Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits

Kliuchnikov, Maslov, and Mosca proved in 2012 that a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:mn>2</mml:mn></mml:math> unitary matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>V</mml:mi></mml:math> can be exactly represented by a single-qubit Clifford+<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math> circuit if and only if the entries of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>V</mml:mi></mml:math> belong to the ring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math>. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math> circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math> circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math> circuits by considering unitary matrices over subrings of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math>. We focus on the subrings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mo stretchy="false">]</mml:mo></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><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>i</mml:mi><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mo stretchy="false">]</mml:mo></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math>, and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>C</mml:mi><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>X</mml:mi><mml:mo fence="false" stretchy="false">}</mml:mo></mml:math> with an analogue of the Hadamard gate and an optional phase gate.