Optimal synthesis into fixed XX interactions

Eric C. Peterson1, Lev S. Bishop2, and Ali Javadi-Abhari2

1IBM Quantum, San Jose, CA, USA
2IBM Quantum, Yorktown Heights, NY, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We describe an optimal procedure, as well as its efficient software implementation, for exact and approximate synthesis of two-qubit unitary operations into any prescribed discrete family of XX-type interactions and local gates. This arises from the analysis and manipulation of certain polyhedral subsets of the space of canonical gates. Using this, we analyze which small sets of XX-type interactions cause the greatest improvement in expected infidelity under experimentally-motivated error models. For the exact circuit synthesis of Haar-randomly selected two-qubit operations, we find an improvement in estimated infidelity by 31.4% when including alongside CX its square- and cube-roots, near to the optimal limit of 36.9% obtained by including all fractional applications of CX.

► BibTeX data

► References

[1] Charles H. Baldwin, Karl Mayer, Natalie C. Brown, Ciarán Ryan-Anderson, and David Hayes. ``Re-examining the quantum volume test: Ideal distributions, compiler optimizations, confidence intervals, and scalable resource estimations'' (2021). arXiv:2110.14808.
arXiv:2110.14808

[2] Madan Lal Mehta. ``Random matrices''. Volume 142 of Pure and Applied Mathematics (Amsterdam), pages xviii+688. Elsevier/​Academic Press, Amsterdam. (2004). Third edition.
https:/​/​doi.org/​10.1016/​C2009-0-22297-5

[3] Qiskit Developers. ``Qiskit: An open-source framework for quantum computing'' (2021). https:/​/​github.com/​Qiskit/​qiskit-terra.
https:/​/​github.com/​Qiskit/​qiskit-terra

[4] Eric C. Peterson. ``monodromy: Computations in the monodromy polytope for quantum gate sets'' (2021). https:/​/​github.com/​Qiskit/​monodromy.
https:/​/​github.com/​Qiskit/​monodromy

[5] Jun Zhang, Jiri Vala, Shankar Sastry, and K. Birgitta Whaley. ``Optimal quantum circuit synthesis from controlled-unitary gates''. Phys. Rev. A 69, 042309 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.69.042309

[6] Lingling Lao, Prakash Murali, Margaret Martonosi, and Dan Browne. ``Designing calibration and expressivity-efficient instruction sets for quantum computing''. 2021 ACM/​IEEE 48th Annual International Symposium on Computer Architecture (ISCA) (2021).
https:/​/​doi.org/​10.1109/​isca52012.2021.00071

[7] Vivek V. Shende, Igor L. Markov, and Stephen S. Bullock. ``Minimal universal two-qubit controlled-not-based circuits''. Physical Review A 69 (2004).
https:/​/​doi.org/​10.1103/​physreva.69.062321

[8] Andrew W. Cross, Lev S. Bishop, Sarah Sheldon, Paul D. Nation, and Jay M. Gambetta. ``Validating quantum computers using randomized model circuits''. Phys. Rev. A 100, 032328 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.032328

[9] Yong-Sheng Zhang, Ming-Yong Ye, and Guang-Can Guo. ``Conditions for optimal construction of two-qubit nonlocal gates''. Phys. Rev. A 71, 062331 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.062331

[10] Ming-Yong Ye, Yong-Sheng Zhang, and Guang-Can Guo. ``Super controlled gates and controlled gates in two-qubit gate simulations'' (2004). arXiv:0407108.
arXiv:quant-ph/0407108

[11] Eric C. Peterson, Gavin E. Crooks, and Robert S. Smith. ``Fixed-Depth Two-Qubit Circuits and the Monodromy Polytope''. Quantum 4, 247 (2020).
https:/​/​doi.org/​10.22331/​q-2020-03-26-247

[12] Nathan Earnest, Caroline Tornow, and Daniel J. Egger. ``Pulse-efficient circuit transpilation for quantum applications on cross-resonance-based hardware''. Phys. Rev. Research 3, 043088 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.043088

[13] Petar Jurcevic, Ali Javadi-Abhari, Lev S Bishop, Isaac Lauer, Daniela F Bogorin, Markus Brink, Lauren Capelluto, Oktay Günlük, Toshinari Itoko, Naoki Kanazawa, Abhinav Kandala, George A Keefe, Kevin Krsulich, William Landers, Eric P Lewandowski, Douglas T McClure, Giacomo Nannicini, Adinath Narasgond, Hasan M Nayfeh, Emily Pritchett, Mary Beth Rothwell, Srikanth Srinivasan, Neereja Sundaresan, Cindy Wang, Ken X Wei, Christopher J Wood, Jeng-Bang Yau, Eric J Zhang, Oliver E Dial, Jerry M Chow, and Jay M Gambetta. ``Demonstration of quantum volume 64 on a superconducting quantum computing system''. Quantum Science and Technology 6, 025020 (2021).
https:/​/​doi.org/​10.1088/​2058-9565/​abe519

[14] Cupjin Huang, Dawei Ding, Qi Ye, Feng Wu, Linghang Kong, Fang Zhang, Xiaotong Ni, Yaoyun Shi, Hui-Hai Zhao, and Jianxin Chen. ``Towards ultra-high fidelity quantum operations: Sqisw gate as a native two-qubit gate'' (2021). arXiv:2105.06074.
arXiv:2105.06074

[15] B. Kraus and J. I. Cirac. ``Optimal creation of entanglement using a two-qubit gate''. Physical Review A 63, 062309 (2001). arXiv:quant-ph/​0011050.
https:/​/​doi.org/​10.1103/​PhysRevA.63.062309
arXiv:quant-ph/0011050

[16] Yuriy Makhlin. ``Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations''. Quantum Information Processing 1, 243–252 (2002).
https:/​/​doi.org/​10.1023/​A:1022144002391

[17] Jun Zhang, Jiri Vala, Shankar Sastry, and K. Birgitta Whaley. ``Geometric theory of nonlocal two-qubit operations''. Phys. Rev. A 67, 042313 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.67.042313

[18] Paul Watts, Jiří Vala, Matthias M. Müller, Tommaso Calarco, K. Birgitta Whaley, Daniel M. Reich, Michael H. Goerz, and Christiane P. Koch. ``Optimizing for an arbitrary perfect entangler. i. functionals''. Phys. Rev. A 91, 062306 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.062306

[19] Paul Watts, Maurice O'Connor, and Jiří Vala. ``Metric structure of the space of two-qubit gates, perfect entanglers and quantum control''. Entropy 15, 1963–1984 (2013).
https:/​/​doi.org/​10.3390/​e15061963

[20] Marcin Musz, Marek Kuś, and Karol Życzkowski. ``Unitary quantum gates, perfect entanglers, and unistochastic maps''. Phys. Rev. A 87, 022111 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.022111

[21] David Avis. ``Living with lrs''. In Discrete and computational geometry (Tokyo, 1998). Volume 1763 of Lecture Notes in Comput. Sci., pages 47–56. Springer, Berlin (2000).
https:/​/​doi.org/​10.1007/​978-3-540-46515-7_4

[22] David Avis and Komei Fukuda. ``Reverse search for enumeration''. Discrete Appl. Math. 65, 21–46 (1996).
https:/​/​doi.org/​10.1016/​0166-218X(95)00026-N

[23] Lovis Anderson and Benjamin Hiller. ``A sweep-plane algorithm for the computation of the volume of a union of polytopes''. In Bernard Fortz and Martine Labbé, editors, Operations Research Proceedings 2018. Pages 87–93. Cham (2019). Springer International Publishing.
https:/​/​doi.org/​10.1007/​978-3-030-18500-8_12

Cited by

[1] Akel Hashim, Rich Rines, Victory Omole, Ravi K. Naik, John Mark Kreikebaum, David I. Santiago, Frederic T. Chong, Irfan Siddiqi, and Pranav Gokhale, "Optimized SWAP networks with equivalent circuit averaging for QAOA", Physical Review Research 4 3, 033028 (2022).

[2] Sophia Fuhui Lin, Sara Sussman, Casey Duckering, Pranav S. Mundada, Jonathan M. Baker, Rohan S. Kumar, Andrew A. Houck, and Frederic T. Chong, "Let Each Quantum Bit Choose Its Basis Gates", arXiv:2208.13380.

[3] Gokul Subramanian Ravi, Kaitlin N. Smith, Pranav Gokhale, Andrea Mari, Nathan Earnest, Ali Javadi-Abhari, and Frederic T. Chong, "VAQEM: A Variational Approach to Quantum Error Mitigation", arXiv:2112.05821.

The above citations are from Crossref's cited-by service (last updated successfully 2022-10-04 15:40:05) and SAO/NASA ADS (last updated successfully 2022-10-04 15:40:06). The list may be incomplete as not all publishers provide suitable and complete citation data.