Abstract We design quantum compression algorithms for parametric families of tensor network states. We first establish an upper bound on the amount of memory needed to store an arbitrary state from a given state family. The bound is determined by the minimum cut of a suitable flow network, and is related to the flow of information from the manifold of parameters that specify the states to the physical systems in which the states are embodied. For given network topology and given edge dimensions, our upper bound is tight when all edge dimensions are powers of the same integer. When this condition is not met, the bound is optimal up to a multiplicative factor smaller than 1.585. We then provide a compression algorithm for general state families, and show that the algorithm runs in polynomial time for matrix product states.
Abstract We design quantum compression algorithms for parametric families of tensor network states. We first establish an upper bound on the amount of memory needed to store an arbitrary state from a given state family. The bound is determined by the minimum cut of a suitable flow network, and is related to the flow of information from the manifold of parameters that specify the states to the physical systems in which the states are embodied. For given network topology and given edge dimensions, our upper bound is tight when all edge dimensions are powers of the same integer. When this condition is not met, the bound is optimal up to a multiplicative factor smaller than 1.585. We then provide a compression algorithm for general state families, and show that the algorithm runs in polynomial time for matrix product states.
We establish proper agnostic learning of matrix product states and tree tensor networks. Given copies of an arbitrary quantum state ρ, our algorithms return a state ∣ψ⟩ of chosen bond dimension such that ⟨ψ∣ρ∣ψ⟩ is within ε of the optimum over the model class, without assuming that ρ itself belongs to or is well approximated by that class. The main idea is to use improper learning to compress the mixed-state objective, reducing proper agnostic learning to optimization against a finite collection of explicitly specified pure states. We then introduce a comparator-dual compression procedure that reduces the bond dimension of these targets while uniformly preserving their overlaps with all bounded-bond comparators, with an error independent of system size. For matrix product states, this gives polynomial copy complexity in the system size, local dimension, bond dimension, and 1/ε, together with polynomial runtime in the system size for fixed local dimension, bond dimension, and accuracy. The same framework yields proper agnostic learning of tree tensor networks on bounded-degree trees, with both copy complexity and runtime polynomial in the system size when the remaining parameters are fixed.
High performance computers produce extreme-scale data sets that require sampling or compression if they are to be used to their full potential. Existing data compression techniques typically exploit features such as sparsity in the data, homogeneity in the data, or {\it a priori} knowledge of what subsets of data are of most interest. Fluid dynamics data in general do not exhibit these features and so are attractive test beds for generic compression techniques that are objective, robust, and tuneable with respect to information lost due to compression. Presented here is a method based on tensor networks, specifically matrix product states or tensor trains, that meets these requirements. The method is demonstrated for compression in one-dimension and is extensible to higher dimensionality. Lossless compression is demonstrated for random Fourier series for sufficiently high bond dimension of the tensor network, with the memory required to store the tensor network scaling directly proportional to the bond dimension. The lossy compression exhibited at lower bond dimension can be well within the relative error of many fluid simulations. The compression algorithm is tested for the time evolution of Burger's equation with excellent results. We additionally demonstrate the capability to perform computations in the compressed form through a tensor network periodic convolution that can be orders of magnitude faster than using fast Fourier transforms and the convolution theorem. In addition to being an attractive method for working with data sets generated by existing computers, the tensor network methods utilised are directly translatable to the emerging paradigm of quantum computing.
Identical preparation creates permutation symmetry that can be used for lossless quantum compression. For n copies of an unknown d-dimensional pure state, the tensor-power input lies in the fully symmetric subspace, whose dimension is polynomial in n for fixed local dimension. Schur--Weyl duality isolates this subspace coherently, allowing the fixed representation labels to be discarded while retaining all information needed for recovery. The resulting encoder and decoder are exact on the input family. Because tensor-power states span the symmetric subspace, the achieved memory dimension is also necessary for reversible coherent compression, so the scheme is space-optimal. The encoder is implemented recursively with Clebsch--Gordan transforms, and only the branches reached by symmetric inputs need to be reproduced. Standard efficient synthesis of these transforms yields polynomial-size circuits for fixed local dimension and arbitrary target accuracy. Thus identically prepared pure states in arbitrary dimension admit lossless, space-optimal compression with an efficient circuit implementation.
Predicting a stochastic process' future lies at the heart of many scientific areas.A predictive model extracts information from a stochastic process' past and uses it to generate future statistics.There has been significant amount of effort expended towards finding optimal predictive models that minimize the required amount of past information.By taking advantage of quantum resources, quantum models have been shown to reduce the memory requirements beyond classical limits.Meanwhile, tensor networks are an extremely useful mathematical tool for understanding quantum many-body systems.In this thesis, we explore the connection between tensor networks and quantum predictive models.A particular class of tensor networks, matrix product states (MPS), is utilized to further improve quantum models.First, we establish a connection between matrix product states (MPS) and the optimal classical predictive models.MPS methods offer a systematic method for constructing quantum predictive models and even an improved method for computing the amount of quantum memory.Second, we show that for some families of stochastic processes our method allows us to construct quantum models with unbounded memory advantages over the optimal classical models.Third, we propose a family of divergence measures to quantify the distance between two stochastic processes.This family of divergence measures overcomes certain weaknesses of the existing measures.Moreover, we propose an efficient means of computing the divergence measure using our MPS methods.Finally, we address the problem of constructing quantum predictive models solely from the output of an observed stochastic process.We use machine learning methods to propose a systematic algorithm for learning the quantum model.This provides an alternative method for constructing quantum models with fixed amounts of amount of memory.
We study the compression of arbitrary parametric families of n identically prepared finite-dimensional quantum states, in a setting that can be regarded as a quantum analogue of population coding. For a family with f free parameters, we propose an asymptotically faithful protocol that requires a memory of overall size (f/2) log n. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of the optimal compression of n identically prepared displaced thermal states. Our protocol achieves the ultimate bound predicted by quantum Shannon theory. In addition, we explore the minimum requirement for quantum memory: On the one hand, the amount of quantum memory used by our protocol can be made arbitrarily small compared to the overall memory cost; on the other hand, any protocol using only classical memory cannot be faithful.
We study the compression of n quantum systems, each prepared in the same state belonging to a given parametric family of quantum states. For a family of states with f independent parameters, we devise an asymptotically faithful protocol that requires a hybrid memory of size (f/2)\log n, including both quantum and classical bits. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of compressing displaced thermal states of n identically prepared modes. In both cases, we show that (f/2)\log n is the minimum amount of memory needed to achieve asymptotic faithfulness. In addition, we analyze how much of the memory needs to be quantum. We find that the ratio between quantum and classical bits can be made arbitrarily small, but cannot reach zero: unless all the quantum states in the family commute, no protocol using only classical bits can be faithful, even if it uses an arbitrarily large number of classical bits.