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arXiv:2304.14212

Coherent and non-unitary errors in ZZ-generated gates

T. Mueller, Tobias Stollenwerk, David Headley, M. Epping, Frank K. Wilhelm

Published Apr 27, 2023• 1 citation
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Abstract

Variational algorithms such as the Quantum Approximate Optimization Algorithm have attracted attention due to their potential for solving problems using near-term quantum computers. The $ZZ$ interaction typically generates the primitive two-qubit gate in such algorithms applied for a time, typically a variational parameter, $γ$. Different compilation techniques exist with respect to the implementation of two-qubit gates. Due to the importance of the $ZZ$-gate, we present an error analysis comparing the continuous-angle controlled phase gate (CP) against the fixed angle controlled $Z$-gate (CZ). We analyze both techniques under the influence of coherent over-rotation and depolarizing noise. We show that CP and CZ compilation techniques achieve comparable $ZZ$-gate fidelities if the incoherent error is below $0.03 \, \%$ and the coherent error is below $0.8 \, \%$. Thus, we argue that for small coherent and incoherent error a non-parameterized two-qubit gate such as CZ in combination with virtual $Z$ decomposition for single-qubit gates could lead to a significant reduction in the calibration required and, therefore, a less error-prone quantum device. We show that above a coherent error of $0.04 π$ ($2 \, \%$), the CZ gate fidelity depends significantly on $γ$.

DOI
10.48550/arxiv.2304.14212

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Variational algorithms such as the Quantum Approximate Optimization Algorithm have attracted attention due to their potential for solving problems using near-term quantum computers. The ZZZZZZ interaction typically generates the primitive two-qubit gate in such algorithms applied for a time, typically a variational parameter, γγγ. Different compilation techniques exist with respect to the implementation of two-qubit gates. Due to the importance of the ZZZZZZ-gate, we present an error analysis comparing the continuous-angle controlled phase gate (CP) against the fixed angle controlled ZZZ-gate (CZ). We analyze both techniques under the influence of coherent over-rotation and depolarizing noise. We show that CP and CZ compilation techniques achieve comparable ZZZZZZ-gate fidelities if the incoherent error is below 0.03 %0.03 \, \%0.03% and the coherent error is below 0.8 %0.8 \, \%0.8%. Thus, we argue that for small coherent and incoherent error a non-parameterized two-qubit gate such as CZ in combination with virtual ZZZ decomposition for single-qubit gates could lead to a significant reduction in the calibration required and, therefore, a less error-prone quantum device. We show that above a coherent error of 0.04π0.04 π0.04π (2 %2 \, \%2%), the CZ gate fidelity depends significantly on γγγ.
Abstract Variational algorithms such as the quantum approximate optimization algorithm have attracted attention due to their potential for solving problems using near-term quantum computers. The ZZ interaction typically generates the primitive two-qubit gate in such algorithms applied for a time, typically a variational parameter, γ. Different compilation techniques exist with respect to the implementation of two-qubit gates. Due to the importance of the ZZ-gate, we present an error analysis comparing the continuous-angle controlled phase gate (CP) against the fixed angle controlled Z-gate (CZ). We analyze both techniques under the influence of coherent over-rotation and depolarizing noise. We show that CP and CZ compilation techniques achieve comparable ZZ-gate fidelities if the incoherent error is below 0.03% and the coherent error is below 0.8%. Thus, we argue that for small coherent and incoherent error a non-parameterized two-qubit gate such as CZ in combination with virtual Z decomposition for single-qubit gates could lead to a significant reduction in the calibration required and, therefore, a less error-prone quantum device. We show that above a coherent error of 0.04π (2%), the CZ gate fidelity depends significantly on γ.
Abstract Two-qubit gates are important components of quantum computing. However, unwanted interactions between qubits (so-called parasitic gates) can be particularly problematic and degrade the performance of quantum applications. In this work, we present two software methods to mitigate parasitic two-qubit gate errors. The first approach is built upon the Cartan’s KAK decomposition and keeps the original unitary decomposition for the error-free native two-qubit gate. It counteracts a parasitic two-qubit gate by only applying single-qubit rotations and therefore has no two-qubit gate overhead. We show the optimal choice of single-qubit mitigation gates. The second approach applies a numerical optimisation algorithm to re-compile a target unitary into the error-parasitic two-qubit gate plus single-qubit gates. We demonstrate these approaches on the CPhase-parasitic iSWAP-like gates. The KAK-based approach helps decrease unitary infidelity by a factor of 3 compared to the noisy implementation without error mitigation. When arbitrary single-qubit rotations are allowed, recompilation could completely mitigate the effect of parasitic errors but may require more native gates than the KAK-based approach. We also compare their average gate fidelity under realistic noise models, including relaxation and depolarising errors. Numerical results suggest that different approaches are advantageous in different error regimes, providing error mitigation guidance for near-term quantum computers.
Abstract Limited by the decoherence of qubits as well as the errors of quantum gates, near-term superconducting quantum computers can only run low-depth quantum circuits to achieve acceptable fidelity. One possible way to overcome these limitations is to construct quantum circuits with additional high-fidelity expressive multi-qubit gates. Recently, a new three-qubit gate, denoted as Controlled-CPHASE-SWAP (CCZS), has been implemented through simultaneous Controlled-Z (CZ) gates. The CCZS gate takes less time than a single CZ gate and can be implemented at the coherence limit. However, how to use the CCZS gate in quantum circuit synthesis remains unexplored. In this paper, we construct the quantum fan-out/parity gates, the controlled-phase gate and the locally fully connected CZ gates with the CCZS gate, respectively. Furthermore, applications of the CCZS gate in quantum error correction, quantum Fourier transform and quantum approximate optimization algorithm are also proposed. We evaluate the performance of the CCZS gate in quantum circuit synthesis through simulation and explore its potential advantages over CZ gates.
The possibility to utilize different types of two-qubit gates on a single quantum computing platform adds flexibility in the decomposition of quantum algorithms.A larger hardware-native gate set may decrease the number of required gates, provided that all gates are realized with high fidelity.Here, we benchmark both controlled-Z (CZ) and exchange-type (iSWAP) gates using a parametrically driven tunable coupler that mediates the interaction between two superconducting qubits.Using randomized benchmarking protocols we estimate an error per gate of 0.9 ± 0.03 and 1.3 ± 0.4% for the CZ and the iSWAP gate, respectively.We argue that spurious ZZ-type couplings are the dominant error source for the iSWAP gate, and that phase stability of all microwave drives is of utmost importance.Such differences in the achievable fidelities for different two-qubit gates have to be taken into account when mapping quantum algorithms to real hardware.
The main source of error for superconducting quantum processors today is two-qubit gates, so understanding what limits their fidelity is central to improving these devices. This thesis studies the controlled-Z (CZ) gate between two transmon qubits, realized by tuning the ∣11⟩|11\rangle∣11⟩ and ∣20⟩|20\rangle∣20⟩ states through an avoided crossing with a flux pulse. There are two primary errors inherent to this approach. The first is leakage into non-computational states like ∣20⟩|20\rangle∣20⟩; the second is an exchange error from population transferring between the single-excitation states ∣01⟩|01\rangle∣01⟩ and ∣10⟩|10\rangle∣10⟩. Using a closed-system simulation, this work compares four flux-pulse shapes (unipolar, net-zero bipolar, Gaussian-filtered bipolar, and adiabatic) and constructs a CNOT from each CZ. The central question is which error channel sets the gate fidelity, and whether the widely used net-zero pulse, known to protect neighboring qubits, also helps inside the gate itself. An important finding of this research is that every type of flux pulse has an average gate fidelity limited by exchange error rather than leakage. While the exchange error is not apparent in the return of the ∣11⟩|11\rangle∣11⟩ population, because excitation-number conservation prevents ∣11⟩|11\rangle∣11⟩ from entering the single-excitation sector, it does appear as an off-diagonal term in the CZ unitary, and subsequently in the CNOT. A second finding concerns the net-zero pulse: although its zero net flux integral cancels the first-order phase imprinted on neighboring qubits, it does not suppress the exchange error. The strong suppression observed in the analytic bipolar gate happens because its idle time leaves the exchange oscillation near a cancellation point at the end of the gate, not because the pulse is net-zero. The flux operating point predicted by the simulation was tested on a physical transmon device accessed remotely, where a square pulse produced a 180∘180^\circ180∘ conditional phase close to the simulated flux and duration. Because the model is closed-system, the reported fidelities describe the coherent error of each pulse rather than the total infidelity of a physical device.
A possible building block for a scalable quantum computer has recently been demonstrated [Mariantoni et al., Science 334, 61 (2011)]. This architecture consists of superconducting qubits capacitively coupled both to individual memory resonators as well as a common bus. In this work we study a natural primitive entangling gate for this and related resonator-based architectures, which consists of a controlled-${\ensuremath{\sigma}}^{z}$ (cz) operation between a qubit and the bus. The cz gate is implemented with the aid of the noncomputational qubit $|2\ensuremath{\rangle}$ state [Strauch et al., Phys. Rev. Lett. 91, 167005 (2003)]. Assuming phase or transmon qubits with 300 MHz anharmonicity, we show that by using only low frequency qubit-bias control it is possible to implement the qubit-bus cz gate with 99.9% (99.99%) fidelity in about 17ns17\phantom{\rule{0.16em}{0ex}}\mathrm{ns}17ns (23ns23\phantom{\rule{0.16em}{0ex}}\mathrm{ns}23ns) with a realistic two-parameter pulse profile, plus two auxiliary zzz rotations. The fidelity measure we refer to here is a state-averaged intrinsic process fidelity, which does not include any effects of noise or decoherence. These results apply to a multiqubit device that includes strongly coupled memory resonators. We investigate the performance of the qubit-bus cz gate as a function of qubit anharmonicity, identify the dominant intrinsic error mechanism and derive an associated fidelity estimator, quantify the pulse shape sensitivity and precision requirements, simulate qubit-qubit cz gates that are mediated by the bus resonator, and also attempt a global optimization of system parameters including resonator frequencies and couplings. Our results are relevant for a wide range of superconducting hardware designs that incorporate resonators and suggest that it should be possible to demonstrate a 99.999.9%99.9 cz gate with existing transmon qubits, which would constitute an important step towards the development of an error-corrected superconducting quantum computer.