Shorter quantum circuits via single-qubit gate approximation
Vadym Kliuchnikov, Kristin Lauter, Romy Minko, Adam Paetznick, Christophe Petit
Abstract
We give a novel procedure for approximating general single-qubit unitaries from a finite universal gate set by reducing the problem to a novel magnitude approximation problem, achieving an immediate improvement in sequence length by a factor of 7/9. Extending the works \cite{Hastings2017} and \cite{Campbell2017}, we show that taking probabilistic mixtures of channels to solve fallback \cite{BRS2015} and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msqrt></mml:math>gate set we achieve an average non-Clifford gate count of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.23</mml:mn><mml:msub><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&#x03B5;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mn>2.13</mml:mn></mml:math>and T-count<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.56</mml:mn><mml:msub><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&#x03B5;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mn>5.3</mml:mn></mml:math>with mixed fallback approximations for diamond norm accuracy<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B5;</mml:mi></mml:math>.This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approximation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T and Clifford+<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msqrt></mml:math>). We also provide detailed numerical results for Clifford+T and Clifford+<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msqrt></mml:math>gate sets. In an effort to keep the paper self-contained, we include an overview of the relevant algorithms for integer point enumeration and relative norm equation solving. We provide a number of further applications of the magnitude approximation problems, as well as improved algorithms for exact synthesis, in the Appendices.