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arXiv:1711.07500

Robust entanglement renormalization on a noisy quantum computer

Isaac H. Kim, Brian G. Swingle

Published Nov 20, 2017• 29 citations
arXivHTMLPDF

Abstract

A method to study strongly interacting quantum many-body systems at and away from criticality is proposed. The method is based on a MERA-like tensor network that can be efficiently and reliably contracted on a noisy quantum computer using a number of qubits that is much smaller than the system size. We prove that the outcome of the contraction is stable to noise and that the estimated energy upper bounds the ground state energy. The stability, which we numerically substantiate, follows from the positivity of operator scaling dimensions under renormalization group flow. The variational upper bound follows from a particular assignment of physical qubits to different locations of the tensor network plus the assumption that the noise model is local. We postulate a scaling law for how well the tensor network can approximate ground states of lattice regulated conformal field theories in d spatial dimensions and provide evidence for the postulate. Under this postulate, a $O(\log^{d}(1/δ))$-qubit quantum computer can prepare a valid quantum-mechanical state with energy density $δ$ above the ground state. In the presence of noise, $δ= O(ε\log^{d+1}(1/ε))$ can be achieved, where $ε$ is the noise strength.

DOI
10.48550/arxiv.1711.07500

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A method to study strongly interacting quantum many-body systems at and away from criticality is proposed. The method is based on a MERA-like tensor network that can be efficiently and reliably contracted on a noisy quantum computer using a number of qubits that is much smaller than the system size. We prove that the outcome of the contraction is stable to noise and that the estimated energy upper bounds the ground state energy. The stability, which we numerically substantiate, follows from the positivity of operator scaling dimensions under renormalization group flow. The variational upper bound follows from a particular assignment of physical qubits to different locations of the tensor network plus the assumption that the noise model is local. We postulate a scaling law for how well the tensor network can approximate ground states of lattice regulated conformal field theories in d spatial dimensions and provide evidence for the postulate. Under this postulate, a O(log⁡d(1/δ))O(\log^{d}(1/δ))O(logd(1/δ))-qubit quantum computer can prepare a valid quantum-mechanical state with energy density δδδ above the ground state. In the presence of noise, δ=O(εlog⁡d+1(1/ε))δ= O(ε\log^{d+1}(1/ε))δ=O(εlogd+1(1/ε)) can be achieved, where εεε is the noise strength.
Abstract We describe a resource-efficient approach to studying many-body quantum states on noisy, intermediate-scale quantum devices. We employ a sequential generation model that allows us to bound the range of correlations in the resulting many-body quantum states. From this, we characterize situations where the estimation of local observables does not require the preparation of the entire state. Instead smaller patches of the state can be generated from which the observables can be estimated. This can potentially reduce circuit size and number of qubits for the computation of physical properties of the states. Moreover, we show that the effect of noise decreases along the computation. Our results apply to a broad class of widely studied tensor network states and can be directly applied to near-term implementations of variational quantum algorithms.
Strongly correlated quantum matter, such as the insufficiently understood high- temperature superconductors, are difficult to simulate numerically. The multiscale entanglement renormalization ansatz (MERA) is a hierarchical class of tensor networks that efficiently captures the typical ground-state entanglement structures. Motivated by recent advances to employ MERA in quantum-classical algorithms, the authors take a fresh look here, analyzing new variations of the approach including tensor Trotterization and importance sampling. Optimal contraction sequences and algorithmic phase diagrams show that these quantum-inspired methods can substantially reduce computational costs.
We aim to rigorously address how local noise affects classical simulability of quantum dynamics benchmarked by tensor-network methods. Using operator entanglement entropy (OEE), we prove the following: (1) For single-qubit depolarizing noise on arbitrary circuits, tensor networks with poly(n)\mathrm{poly}(n)poly(n) bond dimension suffice for fixed absolute Hilbert-Schmidt error after $\order{1}$ depth, while relative error demands $\order{\log n}$ depth; and this bound is optimal. (2) For single-qubit depolarizing noise on 1D local circuits, the existence of whole-trajectory error-bounded matrix product operator (MPO) of poly(n)\mathrm{poly}(n)poly(n) bond dimension at all depths. (3) For general single-qubit noise in 1D brickwall circuits, random two-design gates with contraction coefficient c<1/3c<1/3c<1/3 yield an $\order{1}$ OEE plateau with probability 1−Te−Ω(n)1-Te^{-Ω(n)}1−Te−Ω(n), while arbitrary gates with c<1/48c<1/48c<1/48 give $\order{\log n}$ OEE in the worst case. (4) In higher dimensions, these bounds yield uniform-in-depth poly(n)\mathrm{poly}(n)poly(n) average boundary-bond dimensions for projected entangled pair operators~(PEPO) across every cut -- under depolarizing noise at either absolute or relative accuracy, and under general noise with strong contraction at absolute accuracy. Our results establish a rigorous connection between certain noise models, circuit types, and their classical simulability.
Tensor network states have proven to be powerful tools for investigating quantum many-body systems. Here, the authors propose a tensor network method, based on an adaptation of entanglement renormalization, for investigating strongly disordered systems. The method makes use of the strong disorder renormalization group to determine the order in which lattice sites are coarse grained, which sets the overall structure of the corresponding tensor network ansatz, before optimization using variational energy minimization. This approach leads to a new class of efficiently contractible tensor network ansatz for 1D systems, which may be understood as a generalization of the multiscale entanglement renormalization ansatz for disordered systems.
In this thesis, we use the new numerical method multiscale entanglement renormalization ansatz (MERA) to study the one dimensional spin system. This method, MERA, utilizes a network of tensors to represent quantum many-body systems on a lattice. Other examples of tensor network state are matrix product states (MPSs) for 1D systems, tree tensor networks (TTNs) for systems with a tree shape, and projected entangled-pair states (PEPSs) for 2D systems and be- yond. The three structures dier in the graph that denes how the tensors are interconnected into a network: The graphs for MPSs, TTNs, and 2D PEPSs are, respectively, a chain, a tree, and a 2D lattice. Importantly, from these tensor networks the expectation value of local observables can be computed eciently. Developing algorithms to simulate quantum many-body systems is important for understanding and studying the physical principles. And these may become powerful tools for the analysis of quantum computation and quantum information theory. But in general ecient simulation is dicult, due to the fundamental quantum physical laws. The main method we have studied and worked on is multiscale entangle- ment renormalization ansatz (MERA) which is a variational ansatz for many- body states. MERA is a tree tensor network for 1D, 2D systems and beyond. We have developed code of ternary-MERA to compute the ground state energy, expected values of local observables, and correlators for 1D lattice systems. In a nutshell, MERA is an algorithm for entanglement renormalization which is a numerical technique based on locally reorganizing the Hilbert space of a quantum many-body system with the aim to reduce the amount of entanglement in its wave function. We use this method to study 1D Ising model, XXZ model, and J1 - J2 model, and then comparing the numerical results with exact diagonalization (ED) and density matrix renormalization group (DMRG).
Quantum crosstalk poses a major challenge to scaling up quantum computations as its strength is typically unknown and its effect accumulates exponentially as system size grows. Here, we show that many-body robust control can be utilized to suppress unwanted couplings during multi-qubit gate operations and state preparation. By combining tensor network simulations with the GRAPE algorithm, and leveraging an efficient random sampling over noise ensembles, our method overcomes the exponential scaling of the Hilbert space. We demonstrate its effectiveness for designing control solutions for high-fidelity implementations of parallel X gates and parallel CNOT on a chain of 50 qubits, and for realizing a 30-qubit GHZ state and the ground state of a 20-qubit Heisenberg model. In the presence of many-body quantum crosstalk due to parasitic interaction between neighboring qubits, robust control results in order-of magnitude improvement in fidelity for large system sizes. These findings pave the way for more reliable operations on near-term quantum processors.