Contracting projected entangled pair states is average-case hard

Haferkamp, Jonas; Hangleiter, Dominik; Eisert, Jens; Gluza, Marek

doi:10.1103/physrevresearch.2.013010

Cited by 47 publications

(20 citation statements)

References 90 publications

(89 reference statements)

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“…PEPS intuitively reproduces the structure of the lattice with one tensor for each physical site and the bond indices directly follow the lattice grid. The resulting TN follows the area-law of entanglement but it contains loops, making the contractions for computing expectation values exponentially hard 125 . Furthermore, the computational cost for performing the variational optimization of PEPS, as for instance in the ground state searching, scales as O(χ 10 ) as a function of the bond dimension.…”

Section: Methodsmentioning

confidence: 99%

Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks

et al. 2021

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Gauge theories are of paramount importance in our understanding of fundamental constituents of matter and their interactions. However, the complete characterization of their phase diagrams and the full understanding of non-perturbative effects are still debated, especially at finite charge density, mostly due to the sign-problem affecting Monte Carlo numerical simulations. Here, we report the Tensor Network simulation of a three dimensional lattice gauge theory in the Hamiltonian formulation including dynamical matter: Using this sign-problem-free method, we simulate the ground states of a compact Quantum Electrodynamics at zero and finite charge densities, and address fundamental questions such as the characterization of collective phases of the model, the presence of a confining phase at large gauge coupling, and the study of charge-screening effects.

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Section: Methodsmentioning

confidence: 99%

Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks

et al. 2021

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“…Specifically, it will be worth exploring if the states in 2D noisy circuits can be faithfully represented by a PEPO with a constant bond dimension that depends only on the gate error rate p. However, unlike in the case of MPO, exactly computing an observable from a PEPO is not feasible because exactly contracting PEPSs is #P complete in the worst case [93] and in the average case [94]. Nevertheless, these hardness results do not immediately rule out the possibility of efficient and approximate simulation of 2D noisy RCS because it may be possible to efficiently contract the output PEPOs approximately.…”

Section: Summary and Open Questionsmentioning

confidence: 99%

Efficient classical simulation of noisy random quantum circuits in one dimension

Noh

Jiang

Fefferman

2020

Quantum

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Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers.

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“…[31], we can prepare tensor network representations for arbitrary Gibbs states in the polynomial time of n OðβÞ . However, the classical simulation of the tensor network is #P complete problem [67,68] except in 1D cases. To the best of our knowledge, our result, for the first time, provides the fully-polynomial-time approximation scheme (FPTAS [69]) for the classical simulation of quantum Gibbs states, which is a quantum generalization of the FPTAS for classical Gibbs states [70][71][72].…”

mentioning

confidence: 99%

Clustering of Conditional Mutual Information for Quantum Gibbs States above a Threshold Temperature

2020

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We prove that the quantum Gibbs states of spin systems above a certain threshold temperature are approximate quantum Markov networks, meaning that the conditional mutual information decays rapidly with distance. We demonstrate the exponential decay for short-ranged interacting systems and power-law decay for long-ranged interacting systems. Consequently, we establish the efficiency of quantum Gibbs sampling algorithms, a strong version of the area law, the quasilocality of effective Hamiltonians on subsystems, a clustering theorem for mutual information, and a polynomial-time algorithm for classical Gibbs state simulations.

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Contracting projected entangled pair states is average-case hard

Cited by 47 publications

References 90 publications

Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks

Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks

Efficient classical simulation of noisy random quantum circuits in one dimension

Clustering of Conditional Mutual Information for Quantum Gibbs States above a Threshold Temperature

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