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IEEE International Conference on Quantum Computing and Engineering (QCE)

Stable Toffoli Gate on Fixed-Frequency Superconducting Qutrits

Wonho Jang, Y. Iiyama, Naoki Kanazawa, Toshiaki Inada, R. Sawada, Tamiya Onodera, Koji Terashi

Published Sep 1, 2022• 1 citation
Publisher

Abstract

Entangling gates in superconducting quantum computers suffer from imperfection of control and relaxation of qubits. The Toffoli gate, essential to various quantum algorithms and to certain fast error correction schemes, represents a particular challenge, as it is decomposed into at least six two-qubit gates, whose errors accrue on top of the increased probability of qubit relaxation due to the long gate time. While it is possible to significantly reduce the depth of the Toffoli gate by exploiting three-level, i.e., qutrit states internally, existing methods are susceptible to errors arising from frequency drifts due to charge fluctuation effects in the second excited state of the transmon. In this poster, we propose a novel qutrit-based implementation of the Toffoli gate that is inherently robust against such errors. The method is extensively validated through a pulse-level simulation and experiments performed on IBM Quantum machines. On one device, where the total Toffoli gate time is 2.510μs, we obtained average gate fidelity values of 0.928 ± 0.007 and 0.896 ± 0.036 from multiple measurements performed during the first one and twenty-four hours after calibrations, respectively, where the errors represent the standard deviations of the measurement results.

DOI
10.1109/qce53715.2022.00148

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Entangling gates in superconducting quantum computers suffer from imperfection of control and relaxation of qubits. The Toffoli gate, essential to various quantum algorithms and to certain fast error correction schemes, represents a particular challenge, as it is decomposed into at least six two-qubit gates, whose errors accrue on top of the increased probability of qubit relaxation due to the long gate time. While it is possible to significantly reduce the depth of the Toffoli gate by exploiting three-level, i.e., qutrit states internally, existing methods are susceptible to errors arising from frequency drifts due to charge fluctuation effects in the second excited state of the transmon. In this poster, we propose a novel qutrit-based implementation of the Toffoli gate that is inherently robust against such errors. The method is extensively validated through a pulse-level simulation and experiments performed on IBM Quantum machines. On one device, where the total Toffoli gate time is 2.510μs, we obtained average gate fidelity values of 0.928 ± 0.007 and 0.896 ± 0.036 from multiple measurements performed during the first one and twenty-four hours after calibrations, respectively, where the errors represent the standard deviations of the measurement results.
The fidelity of certain gates on noisy quantum computers may be improved when they are implemented using more than two levels of the involved transmons. The main impediments to achieving this potential are the dynamic gate phase errors that cannot be corrected via calibration. The standard tool for countering such phase errors in two-level qubits is the echo protocol, often referred to as the dynamical decoupling sequence, where the evolution of a qubit is punctuated by an even number of X gates. We introduce basis cycling, which is a direct generalization of the qubit echo to general qudits, and provide an analytic framework for designing gate sequences to produce desired effects using this technique. We then apply basis cycling to a Toffoli gate decomposition incorporating a qutrit and obtain CCZ gate fidelity values up to 93.8$\pm$0.1%, measured by quantum process tomography, on IBM quantum computers. The gate fidelity remains stable without recalibration even while the resonant frequency of the qutrit fluctuates, highlighting the dynamical nature of phase error cancellation through basis cycling. Our results demonstrate that one of the biggest difficulties in implementing qudit-based gate decompositions on superconducting quantum computers can be systematically overcome when certain conditions are met, and thus open a path toward fulfilling the promise of qudits as circuit optimization agents.
The development of noisy intermediate-scale quantum (NISQ) devices has extended the scope of executable quantum circuits with high-fidelity single- and two-qubit gates. Equipping NISQ devices with three-qubit gates will enable the realization of more complex quantum algorithms and efficient quantum error correction protocols with reduced circuit depth. Several three-qubit gates have been implemented for superconducting qubits, but their use in gate synthesis has been limited due to their low fidelity. Here, using fixed-frequency superconducting qubits, we demonstrate a high-fidelity iToffoli gate based on two-qubit interactions, the so-called cross-resonance effect. As with the Toffoli gate, this three-qubit gate can be used to perform universal quantum computation. The iToffoli gate is implemented by simultaneously applying microwave pulses to a linear chain of three qubits, revealing a process fidelity as high as 98.26(2)%. Moreover, we numerically show that our gate scheme can produce additional three-qubit gates which provide more efficient gate synthesis than the Toffoli and Toffoli gates. Our work not only brings a high-fidelity iToffoli gate to current superconducting quantum processors but also opens a pathway for developing multi-qubit gates based on two-qubit interactions.
A single-shot Toffoli, or controlled-controlled-not, gate is desirable for classical and quantum information processing. The Toffoli gate alone is universal for reversible computing and, accompanied by the Hadamard gate, forms a universal gate set for quantum computing. The Toffoli gate is also a key ingredient for (nontopological) quantum error correction. Currently Toffoli gates are achieved by decomposing into sequentially implemented single- and two-qubit gates, which require much longer times and yields lower overall fidelities compared to a single-shot implementation. We develop a quantum-control procedure to construct a single-shot Toffoli gate for three nearest-neighbor-coupled superconducting transmon systems such that the fidelity is 99.9% and is as fast as an entangling two-qubit gate under the same realistic conditions. The gate is achieved by a nongreedy quantum control procedure using our enhanced version of the differential evolution algorithm.
Quantum computation is conventionally performed using quantum operations acting on two-level quantum bits, or qubits. Qubits in modern quantum computers suffer from inevitable detrimental interactions with the environment that cause errors during computation, with multi-qubit operations often being a primary limitation. Most quantum devices naturally have multiple accessible energy levels beyond the lowest two traditionally used to define a qubit. Qudits offer a larger state space to store and process quantum information, reducing complexity of quantum circuits and improving efficiency of quantum algorithms. Here, we experimentally demonstrate a ternary decomposition of a multi-qubit operation on cloud-enabled fixed-frequency superconducting transmons. Specifically, we realize an order-preserving Toffoli gate consisting of four two-transmon operations, whereas the optimal order-preserving binary decomposition uses eight \texttt{CNOT}s on a linear transmon topology. Both decompositions are benchmarked via truth table fidelity where the ternary approach outperforms on most sets of transmons on \texttt{ibmq\_jakarta}, and is further benchmarked via quantum process tomography on one set of transmons to achieve an average gate fidelity of 78.00\% ±\pm± 1.93\%.
Advanced simulations and calculations on quantum computers require high-fidelity implementations of quantum operations. The universal gateset approach builds complex unitaries from a small set of primitive gates, often resulting in a long gate sequence, which is typically a leading factor in the total accumulated error. Compiling a complex unitary for processors with higher-dimensional logical elements, such as qutrits, exacerbates the accumulated error per unitary, since an even longer gate sequence is required. Optimal control methods promise time- and resource-efficient compact gate sequences and, therefore, higher fidelity. These methods generate pulses that can directly implement any complex unitary on a quantum device. In this work, we demonstrate that any arbitrary qubit and qutrit gate can be realized with high fidelity, which can significantly reduce the length of a gate sequence. We generate and test pulses for a large set of randomly selected arbitrary unitaries on several quantum processing units (QPUs): the Lawrence Livermore National Laboratory Quantum Device and Integration Testbed's (QuDIT's) standard QPU and three of Rigetti's QPUs: Ankaa-2, Ankaa-9Q-1, and Aspen-M-3. On the QuDIT platform's standard QPU, the average fidelity of random qutrit gates is $97.9\ifmmode\pm\else\textpm\fi{}0.5\mathrm{%}$ measured with conventional QPT and $98.8\ifmmode\pm\else\textpm\fi{}0.6\mathrm{%}$ from QPT with gate folding. Rigetti's Ankaa-2 achieves random qubit gates with an average fidelity of $98.4\ifmmode\pm\else\textpm\fi{}0.5\mathrm{%}$ (conventional QPT) and $99.7\ifmmode\pm\else\textpm\fi{}0.1\mathrm{%}$ (QPT with gate folding). On Ankaa-9Q-1 and Aspen-M-3, the average fidelities with conventional qubit QPT measurements were higher than 99% (see Appendix). We show that optimal control gates are robust to drift for at least 3 h and that the same calibration parameters can be used for all implemented gates. Our work promises that the calibration overheads for optimal control gates can be made small enough to enable efficient quantum circuits based on this technique.
Superconducting transmon qubits comprise one of the most promising platforms for quantum information processing due to their long coherence times and to their scalability into larger qubit networks. However, their weakly anharmonic spectrum leads to spectral crowding in multiqubit systems, making it challenging to implement fast, high-fidelity gates while avoiding leakage errors. To address this challenge, we use a protocol for speeding up waveforms by inducing phases to harmful transitions (SWIPHT) [S. E. Economou and E. Barnes, Phys. Rev. B 91, 161405(R) (2015)], which yields smooth, simple microwave pulses designed to suppress leakage without sacrificing gate speed through spectral selectivity. Here, we determine the parameter regimes in which SWIPHT is effective and demonstrate that in these regimes, it systematically produces two-qubit gate fidelities for cavity-coupled transmons in the range 99.6%--99.9% with gate times as fast as 23 ns. Our results are obtained from full numerical simulations that include current experimental levels of relaxation and dephasing. These high fidelities persist over a wide range of system parameters that encompass many current experimental setups and are insensitive to small parameter variations and pulse imperfections.