A quantum algorithm for linear differential equations with layerwise parameterized quantum circuits

TL;DR: Researchers propose a quantum algorithm for solving linear differential equations using shallow parameterized quantum circuits, requiring fewer qubits than previous methods, and demonstrate its application in simulating harmonic oscillators and non-Hermitian systems with PT-symmetry.

Abstract: Abstract Solving linear differential equations is a common problem in almost all fields of science and engineering. Here, we present a variational algorithm with shallow circuits for solving such a problem: given an $$N \times N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>×</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> matrix $${\varvec{A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> , an N -dimensional vector $$\varvec{b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> , and an initial vector $$\varvec{x}(0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , how to obtain the solution vector $$\varvec{x}(T)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> at time T according to the constraint $$\textrm{d}\varvec{x}(t)/\textrm{d} t = {\varvec{A}}\varvec{x}(t) + \varvec{b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>d</mml:mtext> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mtext>d</mml:mtext> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> </mml:mrow> </mml:math> . The core idea of the algorithm is to encode the equations into a ground state problem of the Hamiltonian, which is solved via hybrid quantum-classical methods with high fidelities. Compared with the previous works, our algorithm requires the least qubit resources and can restore the entire evolutionary process. In particular, we show its application in simulating the evolution of harmonic oscillators and dynamics of non-Hermitian systems with $$\mathcal{P}\mathcal{T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> -symmetry. Our algorithm framework provides a key technique for solving so many important problems whose essence is the solution of linear differential equations.