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arXiv:2004.12810 · Physical Review Research

High-fidelity composite quantum gates for Raman qubits

Boyan T. Torosov, Nikolay V. Vitanov

Published Nov 6, 2020• 17 citations
arXivHTMLPDF

Abstract

We present a general systematic approach to design robust and high-fidelity quantum logic gates with Raman qubits using the technique of composite pulses.We use two mathematical tools-the Morris-Shore and Majorana decompositions-to reduce the three-state Raman system to an equivalent two-state system.They allow us to exploit the numerous composite pulses designed for two-state systems by extending them to Raman qubits.We construct the NOT, Hadamard, and rotation gates by means of the Morris-Shore transformation with the same uniform approach: sequences of pulses with the same phases for each gate but different ratios of Raman couplings.The phase gate is constructed by using the Majorana decomposition.All composite Raman gates feature very high fidelity, well beyond the quantum computation benchmark values, and significant robustness to experimental errors.All composite phases and pulse areas are given by analytical formulas, which makes the method scalable to any desired accuracy and robustness to errors.

DOI
10.1103/physrevresearch.2.043194

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We present a general systematic approach to design robust and high-fidelity quantum logic gates with Raman qubits using the technique of composite pulses.We use two mathematical tools-the Morris-Shore and Majorana decompositions-to reduce the three-state Raman system to an equivalent two-state system.They allow us to exploit the numerous composite pulses designed for two-state systems by extending them to Raman qubits.We construct the NOT, Hadamard, and rotation gates by means of the Morris-Shore transformation with the same uniform approach: sequences of pulses with the same phases for each gate but different ratios of Raman couplings.The phase gate is constructed by using the Majorana decomposition.All composite Raman gates feature very high fidelity, well beyond the quantum computation benchmark values, and significant robustness to experimental errors.All composite phases and pulse areas are given by analytical formulas, which makes the method scalable to any desired accuracy and robustness to errors.
We present a method to construct high-fidelity quantum phase gates, which are insensitive to errors in various experimental parameters. The phase gates consist of a pair of two sequential broadband composite pulses, with a phase difference $\ensuremath{\pi}+\ensuremath{\alpha}/2$ between them, where $\ensuremath{\alpha}$ is the desired gate phase. By using composite pulses which compensate systematic errors in the pulse area, the frequency detuning, or both the area and the detuning, we thereby construct composite phase gates which compensate errors in the same parameters. Particularly interesting are phase gates which use the recently discovered universal composite pulses, which compensate systematic errors in any parameter of the driving field, which keep the evolution Hermitian (e.g., pulse amplitude and duration, pulse shape, frequency detuning, Stark shifts, residual frequency chirps, etc.).
Systematic control errors remain a primary obstacle to realizing high-fidelity single-qubit gates. We introduce composite pulse sequences that implement X and Hadamard gates while simultaneously compensating amplitude (Rabi-frequency), detuning (frequency), and duration errors. Our construction uses two complementary strategies: (i) derivative-based cancellation of error terms in the full unitary (not just the transition probability), formulated via the Cayley-Klein parametrization, and (ii) direct minimization of the average gate infidelity over prescribed error ranges. We derive symmetric five-pulse solutions with closed-form phases that cancel all first-order terms (including the mixed derivative), and numerically optimize longer sequences -- up to 15 pulses -- to achieve higher-order suppression. We also show that standard ``universal'' five-pulse sequences (U5a/U5b) emerge as simple phase-shifted instances of our symmetric solutions, yielding broad robustness to both detuning and amplitude errors. Finally, we construct variable-area sequences for Rx(π/2)R_x(π/2)Rx​(π/2), which, up to virtual Z rotations, benchmark the Hadamard gate. Across all families we observe the expected trade-off between sequence length and robustness window, with substantial boosts in fidelity over large error domains.
Raman qubits, represented by two ground or metastable quantum states coupled via an intermediate state, hold some advantages over directly coupled qubits, most notably much longer radiative lifetimes, shorter gate duration, and lower radiation intensity due to using electric-dipole allowed optical transitions. They are also relatively simple to implement and control, making them an attractive option for building quantum gates for quantum computers. In this work, we present a simple and fast tomographic method to measure the errors of Raman qubit gates possessing the Morris-Shore dynamic symmetry. The latter occurs when the qubit states are on two-photon resonance and the driving fields have the same time dependence. The method is based on repeating the same gate multiple times, which amplifies the small coherent errors to sufficiently large values, which can be measured with high accuracy and precision. Then the (small) gate errors can be determined from the amplified errors by using the analytical connections between them.
Composite pulses are an efficient tool for robust quantum control. In this work, we derive the form of the composite pulse sequence to implement robust single-qubit gates in a three-level system, where two low-energy levels act as a qubit. The composite pulses can efficiently cancel the systematic errors up to a certain order. We find that the three-pulse sequence cannot completely eliminate the first order of systematic errors, but still availably makes the fidelity resistant to variations in a specific direction. When employing more pulses in the sequence (N>3N>3N>3), the fidelity can be insensitive to the variations in all directions and the robustness region becomes much wider. Finally we demonstrate the applications of composite pulses in quantum information processing, e.g., robust quantum information transfer between two qubits.
We introduce a technique for derivation of high-fidelity composite pulse sequences for two types of multistate quantum systems: systems with the SU(2) and Morris-Shore dynamic symmetries. For the former type, we use the Majorana decomposition to reduce the dynamics to an effective two-state system, which allows us to find the propagator analytically and use the pool of available composite pulses for two-state systems. For the latter type of multistate systems, we use the Morris-Shore decomposition, which reduces the multistate dynamics to a set of two-state systems. We present examples which demonstrate that the multistate composite sequences open a variety of possibilities for coherent control of quantum systems with multiple states.
We propose various composite πππ-pulse sequences for implementing robust z-axis rotation gates widely used in quantum information processing (QIP) scenarios, and discuss their error tolerance of the pulse strength error (PSE) and off-resonance error (ORE), i.e., two typical systematic errors in NMR systems. Compared to existing composite gates, our designed composite pulses exhibit higher gate fidelity and shorter sequence time cost. Furthermore, such short pulse sequences are generalized by simple analytic formulas which can suppress these errors to any desired order simultaneously with linear length scaling, indicating an efficient construction of them with arbitrary accuracy.