Divide-and-conquer verification method for noisy intermediate-scale quantum computation

Several noisy intermediate-scale quantum computations can be regarded as logarithmic-depth quantum circuits on a sparse quantum computing chip, where two-qubit gates can be directly applied on only some pairs of qubits. In this paper, we propose a method to efficiently verify such noisy intermediate-scale quantum computation. To this end, we first characterize small-scale quantum operations with respect to the diamond norm. Then by using these characterized quantum operations, we estimate the fidelity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo fence="false" stretchy="false">&amp;#x27E8;</mml:mo><mml:msub><mml:mi>&amp;#x03C8;</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>&amp;#x03C1;</mml:mi><mml:mo stretchy="false">&amp;#x005E;</mml:mo></mml:mover></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">u</mml:mi><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:msub><mml:mi>&amp;#x03C8;</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo></mml:math> between an actual <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qubit output state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>&amp;#x03C1;</mml:mi><mml:mo stretchy="false">&amp;#x005E;</mml:mo></mml:mover></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">u</mml:mi><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msub></mml:math> obtained from the noisy intermediate-scale quantum computation and the ideal output state (i.e., the target 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:msub><mml:mi>&amp;#x03C8;</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo></mml:math>. Although the direct fidelity estimation method requires <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:mi>n</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> copies of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>&amp;#x03C1;</mml:mi><mml:mo stretchy="false">&amp;#x005E;</mml:mo></mml:mover></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">u</mml:mi><mml:mi mathvariant="normal">t</mml:mi></mml:mrow></mml:msub></mml:math> on average, our method requires only <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:mi>D</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>12</mml:mn><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> copies even in the worst case, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math> is the denseness 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:msub><mml:mi>&amp;#x03C8;</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo></mml:math>. For logarithmic-depth quantum circuits on a sparse chip, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math> is 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:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>n</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math>, and thus <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:mi>D</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>12</mml:mn><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> is a polynomial in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>. By using the IBM Manila 5-qubit chip, we also perform a proof-of-principle experiment to observe the practical performance of our method.