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arXiv:2405.14752

Qudit-Generalization of the Qubit Echo and Its Application to a Qutrit-Based Toffoli Gate

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

Published May 23, 2024• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2405.14752

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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.
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.
Multilevel qudit systems are increasingly being explored as alternatives to traditional qubit systems due to their denser information storage and processing potential. However, qudits are more susceptible to decoherence than qubits due to increased loss channels, noise sensitivity, and crosstalk. To address these challenges, we develop protocols for dynamical decoupling (DD) of qudit systems based on the Heisenberg-Weyl group. We implement and experimentally verify these DD protocols on a superconducting transmon processor that supports qudit operation based on qutrits (d=3) and ququarts (d=4). Specifically, we demonstrate single-qudit DD sequences to decouple qutrits and ququarts from system-bath-induced decoherence. We also introduce two-qudit DD sequences designed to suppress the detrimental cross-Kerr couplings between coupled qudits. This allows us to demonstrate a significant improvement in the fidelity of time-evolved qutrit Bell states. Our results highlight the utility of leveraging DD to enable scalable qudit-based quantum computing.
While quantum computing holds great potential in combinatorial optimization, electronic structure calculation, and number theory, the current era of quantum computing is limited by noisy hardware. Many quantum compilation approaches can mitigate the effects of imperfect hardware by optimizing quantum circuits for objectives such as critical path length. Few approaches consider quantum circuits in terms of the set of vendor-calibrated operations (i.e., native gates) available on target hardware. This manuscript expands the analytical and numerical approaches for optimizing quantum circuits at this abstraction level. We present a procedure for combining the strengths of analytical native gate-level optimization with numerical optimization. Although we focus on optimizing Toffoli gates on the IBMQ native gate set, the methods presented are generalizable to any gate and superconducting qubit architecture. Our optimized Toffoli gate implementation demonstrates an 18% reduction in infidelity compared with the canonical implementation as benchmarked on IBM Jakarta with quantum process tomography. Assuming the inclusion of multi-qubit cross-resonance (MCR) gates in the IBMQ native gate set, we produce Toffoli implementations with only six multi-qubit gates, a 25% reduction from the canonical eight multi-qubit implementations for linearly connected qubits.
Quantum computation is traditionally expressed in terms of quantum bits, or qubits. In this work, we instead consider three-level qutrits. Past work with qutrits has demonstrated only constant factor improvements, owing to the log2(3) binary-to-ternary compression factor. We present a novel technique using qutrits to achieve a logarithmic depth (runtime) decomposition of the Generalized Toffoli gate using no ancilla-a significant improvement over linear depth for the best qubit-only equivalent. Our circuit construction also features a 70x improvement in two-qudit gate count over the qubit-only equivalent decomposition. This results in circuit cost reductions for important algorithms like quantum neurons and Grover search. We develop an open-source circuit simulator for qutrits, along with realistic near-term noise models which account for the cost of operating qutrits. Simulation results for these noise models indicate over 90% mean reliability (fidelity) for our circuit construction, versus under 30% for the qubit-only baseline. These results suggest that qutrits offer a promising path towards scaling quantum computation.
Quantum computation is traditionally expressed in terms of quantum bits, or qubits. In this work, we instead consider three-level qu trits . Past work with qutrits has demonstrated only constant factor improvements, owing to the log 2 (3) binary-to-ternary compression factor. We present a novel technique, intermediate qutrits, to achieve sublinear depth decompositions of the Generalized Toffoli and other arithmetic circuits using no additional ancilla—a significant improvement over linear depth for the best qubit-only equivalents. For example, our Generalized Toffoli construction features a 70× improvement in two-qudit gate count over a qubit-only decomposition. This results in circuit cost reductions for important algorithms like quantum neurons, Grover search, and even Shor’s algorithm. Using a previously developed simulator with near-term noise models, we demonstrate for these models over 90% mean reliability (fidelity) for the Toffoli construction, versus under 30% for the qubit-only baseline. For our other constructions, such as the Incrementer, the A + B adder and the +K adder, we demonstrate the power of intermediate qutrits in producing asymptotic depth improvements with no additional ancilla. Together, these results suggest qutrits offer a promising path toward scaling quantum computation.
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.