Implementing a Fast Unbounded Quantum Fanout Gate Using Power-Law Interactions

Abstract

The standard circuit model for quantum computation presumes the ability to directly perform gates between arbitrary pairs of qubits, which is unlikely to be practical for large-scale experiments. Power-law interactions with strength decaying as 1/r α in the distance r provide an experimentally realizable resource for information processing, whilst still retaining long-range connectivity. We leverage the power of these interactions to implement a fast quantum fanout gate with an arbitrary number of targets. Our implementation allows the quantum Fourier transform (QFT) and Shor's algorithm to be performed on a D-dimensional lattice in time logarithmic in the number of qubits for interactions with α ≤ D. As a corollary, we show that power-law systems with α ≤ D are difficult to simulate classically even for short times, under a standard assumption that factoring is classically intractable. Complementarily, we develop a new technique to give a general lower bound, linear in the size of the system, on the time required to implement the QFT and the fanout gate in systems that are constrained by a linear light cone. This allows us to prove an asymptotically tighter lower bound for long-range systems than is possible with previously available techniques.

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

“…Using quantum state transfer between auxiliary qubits and encoding qubits into large GHZ-like states as subroutines, our protocol leads to optimal implementations of quantum gates between distant qubits in large quantum computers. In particular, the faster encoding of information into a GHZ-like state of ancillary qubits speeds up [7] the implementations of the quantum fanout-a powerful multiqubit quantum gate [46]. At the same time, the faster state transfer speeds up [47] the constructions of multiscale entanglement renormalization ansatz (MERA) states, commonly used to represent highly entangled-including topologically ordered [48]states [49][50][51].…”

“…At the same time, the faster state transfer speeds up [47] the constructions of multiscale entanglement renormalization ansatz (MERA) states, commonly used to represent highly entangled-including topologically ordered [48]states [49][50][51]. Specifically, we can implement a fanout gate [7] on qubits in a hypercube of volume n and prepare a MERA state [47] on these qubits in time t ∼ polylog(n) for α ∈ (d, 2d), t ∼ e γ √ d √ log n for α = 2d-which are both exponential speedups compared to the previous bestand t ∼ n (α−2d)/d for α ∈ (2d, 2d + 1). The optimality of these operations is again guaranteed (up to subpolynomial corrections) by the matching lower limits imposed by the Lieb-Robinson bounds [7,47].…”

“…Specifically, we can implement a fanout gate [7] on qubits in a hypercube of volume n and prepare a MERA state [47] on these qubits in time t ∼ polylog(n) for α ∈ (d, 2d), t ∼ e γ √ d √ log n for α = 2d-which are both exponential speedups compared to the previous bestand t ∼ n (α−2d)/d for α ∈ (2d, 2d + 1). The optimality of these operations is again guaranteed (up to subpolynomial corrections) by the matching lower limits imposed by the Lieb-Robinson bounds [7,47].…”

“…Harnessing entanglement between many particles is key to a quantum advantage in applications including sensing and time-keeping [1,2], secure communication [3], and quantum computing [4,5]. For example, encoding quantum information into a multiqubit Greenberger-Horne-Zeilinger-like (GHZ-like) state is particularly desirable as a subroutine in many quantum applications, including metrology [2], quantum computing [6,7], anonymous quantum communication [8,9], and quantum secret sharing [10].…”

Implementing the fanout operation with simple pairwise interactions

“…More recently, Guo et al [8] presented a method for implementing fanout on a mesh of qubits. Their approach involves a series of modulated long-range Hamiltonians applied to the qubits obeying inverse power laws.…”