Improved upper bounds on the stabilizer rank of magic states

Abstract

In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>copies of the magic state<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence="false" stretchy="false">⟩</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>0</mml:mn><mml:mo fence="false" stretchy="false">⟩</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>i</mml:mi><mml:mi>π</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo fence="false" stretchy="false">⟩</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>in the limit of large<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>. In particular, we show that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:msup><mml:mo fence="false" stretchy="false">⟩</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>⊗</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msup></mml:math>can be exactly expressed as a superposition of at most<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>α</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>stabilizer states, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi><mml:mo>≤</mml:mo><mml:mn>0.3963</mml:mn></mml:math>, improving on the best previously known bound<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi><mml:mo>≤</mml:mo><mml:mn>0.463</mml:mn></mml:math>. This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qubit Clifford + T circuit<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math>with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>uses of the T gate to within a given inverse polynomial relative error using a runtime<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">p</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">l</mml:mi><mml:mi mathvariant="normal">y</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>α</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msup></mml:math>. We also provide improved upper bounds on the stabilizer rank of symmetric product states<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>ψ</mml:mi><mml:msup><mml:mo fence="false" stretchy="false">⟩</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>⊗</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msup></mml:math>more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>instances of any (fixed) single-qubit<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math>-rotation gate with runtime<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>poly</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>m</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.

Topics & Concepts