Synthesis of single-qutrit circuits from Clifford+𝑅 gates

Here, we present two deterministic compilation algorithms for single-qutrit unitaries with <math> <mrow><mi>O</mi><mo>(</mo><mo>log</mo><mn>1</mn><mo>/</mo><mi>ɛ</mi><mo>)</mo></mrow> </math> gate depth. Each algorithm selects a nearby approximation to the target unitary and then exactly synthesizes the approximation over the Clifford <math> <mo>+</mo> </math> <math> <mi mathvariant="bold">R</mi> </math> basis. The first algorithm exhaustively searches over the group; while the second algorithm searches only for Householder reflections. The exhaustive search algorithm yields an average <math> <mi mathvariant="bold">R</mi> </math> count of <math> <mrow> <mn>2.193</mn><mrow><mo>(</mo><mn>11</mn><mo>)</mo></mrow><mo>+</mo><mn>8.621</mn><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow><msub><mo>log</mo><mn>10</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>ɛ</mi><mo>)</mo></mrow> </mrow> </math> , albeit with a time complexity of <math> <mrow><mi>O</mi><mo>(</mo><msup><mi>ɛ</mi><mrow><mo>−</mo><mn>4.4</mn></mrow></msup><mo>)</mo></mrow> </math> . The Householder search algorithm results in a larger average <math> <mi mathvariant="bold">R</mi> </math> count of <math> <mrow> <mn>3.20</mn><mrow><mo>(</mo><mn>13</mn><mo>)</mo></mrow><mo>+</mo><mn>10.77</mn><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><msub><mo>log</mo><mn>10</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>ɛ</mi><mo>)</mo></mrow> </mrow> </math> at a reduced time complexity of <math> <mrow><mi>O</mi><mo>(</mo><msup><mi>ɛ</mi><mrow><mo>−</mo><mn>0.42</mn></mrow></msup><mo>)</mo></mrow> </math> , greatly extending the reach in <math> <mi>ɛ</mi> </math> . These costs correspond asymptotically to 35% and 69% more non-Clifford gates compared with synthesizing the same unitary with two qubits. Such initial results are encouraging for using the <math> <mi mathvariant="bold">R</mi> </math> gate as the nontransversal gate for qutrit-based computation.. Authors: Gustafson, Erik J. [Fermi National Accelerator Laboratory (FNAL), Batavia, IL (United States); NASA Ames Research Center (ARC), Moffett Field, Mountain View, CA (United States); Universities Space Research Association, Mountain View, CA (United States)] (ORCID:0000000172175692); Lamm, Henry [Fermi National Accelerator Laboratory (FNAL), Batavia, IL (United States)] (ORCID:0000000330330791); Liu, Diyi [Univ. of Minnesota-Twin Cities, Minneapolis, MN (United States)] (ORCID:0000000209967686); Murairi, Edison M. [Fermi National Accelerator Laboratory (FNAL), Batavia, IL (United States); George Washington Univ., Washington, DC (United States)] (ORCID:0000000216396308); Zhu, Shuchen [Duke Univ., Durham, NC (United States)] (ORCID:0000000212402002). DOE Contract: 89243024CSC000002; AC02-07CH11359; FG02-95ER40907; SC0025572. Subjects: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS