energy. Each of these terms acts on either plaquettes or edges of the skeleton lattice. The plaquette term gives a ground state without flux through the plaquettes. This is like closing each of the holes: we get vacuum states of anyons on each of the two sheets. This is just the toric code or the Levin-Wen model. It’s a topological phase which is robust at zero-temperature to weak perturbations. This can be solved exactly, even for the excited states. Violations of the vertex constraint are chiral anyonic excitations living on the surface. Consider a finite density of anyons, and let’s pin the anyons into fixed positions. We would like to add a Heisenberg-like interaction Hamiltonian to this. We just introduce an energy difference between the various labels of the particle types. Review of the fusion basis for anyons, specialized to the Fibonacci case. Review of the F-moves ( symbols, Clebsch-Gordan), the pentagon and hexagon equations. So what does a Heisenberg chain for anyons look like? where is the F-move that exchanges two nearest-neighbors in a chain and is a projector onto a maximally entangled state. Ran numerical simulations to compute the gap… but it was gapless! The gap decreased like 1/L, so we guess it is a critical model. We can compute the central charge using the connection with entanglement entropy. It turns out that the model is exactly solvable by using the Temperley-Lieb algebra. They studied these interacting chains for various values of the level . There are lots of equivalences known to CFTs depending on the central charge and the sign of the coupling (antiferromagnatic or ferromagnatic). Plot of the energy spectrum, with very good agreement between exact solution and numerics. When there are finite density interactions, there are gapless edge modes and a nucleated liquid. This foreshadows the situation in 2D. You get a nucleated liquid with a finite bulk gap and edge states. A novel quantum Hall liquid is nucleated inside the parent liquid. Now we move to the Galois conjugation. Review of Galois theory. Different roots of a polynomial are Galois conjugates of each other. The algebraic properties are the same, but the geometric properties are different. As an example, consider again the Fibonacci case. The elements of the F tensor element can be expressed in terms of the golden ratio , or in terms of . However, this second one (Yang-Lee theory) is non-unitary. But if you look at the spectrum of the antiferromagnetic Yang-Lee model, you get a completely real spectrum! The central charge is negative (-3/5). The vacuum state is not the lowest energy state in this model. Similar conclusions for the ferromagnetic case. How does the entanglement entropy scale? It can’t still obey since the central charge is negative, right? Let’s look at the doubled Yang-Lee model. We can redefine the inner product to get a self-adjoint Hamiltonian. (Bender et al., PT-symmetric quantum mechanics). We can solve this doubled YL model exactly, just like with the Levin-Wen models before. Now the code property that defines topological order has a “left-right” eigenvector version, since these are now distinct notions. There are several ideas one might have to turn it into a Hermitian model: Square the plaquette terms, or use just the projector onto the left or right eigenspaces. But these approaches don’t work at giving a topological phase which is robust… they are sensitive to local perturbations. “Hermitianizing” destroys the left-right code property. Might there by some local transformation to a Hermitian model? If not for Fibonacci, then maybe for some other non-unitary TQFT? Norbert: can you use a tensor network description of the ground state to make a Hermitian parent Hamiltonian? Which ground state do you choose? The left or the right? Main result: Galois conjugates of TQFTs where the braid group acts densely in cannot be realized as ground states of quasilocal Hermitian Hamiltonians. Non-Abelian topological phases offer universal quantum computation; the minimal models of CFT are Heisenberg models of non-Abelian anyons; Galois conjugation of topological phases gives intriguing non-unitary theories and non-Hermitian models; but non-unitary phases will not appear in physical systems. Tobias: can’t we add a single extra spin which couples to all the spins and make the model Hermitian?

Renato Renner, Aisenstadt Lecture, An information-theoretic view of thermalization.

Consider a very trivial system of a classical binary random variable (for classical) which takes the values 0 or 1 with equal probability. Now let’s introduce an observer who has some knowledge about the . Suppose that  with probability  and  with probability . Now when we condition the probability of  on the state of  we get a new distribution. A Bayesian would just call this an update. Suppose now that a new observer  has quantum information about the system. So now the conditional state of  is a quantum state. If we try to condition the state of  on the quantum information , this becomes more interesting. For example, in quantum cryptography there is no reason to assume that the adversary is classical so it makes sense to consider this. Now let’s move on to thermodynamics. Consider a Szilard engine, i.e. a box with one particle in it and a piston that can compress to the middle of the box. If we let the “gas” expand adiabatically or compress it, which gives or takes work, then we find that the work satisfies to transition between the two states. From information theory, we know that a two-state system stores information. To erase information costs  units of energy. Landauer showed that in fact this is optimal (Landauer’s principle), and this is independent of the physical representation of the bit. The argument is, if you could erase with a lower energy cost, then you could use the Szilard engine to extract work for free from heat (which violates the second law). If we have conditional knowledge about the state of the Szilard engine, then we can extract a different amount of work, namely where is the binary entropy and describes our (potentially quantum) knowledge about the state of the system. (Technical note: this is the von Neumann entropy in the asymptotic setting and the max entropy in the single-shot setting.) Consider a computer with some registers and . I would like to erase them. If I erase them in sequence, then the total work required to do this would just sum the individual works using the previous formula. If instead, we did some joint operation we would only pay , which is in general less than the sum of the individual entropies. That is, in general we have , so it makes sense to take advantage of the joint information. We could use instead a formula in terms of conditional entropies by just changing the formula in the obvious way. It turns out that this formula is the correct formula, , where is the system and represents the observer. If we take advantage of this conditional information, then we can serially achieve the same work as the optimal joint protocol. This is a manifestation of the so-called chain rule for entropy: . Now let’s discuss thermalization. There is some large system (including an environment). If the total system is in a pure quantum state, how can it thermalize? What does that mean? We can look only at the subsystem, and there is a reduced density operator. Recall Alioscia’s talk for an example of some conditions for when the subsystem thermalizes. However, if two separate subsystems are separately thermal, that does not imply that the joint system is thermal. If instead we condition on the other subsystem instead, we could indeed reach this conclusion. As before, we cannot simply condition on quantum information, but the conditional entropies are well-defined. We can introduce the decoupling lemma, which guarantees that the joint state is nearly separable. It is a sufficient condition, phrased in terms of entropies conditioned on the observer, for decoupling. Namely, decoupling holds if we typically have where the total system has degrees of freedom and the system has . This is tight. If we apply this condition to thermalization. If the subsystem is sufficiently small, then the system will thermalize.

Tobias Osborne, The continuum limit of a quantum circuit: variational classes for quantum fields, arXiv:1102.5524 and arXiv:1005.1268, joint work with Cirac, Eisert, Haegeman, Verschelde, Verstraete.

Review of quantum fields. The most several ways one might “solve” a QFT: perturbation theory, Monte Carlo, and the variational methods. There were problems using the variational method in QFT, namely sensitivity to UV and poor reproduction of correlations. Review of MPS. Recall that MPS can be build from a staircase circuit. Review of area law result due to Hastings. As Tobias himself has proved, log-time evolution can be simulated efficiently, too. Also, a review of MERA. We can view it as a quantum circuit which takes ancillas and prepares a state. How can we pass to the continuum for these variational classes? We can use the quantum circuit preparation prescriptions of MPS and MERA and change the time step size for each gate to an infinitesimal quantity. That’s the main idea. Some nice features of cMPS: it is possible to analytically compute all the correlation functions. There is a built-in UV cutoff. There is generic clustering of correlations. Can build cMERA by alternating scaling transformations and local interactions for infinitesimal time steps, and take the limit. There is one main difference between MERA and cMERA: flexibility in the UV cutoff. The cMERA allow a smooth cutoff, but regular MERA only offer a lattice cutoff. These classes also obey entropy/area laws, rigorously in the case of cMPS, and there is a heuristic argument for cMERA. Gave an example for a simple bosonic system where the expressions are analytic for cMERA.

Day 2 of the conference. By the way, it is pretty difficult to get all this stuff written down correctly the first time through! So, if you see any mistakes please leave a comment so I can fix them.

Quantum Information in Quantum Many-Body Physics

Barbara Terhal, Adiabatic Quantum Computation and Stoquastic Hamiltonians, arXiv:0806.1746, joint work with Sergey Bravyi and quant-ph/0606140, joint with Bravyi, DiVincenzo and Oliveira.

Some quantum Hamiltonians have ground states which are amenable to Monte Carlo techniques because of the absence of the sign problem. We say a Hamiltonian is stoquastic iff, given some product basis, the off diagonal elements of each local term are real and non-positive. (One can apply local unitary operations to potentially transform some Hamiltonian into a stoquastic one.) Examples of stoquastic Hamiltonians: ferromagnets, quantum transverse-field Ising model, Jaynes-Cummings, Spin-boson and, importantly, Josephson-junction flux qubit Hamiltonians such as those engineered by D-wave for adiabatic quantum computation (AQC). Given a stoquastic Hamiltonian, we can define for some sufficiently small and  is a non-negative matrix. By the Perron-Frobenius theorem the largest eigenvalue eigenvector is a probability distribution. Therefore, we can potentially use quantum Monte Carlo (QMC) to sample from it. Theorem 1: Stoquastic frustration-free AQC can be efficiently simulated classically. Recall the matrix  from before. We can sample the matrix elements of  efficiently because the Hamiltonian is local. Since the Hamiltonian is frustration-free (FF), the ground state is annihilated term-by-term. We would like to set up a Markov chain which allows us to sample from the ground state. We can define a stochastic matrix from  and the weights in the ground state , namely where  is the diagonal matrix with the ground state weights on the diagonal. This is a stochastic matrix. Because the Hamiltonian is FF, we can compute the ratio . This doesn’t seem to work in the non-FF case. Now if you iterate and there is a gap you will converge in polynomial time. In the frustrated case, we don’t know how to compute the ratio anymore. An alternative approach: we know the ground state is well-approximated by the Gibbs state at low temperature. The Gibbs matrix is non-negative if the Hamiltonian is stoquastic (but frustrated). The partition function is a sum over non-negative weights now. We can compute each of these weights efficiently, so now we want to estimate the sum with QMC. “World-line QMC.” You can use some kind of Metropolis rule for accepting and rejecting moves.

Spiros Michalakis, Stability of Frustration-free Hamiltonians, arXiv:1109.1588. Joint work with J. Pytel.

The challenge is to find a minimal set of assumptions under which gapped Hamiltonians have a stable spectral gap against weak local perturbations. Not every gapped system is stable. Consider a simple Ising model where the transverse field is a perturbation. This perturbation splits the degeneracy of the ground state. Another example is an Ising model with a defect at the origin. The lesson seems to be that local distinguishability leads to instability of the ground state degeneracy and/or the gap. Now define FF Hamiltonians (FFH). Every FFH on a lattice is the extension of another FFH on some smaller ball . The ground state projectors satisfy where is the global ground state projector. Now define topological quantum order (TQO). Roughly, no sufficiently local operator should be able to distinguish two orthogonal ground states. The toric code is an example of a model with TQO. But TQO is not enough for stability, as a simple counterexample shows (the “Ising toric code with defect”– it’s simpler than it sounds). Let’s define local TQO (LTQO), which says that the local distinguishability between various ground states as measured by the operator norm decays rapidly. This LTQO condition implies an area law (with a possible logarithmic correction, which is what we expect since we haven’t said anything about a spectral gap at this point). Next, we consider another condition called the local gap condition (LG). We want the gap of the local (restricted to some ) Hamiltonians have a gap which decays at most as an inverse polynomial in the size of . Open problem: is LG always satisfied if the global Hamiltonian is a sum of local projections with FF and a spectral gap? The main theorem: If is a FFH satisfying LTQO and LG with sufficiently weak quasi-local perturbations, then the gap is stable and the ground state degeneracy doesn’t split too much. Gave a very brief overview of the proof. Some open problems: how can we prove spectral gaps for 2D FFH? What symmetries of the Hamiltonian imply LTQO? Can we prove LG for all FFH that are sums of local projectors?

Norbert Schuch, Complexity of commuting Hamiltonians for qubits on a square lattice, arXiv:1105.2843.

Commuting Hamiltonians can have highly entangled ground states (e.g. the toric code). They clearly have a constant spectral gap, and we only need to estimate the ground state energy to constant accuracy. Is the complexity of estimating the ground state energy easier than the full quantum problem? Main result: for qubits on a 2D square lattice, the problem of determining if the ground state is FF is in NP. First we need a structure theorem for commuting Hamiltonians due to Bravyi and Vyalyi (2003). This decomposition was used to show that 2-body commuting Hamiltonian problem is in NP. The decomposition does not generally extend to -body for and for qudits. However, if the connectivity is “sparse enough” then the decomposition works. Idea number one: split the lattice into “white” and “black” squares and use the fact that each layer individually can take advantage of the BV decomposition. But we don’t know if these two decompositions are compatible, so we need a new idea. We can tell if the FF ground state exists if we can find states in each layer that have non-trivial overlap… but how can we compute the overlap? We need to take advantage of the structure. The point is that for qubits the overlap can be efficiently computed. When the spins are qubits are only two possibilities for the BV decomposition. One possibility is trivial, whereas the second possibility doesn’t allow any branching structures to form, so the overlap can be computed efficiently. This proof does not provide a constructive procedure for finding the ground state.

Maarten Van den Nest, A monomial matrix formalism to describe quantum many-body states, arXiv:1108.0531.

The motivation is to generalize the Pauli stabilizer formalism in a way which still retains some of the nice properties and captures a large class of states. Some of the nice properties include: efficient description of quantum states via the stabilizers, understanding of properties of the state via manipulation of the stabilizers. We first define a monomial unitary matrix which is a permutation matrix times a diagonal matrix of phases. An M-state is one which is stabilized by (+1 eigenstate of) a set of monomial unitaries. We typically want to restrict to special cases, such as -local, etc. This class of states encompasses e.g. stabilizer states for qudits, all quantum double ground states, AKLT model, W-states, Dicke states, coherent probabilistic computations, coset states of Abelian groups, and so on. We define the monomial stabilizer group to be the group generated by the unitaries. Assume this is finite. There is another group called the permutation group  which comes from “forgetting” the phase structure and keeping only the permutation piece. We define the orbits by the orbit of the computational basis states under the action of the operators in . Some simple theorems: Thm 1, all amplitudes of an M-state are zero outside the orbit. Thm 2, all nonzero amplitudes have equal modulus. If you have an entire M-space, then you can find a basis for the space where each basis state is a uniform-amplitude superposition over a single orbit. The orbits are disjoint, so the dimension of the space is bounded by the total number of orbits. Computing some properties such as sampling the state in the computational basis are NP-hard. However, there are many examples that allow an efficient classical simulation.

Hector Bombin, Structure of 2D topological stabilizer codes, arXiv:1107.2707, joint work with D. Poulin and G. Duclos-Cianci.

Topological order can be thought of as equivalence classes of many-body systems under the action of constant-depth local unitary quantum circuits. We say that a state has short range entanglement if such a circuit can map the state to a product state. Other states are said to have long range entanglement. (Wen, Gu, Chen 2009) We will consider topological stabilizer codes (TSC). These are a subset of the stabilizer code Hamiltonians, in which the local terms are stabilizer code generators. The goal of this talk is two-fold: show rigorously that local equivalence “works” as a way of classifying gapped phases and find the general structure of 2D TSC including subsystem codes. This allows us to extend which are well-known for specific models (such as RG error correction for the toric code) to other models. Let’s recall what defines a theory of anyons: topological charges (particle labels), fusion rules and braiding rules. In 2D, string operators play an important role in these models. In fact, we can recover the data specifying the anyon theory from the action of the string operators. In this talk, we are only considering qubit Hamiltonians. We only consider Hamiltonians on an infinite lattice which are translationally invariant. Then TSC are ones where the centralizer of the stabilizer (in the group of Paulis with finite support) is again the stabilizer. TSC with the same anyon model are equivalent. Since we are going to show that everything is equivalent to some number of copies of the toric code, we can get a complete invariant by considering only the topological entanglement entropy. A sketch of the proof:

1. Topological stabilizer groups admit local, translationally invariant independent generators (only possible in 2D)
2. there are only a finite number of global constraints
3. the number of global constraints equals the number of charges in the anyon theory
4. the charges are topological, so the have string operators, which means we can extract an anyon model
5. we can rule out chirality because the Hamiltonian is commuting (there can be no energy transport along the edge, as can be seen by considering evolution in the Heisenberg picture)
6. build a framework for string segments as in multiple copies of the toric code
7. other stabilizers (non-plaquette) have no charge
8. map string segments to string segments and uncharged stabilizers to single qubit stabilizers
9. ???

(Hector, if you’re reading this, fill in #9 in the comments. You went waaaay too fast!) Now let’s move on to subsystem codes. There are some complications since there is now also a gauge group as well as a stabilizer group. Gauge charges map naturally to stabilizer charges. Although they are not generally isomorphic, these two groups have the same order. In fact, these two charge groups are naturally dual. When all is said and done, every possible anyon model on qubits is a combination of: the toric code (2 bosons, 1 fermion, semionic statistics); TSCC (3 fermions, semionic interactions, chiral); subsystem toric code (1 boson); honeycomb subsystem code (1 fermion).

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 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, . 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 where is the number time steps and is the local site dimension. If we look at the 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 . 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 (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 . Detailed balance amounts to . 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 iff any error path of local operators mapping a ground state to an orthogonal ground state has energy cost at least . Lemma 1, the trace norm distance between the output of the minimum energy decoder and the original ground state is upper bounded by How large does the energy barrier need to be to get self-correcting properties? Clearly a constant is insufficient. What about ? 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 and you get that the storage infidelity decreases like . 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 and the system can’t be self correcting. The memory time is . 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 ; 2) Any neutral cluster of defects C can be created from the vacuum by acting on qubits inside the smallest enclosing box  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 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 such that any logical string segment with aspect ratio greater than 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 .) This bound is tight and gives the 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 , the memory time grows exponentially with . The optimal lattice size grows exponentially with .

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 where  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 . 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 ). 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  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  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,  WZW model. There are some chiral primal fields , and spin values range in and  is the z-component. There is a closed form of the wavefunction for  and where the 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 and when , 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 Heisenberg models on a 2D square lattice. For 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 and it is difficult to extrapolate. For and the sign problem means you can’t use QMC. Focus on the case . 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.