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We propose efficient classical algorithms which (strongly) simulate the action of bosonic linear optics circuits applied to superpositions of Gaussian states. Our approach relies on an augmented covariance matrix formalism to keep track of relative phases between individual terms in a linear combination. This yields an exact simulation algorithm whose runtime is polynomial in the number of modes and the size of the circuit, and quadratic in the number of terms in the superposition. We also present a faster approximate randomized algorithm whose runtime is linear in this number. Our main building blocks are a formula for the triple overlap of three Gaussian states and a fast algorithm for estimating the norm of a superposition of Gaussian states up to a multiplicative error. Our construction borrows from earlier work on simulating quantum circuits in finite-dimensional settings, including, in particular, fermionic linear optics with non-Gaussian initial states and Clifford computations with non-stabilizer initial states. It provides algorithmic access to a practically relevant family of non-Gaussian bosonic circuits., Comment: 43 pages and 11 figures

We show how to realize a general quantum circuit involving gates between arbitrary pairs of qubits by means of geometrically local quantum operations and efficient classical computation. We prove that circuit-level local stochastic noise modeling an imperfect implementation of our derived schemes is equivalent to local stochastic noise in the original circuit. Our constructions incur a constant-factor increase in the quantum circuit depth and a polynomial overhead in the number of qubits: To execute an arbitrary quantum circuit on $n$ qubits, we give a 3D quantum fault-tolerance architecture involving $O(n^{3/2} \log^3 n)$ qubits, and a quasi-2D architecture using $O(n^2 \log^3 n)$ qubits. Applied to recent fault-tolerance constructions, this gives a fault-tolerance threshold theorem for universal quantum computations with local operations, a polynomial qubit overhead and a quasi-polylogarithmic depth overhead. More generally, our transformation dispenses with the need for considering the locality of operations when designing schemes for fault-tolerant quantum information processing., Comment: 38 pages, 4 figures

We present a computational problem with the following properties: (i) Every instance can be solved with near-certainty by a constant-depth quantum circuit using only nearest-neighbor gates in 3D even when its implementation is corrupted by noise. (ii) Any constant-depth classical circuit composed of unbounded fan-in AND, OR, as well as NOT gates, i.e., an AC0-circuit, of size smaller than a certain subexponential, fails to solve a uniformly random instance with probability greater than a certain constant. Such an advantage against unbounded fan-in classical circuits was previously only known in the noise-free case or without locality constraints. We overcome these limitations, proposing a quantum advantage demonstration amenable to experimental realizations. Subexponential circuit-complexity lower bounds have traditionally been referred to as exponential. We use the term colossal since our fault-tolerant 3D architecture resembles a certain Roman monument., Comment: 44 pages, 12 figures

Local update recovery seeks to maintain quantum information by applying local correction maps alternating with and compensating for the action of noise. Motivated by recent constructions based on quantum LDPC codes in the finite-dimensional setting, we establish an analytic upper bound on the fault-tolerance threshold for concatenated GKP-stabilizer codes with local update recovery. Our bound applies to noise channels that are tensor products of one-mode beamsplitters with arbitrary environment states, capturing, in particular, photon loss occurring independently in each mode. It shows that for loss rates above a threshold given explicitly as a function of the locality of the recovery maps, encoded information is lost at an exponential rate. This extends an early result by Razborov from discrete to continuous variable (CV) quantum systems. To prove our result, we study a metric on bosonic states akin to the Wasserstein distance between two CV density functions, which we call the bosonic Wasserstein distance. It can be thought of as a CV extension of a quantum Wasserstein distance of order 1 recently introduced by De Palma et al. in the context of qudit systems, in the sense that it captures the notion of locality in a CV setting. We establish several basic properties, including a relation to the trace distance and diameter bounds for states with finite average photon number. We then study its contraction properties under quantum channels, including tensorization, locality and strict contraction under beamsplitter-type noise channels. Due to the simplicity of its formulation, and the established wide applicability of its finite-dimensional counterpart, we believe that the bosonic Wasserstein distance will become a versatile tool in the study of CV quantum systems., Comment: 30 pages, 2 figures

We propose efficient algorithms for classically simulating fermionic linear optics operations applied to non-Gaussian initial states. By gadget constructions, this provides algorithms for fermionic linear optics with non-Gaussian operations. We argue that this problem is analogous to that of simulating Clifford circuits with non-stabilizer initial states: Algorithms for the latter problem immediately translate to the fermionic setting. Our construction is based on an extension of the covariance matrix formalism which permits to efficiently track relative phases in superpositions of Gaussian states. It yields simulation algorithms with polynomial complexity in the number of fermions, the desired accuracy, and certain quantities capturing the degree of non-Gaussianity of the initial state. We study one such quantity, the fermionic Gaussian extent, and show that it is multiplicative on tensor products when the so-called fermionic Gaussian fidelity is. We establish this property for the tensor product of two arbitrary pure states of four fermions with positive parity., Comment: 68 pages and 15 figures; updated to match published version

Gate-teleportation circuits are arguably among the most basic examples of computations believed to provide a quantum computational advantage: In seminal work [Quantum Inf. Comput., 4(2):134--145], Terhal and DiVincenzo have shown that these circuits elude simulation by efficient classical algorithms under plausible complexity-theoretic assumptions. Here we consider possibilistic simulation [Phys. Rev. A 106, 062430 (2022)], a particularly weak form of this task where the goal is to output any string appearing with non-zero probability in the output distribution of the circuit. We show that even for single-qubit Clifford-gate-teleportation circuits this simulation problem cannot be solved by constant-depth classical circuits with bounded fan-in gates. Our results are unconditional and are obtained by a reduction to the problem of computing the parity, a well-studied problem in classical circuit complexity., Comment: 31 pages 10 figures. Typos fixed. New Section 3.3.3 Other word problems for Clifford modulo Pauli groups

We propose a scheme for fault-tolerant long-range entanglement generation at the ends of a rectangular array of qubits of length $R$ and a square cross section of size $d\times d$ with $d=O(\log R)$. Up to an efficiently computable Pauli correction, the scheme generates a maximally entangled state of two qubits using a depth-$6$ circuit consisting of nearest-neighbor Clifford gates and local measurements only. Compared with existing fault-tolerance schemes for quantum communication, the protocol is distinguished by its low latency: starting from a product state, the entangled state is prepared in a time $O(t_{\textrm{local}})$ determined only by the local gate and measurement operation time $t_{\textrm{local}}$. Furthermore, the requirements on local repeater stations are minimal: Each repeater uses only $\Theta(\log^2 R)$ qubits with a lifetime of order $O(t_{\textrm{local}})$. We prove a converse bound $\Omega(\log R)$ on the number of qubits per repeater among all low-latency schemes for fault-tolerant quantum communication over distance $R$. Furthermore, all operations within a repeater are local when the qubits are arranged in a square lattice. The noise-resilience of our scheme relies on the fault-tolerance properties of the underlying cluster state. We give a full error analysis, establishing a fault-tolerance threshold against general (circuit-level) local stochastic noise affecting preparation, entangling operations and measurements. This includes, in particular, errors correlated in time and space. Our conservative analytical estimates are surprisingly optimistic, suggesting that the scheme is suited for long-range entanglement generation both in and between near-term quantum computing devices., Comment: 68 pages, 11 figures

We consider Kitaev's quantum double model based on a finite group $G$ and describe quantum circuits for (a) preparation of the ground state, (b) creation of anyon pairs separated by an arbitrary distance, and (c) non-destructive topological charge measurement. We show that for any solvable group $G$ all above tasks can be realized by constant-depth adaptive circuits with geometrically local unitary gates and mid-circuit measurements. Each gate may be chosen adaptively depending on previous measurement outcomes. Constant-depth circuits are well suited for implementation on a noisy hardware since it may be possible to execute the entire circuit within the qubit coherence time. Thus our results could facilitate an experimental study of exotic phases of matter with a non-abelian particle statistics. We also show that adaptiveness is essential for our circuit construction. Namely, task (b) cannot be realized by non-adaptive constant-depth local circuits for any non-abelian group $G$. This is in a sharp contrast with abelian anyons which can be created and moved over an arbitrary distance by a depth-$1$ circuit composed of generalized Pauli gates., Comment: 48 pages, 18 figures. Revisions in v2: proof of Theorem 4.1 is extended to all solvable groups by lifting an implicit assumption about the group structure

Proposed hybrid algorithms encode a combinatorial cost function into a problem Hamiltonian and optimize its energy by varying over a set of states with low circuit complexity. Classical processing is typically only used for the choice of variational parameters following gradient descent. As a consequence, these approaches are limited by the descriptive power of the associated states. We argue that for certain combinatorial optimization problems, such algorithms can be hybridized further, thus harnessing the power of efficient non-local classical processing. Specifically, we consider combining a quantum variational ansatz with a greedy classical post-processing procedure for the MaxCut-problem on $3$-regular graphs. We show that the average cut-size produced by this method can be quantified in terms of the energy of a modified problem Hamiltonian. This motivates the consideration of an improved algorithm which variationally optimizes the energy of the modified Hamiltonian. We call this a twisted hybrid algorithm since the additional classical processing step is combined with a different choice of variational parameters. We exemplify the viability of this method using the quantum approximate optimization algorithm (QAOA), giving analytic lower bounds on the expected approximation ratios achieved by twisted QAOA. These show that the necessary non-locality of the quantum ansatz can be reduced compared to the original QAOA: We find that for levels $p=2,\ldots, 6$, the level $p$ can be reduced by one while roughly maintaining the expected approximation ratio. This reduces the circuit depth by $4$ and the number of variational parameters by $2$., Comment: 29 pages, 12 figures

It is known that continuous variable quantum information cannot be protected against naturally occurring noise using Gaussian states and operations only. Noh et al. (PRL 125:080503, 2020) proposed bosonic oscillator-to-oscillator codes relying on non-Gaussian resource states as an alternative, and showed that these encodings can lead to a reduction of the effective error strength at the logical level as measured by the variance of the classical displacement noise channel. An oscillator-to-oscillator code embeds K logical bosonic modes (in an arbitrary state) into N physical modes by means of a Gaussian N-mode unitary and N-K auxiliary one-mode Gottesman-Kitaev-Preskill-states. Here we ask if - in analogy to qubit error-correcting codes - there are families of oscillator-to-oscillator codes with the following threshold property: They allow to convert physical displacement noise with variance below some threshold value to logical noise with variance upper bounded by any (arbitrary) constant. We find that this is not the case if encoding unitaries involving a constant amount of squeezing and maximum likelihood error decoding are used. We show a general lower bound on the logical error probability which is only a function of the amount of squeezing and independent of the number of modes. As a consequence, any physically implementable family of oscillator-to-oscillator codes combined with maximum likelihood error decoding does not admit a threshold., Comment: 33 pages, 5 figures, updated to match published version

We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX-$k$-CUT, the problem of finding an approximate $k$-vertex coloring of a graph. We compare this proposal to the best known classical and hybrid classical-quantum algorithms. First, we show that the standard (non-recursive) QAOA fails to solve this optimization problem for most regular bipartite graphs at any constant level $p$: the approximation ratio achieved by QAOA is hardly better than assigning colors to vertices at random. Second, we construct an efficient classical simulation algorithm which simulates level-$1$ QAOA and level-$1$ RQAOA for arbitrary graphs. In particular, these hybrid algorithms give rise to efficient classical algorithms, and no benefit arising from the use of quantum mechanics is to be expected. Nevertheless, they provide a suitable testbed for assessing the potential benefit of hybrid algorithm: We use the simulation algorithm to perform large-scale simulation of level-$1$ QAOA and RQAOA with up to $300$ qutrits applied to ensembles of randomly generated $3$-colorable constant-degree graphs. We find that level-$1$ RQAOA is surprisingly competitive: for the ensembles considered, its approximation ratios are often higher than those achieved by the best known generic classical algorithm based on rounding an SDP relaxation. This suggests the intriguing possibility that higher-level RQAOA may be a potentially useful algorithm for NISQ devices., Comment: 27 pages, 5 figures

We study the code obtained by concatenating the standard single-mode Gottesman-Kitaev-Preskill (GKP) code with the surface code. We show that the noise tolerance of this surface-GKP code with respect to (Gaussian) displacement errors improves when a single-mode squeezing unitary is applied to each mode assuming that the identification of quadratures with logical Pauli operators is suitably modified. We observe noise-tolerance thresholds of up to $\sigma\approx 0.58$ shift-error standard deviation when the surface code is decoded without using GKP syndrome information. In contrast, prior results by Fukui et al. and Vuillot et al. report a threshold between $\sigma\approx 0.54$ and $\sigma\approx 0.55$ for the standard (toric-, respectively) surface-GKP code. The modified surface-GKP code effectively renders the mode-level physical noise asymmetric, biasing the logical-level noise on the GKP-qubits. The code can thus benefit from the resilience of the surface code against biased noise. We use the approximate maximum likelihood decoding algorithm of Bravyi et al. to obtain our threshold estimates. Throughout, we consider an idealized scenario where measurements are noiseless and GKP states are ideal. Our work demonstrates that Gaussian encodings of individual modes can enhance concatenated codes., Comment: 30 pages, 12 figures

Local Hamiltonians with topological quantum order exhibit highly entangled ground states that cannot be prepared by shallow quantum circuits. Here, we show that this property may extend to all low-energy states in the presence of an on-site $\mathbb{Z}_2$ symmetry. This proves a version of the No Low-Energy Trivial States (NLTS) conjecture for a family of local Hamiltonians with symmetry protected topological order. A surprising consequence of this result is that the Goemans-Williamson algorithm outperforms the Quantum Approximate Optimization Algorithm (QAOA) for certain instances of MaxCut, at any constant level. We argue that the locality and symmetry of QAOA severely limits its performance. To overcome these limitations, we propose a non-local version of QAOA, and give numerical evidence that it significantly outperforms standard QAOA for frustrated Ising models on random 3-regular graphs., Comment: 16 pages

Prior work has shown that there exists a relation problem which can be solved with certainty by a constant-depth quantum circuit composed of geometrically local gates in two dimensions, but cannot be solved with high probability by any classical constant depth circuit composed of bounded fan-in gates. Here we provide two extensions of this result. Firstly, we show that a separation in computational power persists even when the constant-depth quantum circuit is restricted to geometrically local gates in one dimension. The corresponding quantum algorithm is the simplest we know of which achieves a quantum advantage of this type. It may also be more practical for future implementations. Our second, main result, is that a separation persists even if the shallow quantum circuit is corrupted by noise. We construct a relation problem which can be solved with near certainty using a noisy constant-depth quantum circuit composed of geometrically local gates in three dimensions, provided the noise rate is below a certain constant threshold value. On the other hand, the problem cannot be solved with high probability by a noise-free classical circuit of constant depth. A key component of the proof is a quantum error-correcting code which admits constant-depth logical Clifford gates and single-shot logical state preparation. We show that the surface code meets these criteria. To this end, we provide a protocol for single-shot logical state preparation in the surface code which may be of independent interest.

We consider ensembles of bipartite states resulting from a random passive Gaussian unitary applied to a fiducial pure Gaussian state. We show that the symplectic spectra of the reduced density operators concentrate around that of a thermal state with the same energy. This implies, in particular, concentration of the entanglement entropy as well as other entropy measures. Our work extends earlier results on the typicality of entanglement beyond the two ensembles and the reduced purity measure considered in [A. Serafini, O. C. O. Dahlsten, D. Gross, and M. B. Plenio, J. Phys. A: Math. Theor., 40(31):9551 (2007)]., Comment: 19pages. Other materials: one Mathematica notebook, two Python files and one PDF file for the manual. V2: ancillary files uploaded, reference corrected

We provide a computer algebra package called Random Tensor Network Integrator (RTNI). It allows to compute averages of tensor networks containing multiple Haar-distributed random unitary matrices and deterministic symbolic tensors. Such tensor networks are represented as multigraphs, with vertices corresponding to tensors or random unitaries and edges corresponding to tensor contractions. Input and output spaces of random unitaries may be subdivided into arbitrary tensor factors, with dimensions treated symbolically. The algorithm implements the graphical Weingarten calculus and produces a weighted sum of tensor networks representing the average over the unitary group. We illustrate the use of this algorithmic tool on some examples from quantum information theory, including entropy calculations for random tensor network states as considered in toy models for holographic duality. Mathematica and Python implementations are supplied., Comment: Code available (for Mathematica and python) at https://github.com/MotohisaFukuda/RTNI

Motivated by the close relationship between quantum error-correction, topological order, the holographic AdS/CFT duality, and tensor networks, we initiate the study of approximate quantum error-detecting codes in matrix product states (MPS). We first show that using open-boundary MPS to define boundary to bulk encoding maps yields at most constant distance error-detecting codes. These are degenerate ground spaces of gapped local Hamiltonians. To get around this no-go result, we consider excited states, i.e., we use the excitation ansatz to construct encoding maps: these yield error-detecting codes with distance $\Omega(n^{1-\nu})$ for any $\nu\in (0,1)$ and $\Omega(\log n)$ encoded qubits. This shows that gapped systems contain $-$ within isolated energy bands $-$ error-detecting codes spanned by momentum eigenstates. We also consider the gapless Heisenberg-XXX model, whose energy eigenstates can be described via Bethe ansatz tensor networks. We show that it contains $-$ within its low-energy eigenspace $-$ an error-detecting code with the same parameter scaling. All these codes detect arbitrary $d$-local (not necessarily geometrically local) errors even though they are not permutation-invariant. This suggests that a wide range of naturally occurring many-body systems possess intrinsic error-detecting features., Comment: 79 pages, 17 figures. Version 2: added references

We construct efficient deterministic dynamical decoupling schemes protecting continuous variable degrees of freedom. Our schemes target decoherence induced by quadratic system-bath interactions with analytic time-dependence. We show how to suppress such interactions to $N$-th order using only $N$ pulses. Furthermore, we show to homogenize a $2^m$-mode bosonic system using only $(N+1)^{2m+1}$ pulses, yielding - up to $N$-th order - an effective evolution described by non-interacting harmonic oscillators with identical frequencies. The decoupled and homogenized system provides natural decoherence-free subspaces for encoding quantum information. Our schemes only require pulses which are tensor products of single-mode passive Gaussian unitaries and SWAP gates between pairs of modes., Comment: 17 pages, 2 figures

We propose a branch-and-bound algorithm for minimizing a bilinear functional of the form \[ f(X,Y) = \mathrm{tr}((X\otimes Y)Q)+\mathrm{tr}(AX)+\mathrm{tr}(BY) , \] of pairs of Hermitian matrices $(X,Y)$ restricted by joint semidefinite programming constraints. The functional is parametrized by self-adjoint matrices $Q$, $A$ and $B$. This problem generalizes that of a bilinear program, where $X$ and $Y$ belong to polyhedra. The algorithm converges to a global optimum and yields upper and lower bounds on its value in every step. Various problems in quantum information theory can be expressed in this form. As an example application, we compute Dobrushin curves of quantum channels, giving upper bounds on classical coding with energy constraints., Comment: 30 pages, 12 figures

We discuss classical algorithms for approximating the largest eigenvalue of quantum spin and fermionic Hamiltonians based on semidefinite programming relaxation methods. First, we consider traceless $2$-local Hamiltonians $H$ describing a system of $n$ qubits. We give an efficient algorithm that outputs a separable state whose energy is at least $\lambda_{\max}/O(\log n)$, where $\lambda_{\max}$ is the maximum eigenvalue of $H$. We also give a simplified proof of a theorem due to Lieb that establishes the existence of a separable state with energy at least $\lambda_{\max}/9$. Secondly, we consider a system of $n$ fermionic modes and traceless Hamiltonians composed of quadratic and quartic fermionic operators. We give an efficient algorithm that outputs a fermionic Gaussian state whose energy is at least $\lambda_{\max}/O(n\log n)$. Finally, we show that Gaussian states can vastly outperform Slater determinant states commonly used in the Hartree-Fock method. We give a simple family of Hamiltonians for which Gaussian states and Slater determinants approximate $\lambda_{\max}$ within a fraction $1-O(n^{-1})$ and $O(n^{-1})$ respectively., Comment: 17 pages

Tree tensor network descriptions of critical quantum spin chains are empirically known to reproduce correlation functions matching CFT predictions in the continuum limit. It is natural to seek a more complete correspondence, additionally incorporating dynamics. On the CFT side, this is determined by a representation of the diffeomorphism group of the circle. In a remarkable series of papers, Jones outlined a research program where the Thompson group T takes the role of the latter in the discrete setting, and representations of T are constructed from certain elements of a subfactor planar algebra. He also showed that for a particular example of such a construction, this approach only yields - in the continuum limit - a representation which is highly discontinuous and hence unphysical. Here we show that the same issue arises generically when considering tree tensor networks: the set of coarse-graining maps yielding discontinuous representations has full measure in the set of all isometries. This extends Jones' no-go example to typical elements of the so-called tensor planar algebra. We also identify an easily verified necessary condition for a continuous limit to exist. This singles out a particular class of tree tensor networks. Our considerations apply to recent approaches for introducing dynamics in holographic codes., Comment: 5 pages, 3 figures

The additivity problem asks if the use of entanglement can boost the information-carrying capacity of a given channel beyond what is achievable by coding with simple product states only. This has recently been shown not to be the case for phase-insensitive one-mode Gaussian channels, but remains unresolved in general. Here we consider two general classes of bosonic noise channels, which include phase-insensitive Gaussian channels as special cases: these are beamsplitters with general, potentially non-Gaussian environment states and classical noise channels with general probabilistic noise. We show that additivity violations, if existent, are rather minor for all these channels: the maximal gain in classical capacity is bounded by a constant independent of the input energy. Our proof shows that coding by simple classical modulation of coherent states is close to optimal., Comment: 19 pages, 2 figures; updated to match published version

We study how well topological quantum codes can tolerate coherent noise caused by systematic unitary errors such as unwanted $Z$-rotations. Our main result is an efficient algorithm for simulating quantum error correction protocols based on the 2D surface code in the presence of coherent errors. The algorithm has runtime $O(n^2)$, where $n$ is the number of physical qubits. It allows us to simulate systems with more than one thousand qubits and obtain the first error threshold estimates for several toy models of coherent noise. Numerical results are reported for storage of logical states subject to $Z$-rotation errors and for logical state preparation with general $SU(2)$ errors. We observe that for large code distances the effective logical-level noise is well-approximated by random Pauli errors even though the physical-level noise is coherent. Our algorithm works by mapping the surface code to a system of Majorana fermions., Comment: 18 pages, 14 figures

We prove that constant-depth quantum circuits are more powerful than their classical counterparts. To this end we introduce a non-oracular version of the Bernstein-Vazirani problem which we call the 2D Hidden Linear Function problem. An instance of the problem is specified by a quadratic form q that maps n-bit strings to integers modulo four. The goal is to identify a linear boolean function which describes the action of q on a certain subset of n-bit strings. We prove that any classical probabilistic circuit composed of bounded fan-in gates that solves the 2D Hidden Linear Function problem with high probability must have depth logarithmic in n. In contrast, we show that this problem can be solved with certainty by a constant-depth quantum circuit composed of one- and two-qubit gates acting locally on a two-dimensional grid.

We show that recent multivariate generalizations of the Araki-Lieb-Thirring inequality and the Golden-Thompson inequality [Sutter, Berta, and Tomamichel, Comm. Math. Phys. (2016)] for Schatten norms hold more generally for all unitarily invariant norms and certain variations thereof. The main technical contribution is a generalization of the concept of log-majorization which allows us to treat majorization with regards to logarithmic integral averages of vectors of singular values., Comment: 17 pages

We give necessary and sufficient conditions for a Gaussian quantum channel to have a dilation involving a passive, i.e., number-preserving unitary. We then establish a normal form of such channels: any passively dilatable channel is the result of applying passive unitaries to the input and output of a Gaussian additive channel. The latter combine the state of the system with that of the environment by means of a multi-mode beamsplitter., Comment: 20 pages, 1 figure, updated to match published version

We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results from a certain convolution operation: the latter maps a classical probability distribution on phase space and a quantum state to a quantum state. We show that this inequality also gives rise to several related inequalities whose counterparts are well-known in the classical setting: in particular, it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality, and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup. As an application, we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck semigroup, and argue that it implies fast convergence towards the fixed point for a large class of initial states., Comment: 37 pages; updated to match published version

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., Comment: 77 pages

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., Comment: 5 pages, 1 figure. See long companion paper arXiv:1509.07414 for full technical details

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., Comment: 105 pages, 2 figures

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., Comment: 50 pages, many figures, v3: updated to match published version

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 $O(n^3)$., Comment: 7 pages plus 4 1/2 pages of python code

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 $\Omega(d^{1/(D-1)})$, 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., Comment: 7 pages, 1 figure

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 $\Omega(L^2)$ and require active time-dependent control., Comment: 12 pages, 2 figures, v2: updated to match published version

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., Comment: 27 pages, 4 figures

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., Comment: Two pages plus a bit

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 \pi/8 rotation. Our classification applies to any stabilizer code with geometrically local stabilizers and sufficiently large code distance., Comment: 6 pages, 2 figures

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., Comment: v2: slightly rewritten, fixed typos, updated references, fixed figures. High-level description of proof with emphasis on applications to classical capacity. 10 pages, 4 figures

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., Comment: Mathematical exposition with detailed proofs. 14 pages. v2: updated to match published version

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 $O(N^3)$., Comment: 19 pages

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., Comment: 50 pages, 7 figures

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., Comment: 18 pages, 2 figures. V2: clarified results of the crypto section, improved readability