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arXiv:2010.02930 · Physical Review X

Optimal State Transfer and Entanglement Generation in Power-Law Interacting Systems

Minh C. Tran, Andrew Y. Guo, Abhinav Deshpande, Andrew Lucas, Alexey Vyacheslavovich Gorshkov

Published Jul 21, 2021• 41 citations
arXivHTMLPDF

Abstract

We present an optimal protocol for encoding an unknown qubit state into a multiqubit Greenberger-Horne-Zeilinger-like state and, consequently, transferring quantum information in large systems exhibiting power-law (1=r α ) interactions.For all power-law exponents α between d and 2d þ 1, where d is the dimension of the system, the protocol yields a polynomial speed-up for α > 2d and a superpolynomial speed-up for α ≤ 2d, compared to the state of the art.For all α > d, the protocol saturates the Lieb-Robinson bounds (up to subpolynomial corrections), thereby establishing the optimality of the protocol and the tightness of the bounds in this regime.The protocol has a wide range of applications, including in quantum sensing, quantum computing, and preparation of topologically ordered states.In addition, the protocol provides a lower bound on the gate count in digital simulations of power-law interacting systems.

DOI
10.1103/physrevx.11.031016

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We present an optimal protocol for encoding an unknown qubit state into a multiqubit Greenberger-Horne-Zeilinger-like state and, consequently, transferring quantum information in large systems exhibiting power-law (1=r α ) interactions.For all power-law exponents α between d and 2d þ 1, where d is the dimension of the system, the protocol yields a polynomial speed-up for α > 2d and a superpolynomial speed-up for α ≤ 2d, compared to the state of the art.For all α > d, the protocol saturates the Lieb-Robinson bounds (up to subpolynomial corrections), thereby establishing the optimality of the protocol and the tightness of the bounds in this regime.The protocol has a wide range of applications, including in quantum sensing, quantum computing, and preparation of topologically ordered states.In addition, the protocol provides a lower bound on the gate count in digital simulations of power-law interacting systems.
In short-range interacting systems, the speed at which entanglement can be established between two separated points is limited by a constant Lieb-Robinson velocity. Long-range interacting systems are capable of faster entanglement generation, but the degree of the speedup possible is an open question. In this Letter, we present a protocol capable of transferring a quantum state across a distance L in d dimensions using long-range interactions with a strength bounded by 1/r^{α}. If α<d, the state transfer time is asymptotically independent of L; if α=d, the time scales logarithmically with the distance L; if d<α<d+1, the transfer occurs in a time proportional to L^{α-d}; and if α≥d+1, it occurs in a time proportional to L. We then use this protocol to upper bound the time required to create a state specified by a multiscale entanglement renormalization ansatz (MERA) tensor network and show that if the linear size of the MERA state is L, then it can be created in a time that scales with L identically to the state transfer up to logarithmic corrections. This protocol realizes an exponential speedup in cases of α=d, which could be useful in creating large entangled states for dipole-dipole (1/r^{3}) interactions in three dimensions.
The N-qubit Greenberger-Horne-Zeilinger (GHZ) state is an important resource for quantum technologies. We consider the task of GHZ encoding using all-to-all interactions, which prepares the GHZ state in a special case, and is furthermore useful for quantum error correction, interaction-rate enhancement, and transmitting information using power-law interactions. The naive protocol based on parallelizing cnot gates takes O(1)-time of Hamiltonian evolution. In this work, we propose a fast protocol that achieves GHZ encoding with high accuracy. The evolution time O(log^{2}N/N) almost saturates the theoretical limit Ω(logN/N). Moreover, the final state is close to the ideal encoded one with high fidelity >1-10^{-3}, up to large system sizes N≲2000. The protocol only requires a few stages of time-independent Hamiltonian evolution; the key idea is to use the data qubit as control, and to use fast spin-squeezing dynamics generated by e.g., two-axis twisting.
We investigate the fundamental time complexity, as constrained by Lieb-Robinson bounds, for preparing entangled states useful in quantum metrology. We relate the minimum time to the Quantum Fisher Information (FQF_QFQ​) for a system of NNN quantum spins on a ddd-dimensional lattice with 1/rα1/r^α1/rα interactions with rrr being the distance between two interacting spins. We focus on states with FQ∼N1+γF_Q \sim N^{1+γ}FQ​∼N1+γ where γ∈(0,1]γ\in (0,1]γ∈(0,1], i.e., scaling from the standard quantum limit to the Heisenberg limit. For short-range interactions (α>2d+1α> 2d+1α>2d+1), we prove the minimum time ttt scales as t≳Lγt \gtrsim L^γt≳Lγ, where L∼N1/dL \sim N^{1/d}L∼N1/d. For long-range interactions, we find a hierarchy of possible speedups: t≳Lγ(α−2d)t \gtrsim L^{γ(α-2d)}t≳Lγ(α−2d) for 2d<α<2d+12d < α< 2d+12d<α<2d+1, t≳log⁡Lt \gtrsim \log Lt≳logL for (2−γ)d<α<2d(2-γ)d < α< 2d(2−γ)d<α<2d, and ttt may even vanish algebraically in 1/L1/L1/L for α<(2−γ)dα< (2-γ)dα<(2−γ)d. These bounds extend to the minimum circuit depth required for state preparation, assuming two-qubit gate speeds scale as 1/rα1/r^α1/rα. We further show that these bounds are saturable, up to sub-polynomial corrections, for all ααα at the Heisenberg limit (γ=1γ=1γ=1) and for α>(2−γ)dα> (2-γ)dα>(2−γ)d when γ<1γ<1γ<1. Our results establish a benchmark for the time-optimality of protocols that prepare metrologically useful quantum states.
In this thesis, we focus on the questions of how quantum entanglement can be generated between two or more spatially separated systems and, once generated, how it can be applied in quantum technology. First we will discuss a protocol, which we conjecture to be optimal in some regimes, that quickly creates entangled states across long distances in systems with power-law interactions. We will discuss how this protocol compares with currently known bounds on entangled state generation and how it might be implemented in a three-dimensional lattice of Rydberg atoms. Next, we will turn our attention to more general questions of how the Lieb-Robinson bound and other limitations on entanglement can be used to inform the design of quantum computers. Quantum computers will be required to create entanglement if they are to realize significant advantages over classical computers, meaning that the generation of entanglement is an important question. First, we will discuss the implications of the Lieb-Robinson bound on graph descriptions of quantum computer architectures, and how the relevant graph parameter (diameter) compares to likely cost functions for architectures, such as maximum graph degree and total number of necessary connections. We will present a proposed graph architecture, the hierarchical product, which we believe provides excellent balance between these considerations. We will then introduce new methods of evaluating graphs that allow us to include quantum architectures capable of measurement and feedback operations. After doing so, we will show that the generation of entanglement entropy becomes a limit on computation. We will show that, for several possible physical models of computation, the generation of entanglement can be bounded by simple graph properties. We demonstrate a connection between worst-case scenarios for entanglement generation and a graph quantity called the Cheeger constant or isoperimetric number, which we evaluate for several proposed quantum computing architectures. Finally, we will look at the scenario of quantum sensing. In particular, we will examine protocols for quantum function estimation, where quantum sensors are available to measure all of the inputs to the function. We will demonstrate that entangled sensors are more capable than non-entangled ones by first deriving a new lower bound on measurement error and then presenting protocols that saturate these bounds. We will first do so for linear functions of the measured quantities and then extend this to general functions using a two-step linearizing protocol.
Abstract The speed limit of quantum state transfer (QST) in a system of interacting particles is not only important for quantum information processing, but also directly linked to Lieb–Robinson-type bounds that are crucial for understanding various aspects of quantum many-body physics. For strongly long-range interacting systems such as a fully-connected quantum computer, such a speed limit is still unknown. Here we develop a new quantum brachistochrone method that can incorporate inequality constraints on the Hamiltonian. This method allows us to prove an exactly tight bound on the speed of QST on a subclass of Hamiltonians experimentally realizable by a fully-connected quantum computer.