arXiv subjects

Robert Koenig

Publications and source records attributed to Robert Koenig.

At least 55 records · Page 3Linked to original sources

arXiv2016

Preparing topologically ordered states by Hamiltonian interpolation

We study the preparation of topologically ordered states by interpolating between an initial Hamiltonian with a unique product ground state and a Hamiltonian with a topologically degenerate ground state space. By simulating the dynamics for small systems, we numerically observe a certain stability of the prepared state as a function of the initial Hamiltonian. For small systems or long interpolation times, we argue that the resulting state can be identified by computing suitable effective Hamiltonians. For effective anyon models, this analysis singles out the relevant physical processes and extends the study of the splitting of the topological degeneracy by Bonderson. We illustrate our findings using Kitaev's Majorana chain, effective anyon chains, the toric code and Levin-Wen string-net models.

quant-ph↗

arXiv2016

Protected gates for topological quantum field theories

We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators --- for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically-local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons; in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group.

quant-ph↗

arXiv2016

Matrix product approximations to multipoint functions in two-dimensional conformal field theory

Matrix product states (MPS) illustrate the suitability of tensor networks for the description of interacting many-body systems: ground states of gapped $1$-D systems are approximable by MPS as shown by Hastings [J. Stat. Mech. Theor. Exp., P08024 (2007)]. In contrast, whether MPS and more general tensor networks can accurately reproduce correlations in critical quantum systems, respectively quantum field theories, has not been established rigorously. Ample evidence exists: entropic considerations provide restrictions on the form of suitable Ansatz states, and numerical studies show that certain tensor networks can indeed approximate the associated correlation functions. Here we provide a complete positive answer to this question in the case of MPS and $2D$ conformal field theory: we give quantitative estimates for the approximation error when approximating correlation functions by MPS. Our work is constructive and yields an explicit MPS, thus providing both suitable initial values as well as a rigorous justification of variational methods.

quant-ph↗

arXiv2015

The operational meaning of min- and max-entropy

We show that the conditional min-entropy Hmin(A|B) of a bipartite state rho_AB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B-part of rho_AB are allowed. In the special case where A is classical, this overlap corresponds to the probability of guessing A given B. In a similar vein, we connect the conditional max-entropy Hmax(A|B) to the maximum fidelity of rho_AB with a product state that is completely mixed on A. In the case where A is classical, this corresponds to the security of A when used as a secret key in the presence of an adversary holding B. Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessing A given B is a lower bound on the number of uniform secret bits that can be extracted from A relative to an adversary holding B.

quant-ph↗

arXiv2015

Matrix product approximations to conformal field theories

We establish rigorous error bounds for approximating correlation functions of conformal field theories (CFTs) by certain finite-dimensional tensor networks. For chiral CFTs, the approximation takes the form of a matrix product state. For full CFTs consisting of a chiral and an anti-chiral part, the approximation is given by a finitely correlated state. We show that the bond dimension scales polynomially in the inverse of the approximation error and sub-exponentially in the ultraviolett cutoff. We illustrate our findings using Wess-Zumino-Witten models, and show that there is a one-to-one correspondence between group-covariant MPS and our approximation.

math-ph↗

arXiv2014

How to efficiently select an arbitrary Clifford group element

We give an algorithm which produces a unique element of the Clifford group $\mathcal{C}_n$ on $n$ qubits from an integer $0\le i < |\mathcal{C}_n|$ (the number of elements in the group). The algorithm involves $O(n^3)$ operations. It is a variant of the subgroup algorithm by Diaconis and Shahshahani which is commonly applied to compact Lie groups. We provide an adaption for the symplectic group $Sp(2n,\mathbb{F}_2)$ which provides, in addition to a canonical mapping from the integers to group elements $g$, a factorization of $g$ into a sequence of at most $4n$ symplectic transvections. The algorithm can be used to efficiently select random elements of $\mathcal{C}_n$ which is often useful in quantum information theory and quantum computation. We also give an algorithm for the inverse map, indexing a group element in time $O(n^3)$.

quant-ph↗

arXiv2014

An optimal dissipative encoder for the toric code

We consider the problem of preparing specific encoded resource states for the toric code by local, time-independent interactions with a memoryless environment. We propose a construction of such a dissipative encoder which converts product states to topologically ordered ones while preserving logical information. The corresponding Liouvillian is made up of four-local Lindblad operators. For a qubit lattice of size $L\times L$, we show that this process prepares encoded states in time $O(L)$, which is optimal. This scaling compares favorably with known local unitary encoders for the toric code which take time of order $Ω(L^2)$ and require active time-dependent control.

quant-ph↗

arXiv2014

The entropy power inequality for quantum systems

When two independent analog signals, X and Y are added together giving Z=X+Y, the entropy of Z, H(Z), is not a simple function of the entropies H(X) and H(Y), but rather depends on the details of X and Y's distributions. Nevertheless, the entropy power inequality (EPI), which states that exp [2H(Z)] \geq exp[2H(X)] + exp[2H(Y)], gives a very tight restriction on the entropy of Z. This inequality has found many applications in information theory and statistics. The quantum analogue of adding two random variables is the combination of two independent bosonic modes at a beam splitter. The purpose of this work is to give a detailed outline of the proof of two separate generalizations of the entropy power inequality to the quantum regime. Our proofs are similar in spirit to standard classical proofs of the EPI, but some new quantities and ideas are needed in the quantum setting. Specifically, we find a new quantum de Bruijin identity relating entropy production under diffusion to a divergence-based quantum Fisher information. Furthermore, this Fisher information exhibits certain convexity properties in the context of beam splitters.

quant-ph↗

arXiv2013

Generating topological order: no speedup by dissipation

We consider the problem of preparing topologically ordered states using unitary and non-unitary circuits, as well as local time-dependent Hamiltonian and Liouvillian evolutions. We prove that for any topological code in $D$ dimensions, the time required to encode logical information into the ground space is at least $Ω(d^{1/(D-1)})$, where $d$ is the code distance. This result is tight for the toric code, giving a scaling with the linear system size. More generally, we show that the linear scaling is necessary even when dropping the requirement of encoding: preparing any state close to the ground space using dissipation takes an amount of time proportional to the diameter of the system in typical 2D topologically ordered systems, as well as for example the 3D and 4D toric codes.

quant-ph↗

arXiv2013

The conditional entropy power inequality for Gaussian quantum states

We propose a generalization of the quantum entropy power inequality involving conditional entropies. For the special case of Gaussian states, we give a proof based on perturbation theory for symplectic spectra. We discuss some implications for entanglement-assisted classical communication over additive bosonic noise channels.

quant-ph↗

arXiv2012

The classical capacity of quantum thermal noise channels to within 1.45 bits

We find a tight upper bound for the classical capacity of quantum thermal noise channels that is within $1/\ln 2$ bits of Holevo's lower bound. This lower bound is achievable using unentangled, classical signal states, namely displaced coherent states. Thus, we find that while quantum tricks might offer benefits, when it comes to classical communication they can only help a bit.

quant-ph↗

arXiv2012

Classification of topologically protected gates for local stabilizer codes

Given a quantum error correcting code, an important task is to find encoded operations that can be implemented efficiently and fault-tolerantly. In this Letter we focus on topological stabilizer codes and encoded unitary gates that can be implemented by a constant-depth quantum circuit. Such gates have a certain degree of protection since propagation of errors in a constant-depth circuit is limited by a constant size light cone. For the 2D geometry we show that constant-depth circuits can only implement a finite group of encoded gates known as the Clifford group. This implies that topological protection must be "turned off" for at least some steps in the computation in order to achieve universality. For the 3D geometry we show that an encoded gate U is implementable by a constant-depth circuit only if the image of any Pauli operator under conjugation by U belongs to the Clifford group. This class of gates includes some non-Clifford gates such as the π/8 rotation. Our classification applies to any stabilizer code with geometrically local stabilizers and sufficiently large code distance.

quant-ph↗

arXiv2012

Limits on classical communication from quantum entropy power inequalities

Almost all modern communication systems rely on electromagnetic fields as a means of information transmission, and finding the capacities of these systems is a problem of significant practical importance. The Additive White Gaussian Noise (AWGN) channel is often a good approximate description of such systems, and its capacity is given by a simple formula. However, when quantum effects are important, estimating the capacity becomes difficult: a lower bound is known, but a similar upper bound is missing. We present strong new upper bounds for the classical capacity of quantum additive noise channels, including quantum analogues of the AWGN channel. Our main technical tool is a quantum entropy power inequality that controls the entropy production as two quantum signals combine at a beam splitter. Its proof involves a new connection between entropy production rates and a quantum Fisher information, and uses a quantum diffusion that smooths arbitrary states towards gaussians.

quant-ph↗

arXiv2011

Classical simulation of dissipative fermionic linear optics

Fermionic linear optics is a limited form of quantum computation which is known to be efficiently simulable on a classical computer. We revisit and extend this result by enlarging the set of available computational gates: in addition to unitaries and measurements, we allow dissipative evolution governed by a Markovian master equation with linear Lindblad operators. We show that this more general form of fermionic computation is also simulable efficiently by classical means. Given a system of $N$ fermionic modes, our algorithm simulates any such gate in time $O(N^3)$ while a single-mode measurement is simulated in time $O(N^2)$. The steady state of the Lindblad equation can be computed in time $O(N^3)$.

quant-ph↗

arXiv2011

Simplified instantaneous non-local quantum computation with applications to position-based cryptography

Instantaneous measurements of non-local observables between space-like separated regions can be performed without violating causality. This feat relies on the use of entanglement. Here we propose novel protocols for this task and the related problem of multipartite quantum computation with local operations and a single round of classical communication. Compared to previously known techniques, our protocols reduce the entanglement consumption by an exponential amount. We also prove a linear lower bound on the amount of entanglement required for the implementation of a certain non-local measurement. These results relate to position-based cryptography: an amount of entanglement scaling exponentially in the number of communicated qubits is sufficient to render any such scheme insecure. Furthermore, we show that certain schemes are secure under the assumption that the adversary has less entanglement than a given linear bound and is restricted to classical communication.

quant-ph↗

arXiv2011

Unconditional security from noisy quantum storage

We consider the implementation of two-party cryptographic primitives based on the sole assumption that no large-scale reliable quantum storage is available to the cheating party. We construct novel protocols for oblivious transfer and bit commitment, and prove that realistic noise levels provide security even against the most general attack. Such unconditional results were previously only known in the so-called bounded-storage model which is a special case of our setting. Our protocols can be implemented with present-day hardware used for quantum key distribution. In particular, no quantum storage is required for the honest parties.

quant-ph↗

arXiv2011

Disorder-assisted error correction in Majorana chains

It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions - the simplest toy model of a quantum memory. Disorder takes the form of a random site-dependent chemical potential. The corresponding one-particle problem is a one-dimensional Anderson model with disorder in the hopping amplitudes. We focus on the zero-temperature storage of a qubit encoded in the ground state of the Majorana chain. Storage and retrieval are modeled by a unitary evolution under the memory Hamiltonian with an unknown weak perturbation followed by an error-correction step. Assuming dynamical localization of the one-particle problem, we show that the storage time grows exponentially with the system size. We give supporting evidence for the required localization property by estimating Lyapunov exponents of the one-particle eigenfunctions. We also simulate the storage process for chains with a few hundred sites. Our numerical results indicate that in the absence of disorder, the storage time grows only as a logarithm of the system size. We provide numerical evidence for the beneficial effect of disorder on storage times and show that suitably chosen pseudorandom potentials can outperform random ones.

quant-ph↗

arXiv2010

Anyonic entanglement renormalization

We introduce a family of variational ansatz states for chains of anyons which optimally exploits the structure of the anyonic Hilbert space. This ansatz is the natural analog of the multi-scale entanglement renormalization ansatz for spin chains. In particular, it has the same interpretation as a coarse-graining procedure and is expected to accurately describe critical systems with algebraically decaying correlations. We numerically investigate the validity of this ansatz using the anyonic golden chain and its relatives as a testbed. This demonstrates the power of entanglement renormalization in a setting with non-abelian exchange statistics, extending previous work on qudits, bosons and fermions in two dimensions.

cond-mat.str-el↗