Quantum Phase Estimation by Compressed Sensing

As a signal recovery algorithm, compressed sensing is particularly effective when the data has low complexity and samples are scarce, which aligns natually with the task of quantum phase estimation (QPE) on early fault-tolerant quantum computers. In this work, we present a new Heisenberg-limited, robust QPE algorithm based on compressed sensing, which requires only sparse and discrete sampling of times. Specifically, given multiple copies of a suitable initial state and queries to a specific unitary matrix, our algorithm can recover the phase with a total runtime of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>&amp;#x03F5;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mtext>poly</mml:mtext><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>&amp;#x03F5;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> is the desired accuracy. Additionally, the maximum runtime satisfies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo movablelimits="true" form="prefix">max</mml:mo></mml:mrow></mml:msub><mml:mi>&amp;#x03F5;</mml:mi><mml:mo>&amp;#x226A;</mml:mo><mml:mi>&amp;#x03C0;</mml:mi></mml:math>, making it comparable to state-of-the-art algorithms. Furthermore, our result resolves the basis mismatch problem in certain cases by introducing an additional parameter to the traditional compressed sensing framework.