Fault-tolerant Quantum Error Correction Using a Linear Array of Emitters

We propose a fault-tolerant quantum error correction architecture consisting of a linear array of emitters and delay lines. In our scheme, a resource state for fault-tolerant quantum computation is generated by letting the emitters interact with a stream of photons and their neighboring emitters. Depending on the number of emitters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math>, we study the effect of delay line errors in two regimes: when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math> is a small constant of order unity and when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math> scales with the code distance. Between these two regimes, the logical error rate steadily decreases as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math> increases, from a scaling of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>exp</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>&amp;#x2212;</mml:mo><mml:mi>c</mml:mi><mml:msup><mml:mi>&amp;#x03B7;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&amp;#x2212;</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:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>exp</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>&amp;#x2212;</mml:mo><mml:msup><mml:mi>c</mml:mi><mml:mo>&amp;#x2032;</mml:mo></mml:msup><mml:msup><mml:mi>&amp;#x03B7;</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:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03B7;</mml:mi></mml:math> is the error rate per unit length in the delay line, for some constants <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>c</mml:mi><mml:mo>&amp;#x2032;</mml:mo></mml:msup><mml:mo>&amp;#x003E;</mml:mo><mml:mn>0</mml:mn></mml:math>. We also carry out a detailed study of the break-even point and the fault-tolerance overhead. These studies suggest that the multi-emitter architecture, using the state-of-the-art delay lines, can be used to demonstrate error suppression, assuming other sources of errors are sufficiently small.