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arXiv:1912.07780 · Physical Review A

Optimized fast gates for quantum computing with trapped ions

Evan P. G. Gale, Zain Mehdi, Lachlan M. Oberg, Alexander K. Ratcliffe, Simon A. Haine, J. J. Hope

Published May 18, 2020• 30 citations
arXivHTMLPDF

Abstract

We present an efficient approach to optimizing pulse sequences for implementing fast entangling two-qubit gates on trapped ion quantum information processors. We employ a two-phase procedure for optimizing gate fidelity, which we demonstrate for multi-ion systems in linear Paul trap and microtrap architectures. The first phase involves a global optimization over a computationally inexpensive cost function constructed under strong approximations of the gate dynamics. The second phase involves local optimizations that utilize a more precise ordinary differential equation description of the gate dynamics, which captures the nonlinearity of the Coulomb interaction and the effects of finite laser repetition rate. We propose two gate schemes that are compatible with this approach, and we demonstrate that they outperform existing schemes in terms of achievable gate speed and fidelity for feasible laser repetition rates. In optimizing sub-microsecond gates in microtrap architectures, the proposed schemes achieve orders-of-magnitude-higher fidelities than previous proposals. Finally, we investigate the impact of pulse imperfections on gate fidelity and evaluate error bounds for a range of gate speeds.

DOI
10.1103/physreva.101.052328

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We present an efficient approach to optimizing pulse sequences for implementing fast entangling two-qubit gates on trapped ion quantum information processors. We employ a two-phase procedure for optimizing gate fidelity, which we demonstrate for multi-ion systems in linear Paul trap and microtrap architectures. The first phase involves a global optimization over a computationally inexpensive cost function constructed under strong approximations of the gate dynamics. The second phase involves local optimizations that utilize a more precise ordinary differential equation description of the gate dynamics, which captures the nonlinearity of the Coulomb interaction and the effects of finite laser repetition rate. We propose two gate schemes that are compatible with this approach, and we demonstrate that they outperform existing schemes in terms of achievable gate speed and fidelity for feasible laser repetition rates. In optimizing sub-microsecond gates in microtrap architectures, the proposed schemes achieve orders-of-magnitude-higher fidelities than previous proposals. Finally, we investigate the impact of pulse imperfections on gate fidelity and evaluate error bounds for a range of gate speeds.
Non-adiabatic two-qubit gate proposals for trapped-ion systems offer superior performance and flexibility over adiabatic schemes at the cost of increased laser control requirements. Existing fast gate schemes are limited by single-qubit transition errors, which constrain the total number of pulses in high-fidelity solutions. We introduce an improved gate search scheme that enables both local and non-local two-qubit gates in chains containing tens of ions. These protocols use a multi-objective machine design approach that incorporates dominant sources of error in the design to ensure the solutions are compatible with existing fast laser controls. We also generalize previous schemes by allowing for unpaired pulses during the gate evolution. By imposing symmetries on the pulse sequences, we eliminate susceptibility to laser phase noise and further simplify the multi-mode control over the state-dependent motion of the ion crystal. We perform a comprehensive analysis of expected gate performance in the presence of random and systematic experimental errors to demonstrate the feasibility of performing microsecond two-qubit gates between arbitrary ion pairs in current linear ion-trap processors of up to 505050 ions with fidelities approaching 99.9%99.9\%99.9%.
We theoretically investigate the use of fast pulsed two-qubit gates for trapped ion quantum computing in a two-dimensional microtrap architecture. In one dimension, such fast gates are optimal when employed between nearest neighbors, and we examine the generalization to a two-dimensional geometry. We demonstrate that fast pulsed gates are capable of implementing high-fidelity entangling operations between ions in neighboring traps faster than the trapping period, with experimentally demonstrated laser repetition rates. Notably, we find that without increasing the gate duration high-fidelity gates are achievable even in large arrays with hundreds of ions. To demonstrate the usefulness of this proposal, we investigate the application of these gates to the digital simulation of a 40-mode Fermi-Hubbard model. This also demonstrates why shorter chains of gates required to connect arbitrary pairs of ions make this geometry well suited for large-scale computation.
Realising large-scale quantum computation in the near future will require increasing the number of low-error two-qubit gates that can be implemented on a quantum computer before decoherence. One of the biggest challenges facing current trapped ion quantum computers is implementing high-speed two-qubit operations, whilst increasing the number of qubits. One of the most promising proposals for overcoming current limitations is the use of ultra-fast pulses to implement fast two-qubit gates between nearest-neighbour pairs of ions. In this thesis, I investigate these ‘fast gates’ in two-dimensional arrays of microtraps, each containing a single ion. I argue that two-dimensional architectures allow for a significant reduction in the number of two-qubit gates required for a particular computation, as compared to one-dimensional ion chains. I demonstrate this reduction for a quantum simulation of a 40-mode Fermi-Hubbard Hamiltonian. I develop an efficient scheme that allows fast gates to be numerically optimised for two-dimensional geometries. I find that this optimisation scheme is capable of designing gates that are faster, higher fidelity, and require lower laser repetition rates. Using this scheme, I find that high-speed two-qubit gates can be optimised for two-dimensional architectures, with fidelities well above thresholds required for fault tolerant error correction, around 99.99%. Furthermore, I find that fast gates in these architectures are robust to the presence of large numbers of surrounding ions. Following previous studies [1, 2] which have identified pulse imperfections as a dominant source of error in fast gates, I perform a worst-case error analysis. I find the fast gates presented in this thesis to require very small errors in single-qubit rotations, and I present recommendations for achieving those requirements. I also investigate other experimental considerations, and make recommendations for overcoming other technical challenges in realising fast gates.
This thesis describes schemes for both a fast two-qubit gate operation and the steady-state preparation of a Bell state with trapped ions. A critical figure of merit for quantum computing with trapped ions is the gate duration relative to the decoherence timescale. We propose a fast gate scheme that off ers improvements in time, fidelity and simplicity of implementation over existing fast gate proposals. Our scheme can operate on both neighbouring and non-neighbouring ions in a long ion crystal. This provides a simpler and faster mechanism than traditional gates for complex quantum computing operations on large numbers of ions. The scheme achieves fidelities well above quantum error correction thresholds around 0.0001, and operates arbitrarily fast given arbitrary laser repetition rates. The production of these ultra-fast pulses is an experimental challenge, and fast gates have not yet been implemented; we present an implementation scheme using pulse splitting to provide a higher repetition rate and the pulse timing freedoms required for the gate scheme. We also analyse the effects of errors in the pulses on the gate operation. We analyse another strategy to generate entanglement using a driven dissipative process. Typically, environmental couplings cause decoherence. However, by combining dissipative dynamics with suitably chosen Hamiltonian evolution, the system can be steered to the desired steady states. Our steady-state scheme prepares a maximally-entangled Bell state with fidelity above 0.99, much higher than for schemes implemented with trapped ions. The driven dissipation continuously pumps the system towards the antisymmetric Bell steady-state, which is dark to the system dynamics and robust to parameter variations. The dominant loss mechanism is anomalous heating of the motional modes, reducing our fidelity by less than 0.01 for current experimental rates. Our scheme jointly addresses the ions and does not use sympathetic cooling. We enhance our scheme by combining the dissipative state preparation with the detection of photons, and obtain a significant fidelity enhancement.
Fast entangling gates for trapped ion pairs offer vastly improved gate operation times relative to implemented gates, as well as approaches to trap scaling. Gates on a neighbouring ion pair only involve local ions when performed sufficiently fast, and we find that even a fast gate between a pair of distant ions with few degrees of freedom restores all the motional modes given more stringent gate speed conditions. We compare pulsed fast gate schemes, defined by a timescale faster than the trap period, and find that our proposed scheme has less stringent requirements on laser repetition rate for achieving arbitrary gate time targets and infidelities well below 10 −4 . By extending gate schemes to ion crystals, we explore the effect of ion number on gate fidelity for coupling two neighbouring ions in large crystals. Inter-ion distance determines the gate time, and a factor of five increase in repetition rate, or correspondingly the laser power, reduces the infidelity by almost two orders of magnitude. We also apply our fast gate scheme to entangle the first and last ions in a crystal. As the number of ions in the crystal increases, significant increases in the laser power are required to provide the short gate times corresponding to fidelity above 0.99.
Abstract We propose and study ways speeding up of the entangling operations in the trapped ions system with high fidelity. First, we find a scheme to increase the speed of a two-qubit gate without the limitation of trap frequency, which was considered as the fundamental limit. Second, we study the fast gate scheme for entangling more than two qubits simultaneously. We apply the method of applying multiple frequency components on laser beams for the gate operations. In particular, in order to avoid infinite terms from the coupling to carrier transition, we focus on the phase-insensitive gate scheme here. We carefully study the effect of large excitation of motional mode beyond the limit of Lamb–Dicke approximation by including up to second order terms of the Lamb–Dicke parameter. We study the speed limit of multi-qubit global entangling gates without individual addressing requirements. Furthermore, our gates can be made insensitive to the fluctuation of initial motional phases which are difficult to stabilise in the phase-insensitive gate scheme.