Variational Quantum Linear Solver

Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare <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>x</mml:mi><mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo></mml:math> such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>x</mml:mi><mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo><mml:mo>&amp;#x221D;</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>b</mml:mi><mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo></mml:math>. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> is achieved. Specifically, we prove that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi><mml:mo>&amp;#x2A7E;</mml:mo><mml:msup><mml:mi>&amp;#x03F5;</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>&amp;#x03BA;</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> is the VQLS cost function and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03BA;</mml:mi></mml:math> is the condition number of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>. We present efficient quantum circuits to estimate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math>, while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1024</mml:mn><mml:mo>&amp;#x00D7;</mml:mo><mml:mn>1024</mml:mn></mml:math>. Finally, we numerically solve non-trivial problems of size up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>50</mml:mn></mml:mrow></mml:msup><mml:mo>&amp;#x00D7;</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>50</mml:mn></mml:mrow></mml:msup></mml:math>. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03F5;</mml:mi></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03BA;</mml:mi></mml:math>, and the system size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>.