Variational Quantum Linear Solver

Abstract

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>.