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# Parallel entangling operations on a universal ion-trap quantum computer

## Abstract

The circuit model of a quantum computer consists of sequences of gate operations between quantum bits (qubits), drawn from a universal family of discrete operations1. The ability to execute parallel entangling quantum gates offers efficiency gains in numerous quantum circuits2,3,4, as well as for entire algorithms—such as Shor’s factoring algorithm5—and quantum simulations6,7. In circuits such as full adders and multiple-control Toffoli gates, parallelism can provide an exponential improvement in overall execution time through the divide-and-conquer technique8. More importantly, quantum gate parallelism is essential for fault-tolerant error correction of qubits that suffer from idle errors9,10. However, the implementation of parallel quantum gates is complicated by potential crosstalk, especially between qubits that are fully connected by a common-mode bus, such as in Coulomb-coupled trapped atomic ions11,12 or cavity-coupled superconducting transmons13. Here we present experimental results for parallel two-qubit entangling gates in an array of fully connected trapped 171Yb+ ion qubits. We perform a one-bit full-addition operation on a quantum computer using a depth-four quantum circuit4,14,15, where circuit depth denotes the number of runtime steps required. Our method exploits the power of highly connected qubit systems using classical control techniques and will help to speed up quantum circuits and achieve fault tolerance in trapped-ion quantum computers.

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## Data availability

All relevant data are available from the corresponding author upon request.

## References

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## Acknowledgements

We thank S.-T. Wang, Z. Gong, S. Debnath, P. H. Leung, Y. Wu and L. Duan for discussions. Circuits were drawn using the qcircuit.tex package. This work was supported by the ARO with funds from the IARPA LogiQ programme, the AFOSR MURI programme and the NSF Physics Frontier Center at JQI. This material is partially based on work supported by the National Science Foundation during D.M.’s assignment at the Foundation. Any opinion, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

## Author information

Authors

### Contributions

C.F., A.O., N.M.L., K.A.L., D.M. and C.M. designed the research; C.F., N.M.L., K.A.L., D.Z. and C.M. collected and analysed the data; C.F. performed the theory derivations; A.O. performed the pulse sequence optimizations; C.F., A.O., N.M.L., K.A.L., D.Z., D.M. and C.M. contributed to the manuscript.

### Corresponding author

Correspondence to C. Figgatt.

## Ethics declarations

### Competing interests

C.M. is co-founder and Chief Scientist at IonQ, Inc.

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## Extended data figures and tables

### Extended Data Fig. 1 Additional experimental gate fidelities for parallel two-qubit entangling gates.

ad, Parity curves used to calculate fidelities for parallel XX gates applied on several sets of ions. Circles indicate data, with matching-colour lines indicating calculated fits. The key specifies the ion pair corresponding to each parity curve. The six parity curves shown in each plot include the two gate ion pairs (the first two ion pairs in the key) and the four crosstalk ion pairs. a, Ions (1, 2) and (3, 4) yield fidelities of 98.4(3)% and 97.7(3)% for the respective entangled pairs, with an average crosstalk error of 0.6(3)%. b, Ions (1, 5) and (2, 4) yield fidelities of 96.8(3)% and 98.1(2)% for the corresponding entangled pairs, with an average crosstalk error of 1.7(3)%. c, Ions (1, 3) and (2, 5) yield fidelities of 98.3(3)% and 97.5(2)% for the respective entangled pairs, with an average crosstalk error of 0.8(4)%. d, Ions (1, 2) and (4, 5) yield fidelities of 97.2(3)% and 91.9(3)% for the corresponding entangled pairs, with an average crosstalk error of 0.9(3)%.

### Extended Data Fig. 2 Experimental gate fidelities for parallel two-qubit partially entangling gates.

Parity curve for parallel XX(χ) gates on ions (1, 5) and (2, 4), where an XX(π/4) gate is applied on ions (1, 5) and an XX(π/8) gate on ions (2, 4). Circles indicate data, with matching-colour lines indicating calculated fits. The key specifies the ion pair corresponding to each parity curve. The six parity curves shown include the two gate ion pairs (the first two ion pairs in the key) and the four crosstalk ion pairs. The data yield fidelities of 96.4(3)% and 99.4(3)% for the respective entangled pairs, with an average crosstalk error of 2.2(3)%.

### Extended Data Fig. 3 Independence of parallel-gate calibration.

Parallel gates can be calibrated independently. ad, Data obtained by applying a pair of entangling gates in parallel and observing the change in population for each pair as the scaling factor for one of the ions or gates is varied. The key specifies the ion pair state corresponding to each dataset; for example, ‘(1, 2) 00’ indicates the 00 population for ions (1, 2). The 01 and 10 populations are very close to 0 and hence not always visible. The error bars are statistical. a, Scan of the scaling factor on ion 1 with an entangling gate on ions (3, 4). b, Scan of the scaling factor on ions (1, 2) with an entangling gate on ions (3, 4). c, Scan of the scaling factor on ion 2 with no light on ions (3, 4). d, Scan of the scaling factor on ions (1, 2) with no light on ions (3, 4).

### Extended Data Fig. 4 Full-adder implementation.

Application-optimized full-adder implementation using XX(χ), Rx(θ) and Ry(θ) gates, where θ is the rotation angle applied by the single-qubit R gate. The two parallel two-qubit operations are outlined in dashed boxes.

### Extended Data Fig. 5 C(V) gate implementation.

Implementation of the $$C(V)=\sqrt{{\rm{CNOT}}}$$ gate using XX(χ), Rx(θ) and Ry(θ) gates. The gate is used to construct the full adder used in this work.

## Supplementary information

### Supplementary Information

This file contains the following sections: Constraint problem for optimal parallel operations; Optimization methods; Fidelity of parallel xx operations; Toward a single-operation GHZ state; calculating fidelities of 2-qubit entangling gates.

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Figgatt, C., Ostrander, A., Linke, N.M. et al. Parallel entangling operations on a universal ion-trap quantum computer. Nature 572, 368–372 (2019). https://doi.org/10.1038/s41586-019-1427-5

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• DOI: https://doi.org/10.1038/s41586-019-1427-5

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