arXiv subjects

Johannes Bausch

Publications and source records attributed to Johannes Bausch.

33 records · Page 2Linked to original sources

arXiv2020

The Complexity of Translationally Invariant Problems beyond Ground State Energies

It is known that three fundamental questions regarding local Hamiltonians -- approximating the ground state energy (the Local Hamiltonian problem), simulating local measurements on the ground space (APX-SIM), and deciding if the low energy space has an energy barrier (GSCON) -- are $\mathsf{QMA}$-hard, $\mathsf{P}^{\mathsf{QMA}[log]}$-hard and $\mathsf{QCMA}$-hard, respectively, meaning they are likely intractable even on a quantum computer. Yet while hardness for the Local Hamiltonian problem is known to hold even for translationally-invariant systems, it is not yet known whether APX-SIM and GSCON remain hard in such "simple" systems. In this work, we show that the translationally invariant versions of both APX-SIM and GSCON remain intractable, namely are $\mathsf{P}^{\mathsf{QMA}_{\mathsf{EXP}}}$- and $\mathsf{QCMA}_{\mathsf{EXP}}$-complete, respectively. Each of these results is attained by giving a respective generic "lifting theorem" for producing hardness results. For APX-SIM, for example, we give a framework for "lifting" any abstract local circuit-to-Hamiltonian mapping $H$ (satisfying mild assumptions) to hardness of APX-SIM on the family of Hamiltonians produced by $H$, while preserving the structural and geometric properties of $H$ (e.g. translation invariance, geometry, locality, etc). Each result also leverages counterintuitive properties of our constructions: for APX-SIM, we "compress" the answers to polynomially many parallel queries to a QMA oracle into a single qubit. For GSCON, we give a hardness construction robust against highly non-local unitaries, i.e. even if the adversary acts on all but one qudit in the system in each step.

quant-ph↗

arXiv2020

Recurrent Quantum Neural Networks

Recurrent neural networks are the foundation of many sequence-to-sequence models in machine learning, such as machine translation and speech synthesis. In contrast, applied quantum computing is in its infancy. Nevertheless there already exist quantum machine learning models such as variational quantum eigensolvers which have been used successfully e.g. in the context of energy minimization tasks. In this work we construct a quantum recurrent neural network (QRNN) with demonstrable performance on non-trivial tasks such as sequence learning and integer digit classification. The QRNN cell is built from parametrized quantum neurons, which, in conjunction with amplitude amplification, create a nonlinear activation of polynomials of its inputs and cell state, and allow the extraction of a probability distribution over predicted classes at each step. To study the model's performance, we provide an implementation in pytorch, which allows the relatively efficient optimization of parametrized quantum circuits with thousands of parameters. We establish a QRNN training setup by benchmarking optimization hyperparameters, and analyse suitable network topologies for simple memorisation and sequence prediction tasks from Elman's seminal paper (1990) on temporal structure learning. We then proceed to evaluate the QRNN on MNIST classification, both by feeding the QRNN each image pixel-by-pixel; and by utilising modern data augmentation as preprocessing step. Finally, we analyse to what extent the unitary nature of the network counteracts the vanishing gradient problem that plagues many existing quantum classifiers and classical RNNs.

cs.LG↗

arXiv2020

Mitigating Errors in Local Fermionic Encodings

Quantum simulations of fermionic many-body systems crucially rely on mappings from indistinguishable fermions to distinguishable qubits. The non-local structure of fermionic Fock space necessitates encodings that either map local fermionic operators to non-local qubit operators, or encode the fermionic representation in a long-range entangled code space. In this latter case, there is an unavoidable trade-off between two desirable properties of the encoding: low weight representations of local fermionic operators, and a high distance code space. Here it is argued that despite this fundamental limitation, fermionic encodings with low-weight representations of local fermionic operators can still exhibit error mitigating properties which can serve a similar role to that played by high code distances. In particular when undetectable errors correspond to "natural" fermionic noise. We illustrate this point explicitly for two fermionic encodings: the Verstraete-Cirac encoding, and an encoding appearing in concurrent work by Derby and Klassen. In these encodings many, but not all, single-qubit errors can be detected. However we show that the remaining undetectable single-qubit errors map to local, low-weight fermionic phase noise. We argue that such noise is natural for fermionic lattice models. This suggests that even when employing low-weight fermionic encodings, error rates can be suppressed in a similar fashion to high distance codes, provided one is willing to accept simulated natural fermionic noise in their simulated fermionic system.

quant-ph↗

arXiv2020

A Quantum Search Decoder for Natural Language Processing

Probabilistic language models, e.g. those based on an LSTM, often face the problem of finding a high probability prediction from a sequence of random variables over a set of tokens. This is commonly addressed using a form of greedy decoding such as beam search, where a limited number of highest-likelihood paths (the beam width) of the decoder are kept, and at the end the maximum-likelihood path is chosen. In this work, we construct a quantum algorithm to find the globally optimal parse (i.e. for infinite beam width) with high constant success probability. When the input to the decoder is distributed as a power-law with exponent $k>0$, our algorithm has runtime $R^{n f(R,k)}$, where $R$ is the alphabet size, $n$ the input length; here $f<1/2$, and $f\rightarrow 0$ exponentially fast with increasing $k$, hence making our algorithm always more than quadratically faster than its classical counterpart. We further modify our procedure to recover a finite beam width variant, which enables an even stronger empirical speedup while still retaining higher accuracy than possible classically. Finally, we apply this quantum beam search decoder to Mozilla's implementation of Baidu's DeepSpeech neural net, which we show to exhibit such a power law word rank frequency.

quant-ph↗

arXiv2020

Undecidability of the Spectral Gap in One Dimension

The spectral gap problem - determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations - pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum spin systems in two (or more) spatial dimensions: there exists no algorithm that determines in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one dimensional spin systems are simpler than their higher-dimensional counterparts: for example, they cannot have thermal phase transitions or topological order, and there exist highly-effective numerical algorithms such as DMRG - and even provably polynomial-time ones - for gapped 1D systems, exploiting the fact that such systems obey an entropy area-law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are not possible in 1D. So does the spectral gap problem become decidable in 1D? In this paper we prove this is not the case, by constructing a family of 1D spin chains with translationally-invariant nearest neighbour interactions for which no algorithm can determine the presence of a spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable as in higher dimensions, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and non-degenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour with dense spectrum.

quant-ph↗

arXiv2020

Universal Translationally-Invariant Hamiltonians

In this work we extend the notion of universal quantum Hamiltonians to the setting of translationally-invariant systems. We present a construction that allows a two-dimensional spin lattice with nearest-neighbour interactions, open boundaries, and translational symmetry to simulate any local target Hamiltonian---i.e. to reproduce the whole of the target system within its low-energy subspace to arbitrarily-high precision. Since this implies the capability to simulate non-translationally-invariant many-body systems with translationally-invariant couplings, any effect such as characteristics commonly associated to systems with external disorder, e.g. many-body localization, can also occur within the low-energy Hilbert space sector of translationally-invariant systems. Then we sketch a variant of the universal lattice construction optimized for simulating translationally-invariant target Hamiltonians. Finally we prove that qubit Hamiltonians consisting of Heisenberg or XY interactions of varying interaction strengths restricted to the edges of a connected translationally-invariant graph embedded in $\mathbb{R}^D$ are universal, and can efficiently simulate any geometrically local Hamiltonian in $\mathbb{R}^D$.

quant-ph↗

arXiv2019

Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

In this work we propose a many-body Hamiltonian construction which introduces only a single separate energy scale of order $Θ(1/N^{2+δ})$, for a small parameter $δ>0$, and for $N$ terms in the target Hamiltonian. In its low-energy subspace, the construction can approximate any normalized target Hamiltonian $H_\mathrm{t}=\sum_{i=1}^N h_i$ with norm ratios $r=\max_{i,j\in\{1,\ldots,N\}}\|h_i\| / \| h_j \|=O(\exp(\exp(\mathrm{poly} n)))$ to within relative precision $O(N^{-δ})$. This comes at the expense of increasing the locality by at most one, and adding an at most poly-sized ancilliary system for each coupling; interactions on the ancilliary system are geometrically local, and can be translationally-invariant. As an application, we discuss implications for QMA-hardness of the local Hamiltonian problem, and argue that "almost" translational invariance-defined as arbitrarily small relative variations of the strength of the local terms-is as good as non-translational-invariance in many of the constructions used throughout Hamiltonian complexity theory. We furthermore show that the choice of geared limit of many-body systems, where e.g. width and height of a lattice are taken to infinity in a specific relation, can have different complexity-theoretic implications: even for translationally-invariant models, changing the geared limit can vary the hardness of finding the ground state energy with respect to a given promise gap from computationally trivial, to QMAEXP-, or even BQEXPSPACE-complete.

quant-ph↗

arXiv2019

Quantum Codes from Neural Networks

We examine the usefulness of applying neural networks as a variational state ansatz for many-body quantum systems in the context of quantum information-processing tasks. In the neural network state ansatz, the complex amplitude function of a quantum state is computed by a neural network. The resulting multipartite entanglement structure captured by this ansatz has proven rich enough to describe the ground states and unitary dynamics of various physical systems of interest. In the present paper, we initiate the study of neural network states in quantum information-processing tasks. We demonstrate that neural network states are capable of efficiently representing quantum codes for quantum information transmission and quantum error correction, supplying further evidence for the usefulness of neural network states to describe multipartite entanglement. In particular, we show the following main results: a) Neural network states yield quantum codes with a high coherent information for two important quantum channels, the generalized amplitude damping channel and the dephrasure channel. These codes outperform all other known codes for these channels, and cannot be found using a direct parametrization of the quantum state. b) For the depolarizing channel, the neural network state ansatz reliably finds the best known codes given by repetition codes. c) Neural network states can be used to represent absolutely maximally entangled states, a special type of quantum error-correcting codes. In all three cases, the neural network state ansatz provides an efficient and versatile means as a variational parametrization of these highly entangled states.

quant-ph↗

arXiv2018

Analysis and limitations of modified circuit-to-Hamiltonian constructions

Feynman's circuit-to-Hamiltonian construction connects quantum computation and ground states of many-body quantum systems. Kitaev applied this construction to demonstrate QMA-completeness of the local Hamiltonian problem, and Aharanov et al. used it to show the equivalence of adiabatic computation and the quantum circuit model. In this work, we analyze the low energy properties of a class of modified circuit Hamiltonians, which include features like complex weights and branching transitions. For history states with linear clocks and complex weights, we develop a method for modifying the circuit propagation Hamiltonian to implement any desired distribution over the time steps of the circuit in a frustration-free ground state, and show that this can be used to obtain a constant output probability for universal adiabatic computation while retaining the $Ω(T^{-2})$ scaling of the spectral gap, and without any additional overhead in terms of numbers of qubits. Furthermore, we establish limits on the increase in the ground energy due to input and output penalty terms for modified tridiagonal clocks with non-uniform distributions on the time steps by proving a tight $O(T^{-2})$ upper bound on the product of the spectral gap and ground state overlap with the endpoints of the computation. Using variational techniques which go beyond the $Ω(T^{-3})$ scaling that follows from the usual geometrical lemma, we prove that the standard Feynman-Kitaev Hamiltonian already saturates this bound. We review the formalism of unitary labeled graphs which replace the usual linear clock by graphs that allow branching and loops, and we extend the $O(T^{-2})$ bound from linear clocks to this more general setting. In order to achieve this, we apply Chebyshev polynomials to generalize an upper bound on the spectral gap in terms of the graph diameter to the context of arbitrary Hermitian matrices.

quant-ph↗

arXiv2018

Classifying Data with Local Hamiltonians

The goal of this work is to define a notion of a quantum neural network to classify data, which exploits the low energy spectrum of a local Hamiltonian. As a concrete application, we build a binary classifier, train it on some actual data and then test its performance on a simple classification task. More specifically, we use Microsoft's quantum simulator, Liquid, to construct local Hamiltonians that can encode trained classifier functions in their ground space, and which can be probed by measuring the overlap with test states corresponding to the data to be classified. To obtain such a classifier Hamiltonian, we further propose a training scheme based on quantum annealing which is completely closed-off to the environment and which does not depend on external measurements until the very end, avoiding unnecessary decoherence during the annealing procedure. For a network of size n, the trained network can be stored as a list of O(n) coupling strengths. We address the question of which interactions are most suitable for a given classification task, and develop a qubit-saving optimization for the training procedure on a simulated annealing device. Furthermore, a small neural network to classify colors into red vs. blue is trained and tested, and benchmarked against the annealing parameters.

quant-ph↗

arXiv2018

Size-Driven Quantum Phase Transitions

Can the properties of the thermodynamic limit of a many-body quantum system be extrapolated by analysing a sequence of finite-size cases? We present a model for which such an approach gives completely misleading results: a translationally invariant, local Hamiltonian on a square lattice with open boundary conditions and constant spectral gap, which has a classical product ground state for all system sizes smaller than a particular threshold size, but a ground state with topological degeneracy for all system sizes larger than this threshold. Starting from a minimal case with spins of dimension 6 and threshold lattice size 15 x 15, we show that the latter grows faster than any computable function with increasing local spin dimension. The resulting effect may be viewed as a new type of quantum phase transition that is driven by the size of the system rather than by an external field or coupling strength. We prove that the construction is thermally robust, opening the possibility that these effects are accessible to experimental observation.

quant-ph↗

arXiv2017

The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension

We prove that estimating the ground state energy of a translationally-invariant, nearest-neighbour Hamiltonian on a 1D spin chain is QMAEXP-complete, even for systems of low local dimension (roughly 40). This is an improvement over the best previously-known result by several orders of magnitude, and it shows that spin-glass-like frustration can occur in translationally-invariant quantum systems with a local dimension comparable to the smallest-known non-translationally-invariant systems with similar behaviour. While previous constructions of such systems rely on standard models of quantum computation, we construct a new model that is particularly well-suited for encoding quantum computation into the ground state of a translationally-invariant system. This allows us to shift the proof burden from optimizing the Hamiltonian encoding a standard computational model to proving universality of a simple model. Previous techniques for encoding quantum computation into the ground state of a local Hamiltonian allow only a linear sequence of gates, hence only a linear (or nearly linear) path in the graph of all computational states. We extend these techniques by allowing significantly more general paths, including branching and cycles, thus enabling a highly efficient encoding of our computational model. However, this requires more sophisticated techniques for analysing the spectrum of the resulting Hamiltonian. To address this, we introduce a framework of graphs with unitary edge labels. After relating our Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its spectrum by combining matrix analysis and spectral graph theory techniques.

quant-ph↗

arXiv2017

The Complexity of Translationally-Invariant Low-Dimensional Spin Lattices in 3D

In this paper, we consider spin systems in three spatial dimensions, and prove that the local Hamiltonian problem for 3D lattices with face-centered cubic unit cells, 4-local translationally-invariant interactions between spin-3/2 particles and open boundary conditions is QMAEXP-complete. We go beyond a mere embedding of past hard 1D history state constructions, and utilize a classical Wang tiling problem as binary counter in order to translate one cube side length into a binary description for the verifier input. We further make use of a recently-developed computational model especially well-suited for history state constructions, and combine it with a specific circuit encoding shown to be universal for quantum computation. These novel techniques allow us to significantly lower the local spin dimension, surpassing the best translationally-invariant result to date by two orders of magnitude (in the number of degrees of freedom per coupling). This brings our models en par with the best non-translationally-invariant construction.

quant-ph↗

arXiv2016

The Complexity of Divisibility

We address two sets of long-standing open questions in probability theory, from a computational complexity perspective: divisibility of stochastic maps, and divisibility and decomposability of probability distributions. We prove that finite divisibility of stochastic maps is an NP-complete problem, and extend this result to nonnegative matrices, and completely-positive trace-preserving maps, i.e. the quantum analogue of stochastic maps. We further prove a complexity hierarchy for the divisibility and decomposability of probability distributions, showing that finite distribution divisibility is in P, but decomposability is NP-hard. For the former, we give an explicit polynomial-time algorithm. All results on distributions extend to weak-membership formulations, proving that the complexity of these problems is robust to perturbations.

math.PR↗

arXiv2013

On the Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua

Linear combinations of chi square random variables occur in a wide range of fields. Unfortunately, a closed, analytic expression for the pdf is not yet known. As a first result of this work, an explicit analytic expression for the density of the sum of two gamma random variables is derived. Then a computationally efficient algorithm to numerically calculate the linear combination of chi square random variables is developed. An explicit expression for the error bound is obtained. The proposed technique is shown to be computationally efficient, i.e. only polynomial in growth in the number of terms compared to the exponential growth of most other methods. It provides a vast improvement in accuracy and shows only logarithmic growth in the required precision. In addition, it is applicable to a much greater number of terms and currently the only way of computing the distribution for hundreds of terms. As an application, the exponential dependence of the eigenvalue fluctuation probability of a random matrix model for 4d supergravity with N scalar fields is found to be of the asymptotic form exp(-0.35N).

math.PR↗