Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates

We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:mo>&amp;#x2061;</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> non-Clifford gates. Specifically, for an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -qubit 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>&amp;#x03C8;</mml:mi> <mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo> </mml:math> prepared with at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> </mml:math> non-Clifford gates, our algorithms use <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">p</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">l</mml:mi> <mml:mi mathvariant="sans-serif">y</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>t</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>&amp;#x03B5;</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> time and copies of <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>&amp;#x03C8;</mml:mi> <mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo> </mml:math> to learn <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>&amp;#x03C8;</mml:mi> <mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo> </mml:math> to trace distance at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>&amp;#x03B5;</mml:mi> </mml:math> .The first algorithm for this task is more efficient, but requires entangled measurements across two copies of <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>&amp;#x03C8;</mml:mi> <mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo> </mml:math> . The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently large stabilizer dimension, where a quantum state has stabilizer dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> if it is stabilized by an abelian group of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> </mml:msup> </mml:math> Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest.