# How would a quantum computer be used to solve Partial Differential Equations?

Say you have a PDE you want to solve.

What kind of quantum algorithms would you use to solve it? How do we input our problem on a quantum computer? What will be the output and in what form?

I know that quantum algorithms for solving linear systems (often named HHL but actually this is a bad name as other versions are not from the HHL authors) were listed before but maybe other methods are out there. Also as it is considered as a subroutine, the output is quantum and then unless you want statistics from it or use it as an input of another quantum algorithm, it is limiting.

• How general do you want your PDE to be? Is it linear? – AHusain Aug 16 '18 at 3:37
• If you have different setups of PDEs in mind, I would like to know for each. Say linear for instance first because I guess non-linear may be harder to do. – cnada Aug 16 '18 at 14:57

I don't have an exact answer to your question (if it actually exists); but I can answer part of your question concerned with the I/O to a quantum processor.

As a general rule of thumb; Quantum Algorithms (currently) cannot provide direct answers to problem statements. At least for now, quantum processors exists as heterogeneous accelerators with a classical computing unit. The 'quantum accelerator' is concerned with only that part of the overall algorithm that is not trivial (or exponential in complexity) to solve on a classical computer. In the end, only a sub portion of the program is actually computed on the quantum processor. (Eg. Shor's Factoring Algorithm is actually a period finding algorithm. Period finding is a non-trivial task.)

Among several other reasons, of the main problems is input and output operation with a quantum processor. The problem 'must' be expressible in a concise form (eg. an equation). This equation is expressed as a quantum circuit in the 'oracle' which is primarily concerned with solving the equation and measurement outcome are recorded (tomography). The output too needs post processing to actually make sense (which is again performed by the classical counterpart).

p.s. I would be very interested to know more about PDE solving quantum algorithms; if there is an efficient one.

• I understand the "general" point of view. It is just not trivial to me how we model PDE solving on a quantum computer. This is direct in HHL cause your problem can be expressed as a linear system Ax=f when you do discretization. You just express your f as a quantum state (your first input), use A in an Hermitian form for phase estimation for instance (second input) and by using the subroutine that uses controlled rotation and uncomputation (at least for the original version of HHL) you have your output as a quantum state. – cnada Aug 17 '18 at 16:46
• This becomes somehow efficient in the size of the problem because you use the exponential dimensionality of the Hilbert space for encoding in the probability amplitudes of the wavefunction. – cnada Aug 17 '18 at 16:54
• But I would wonder if there are other ways/algorithms for PDEs. – cnada Aug 17 '18 at 16:54

I came across an approach to solve differential equations using D-wave quantum annealer. The link is here: https://arxiv.org/abs/1812.10572.

The basic method is to derive the energy functional for the differential equation that is then minimized on a quantum annealer. The minimization can use finite element basis to map the energy to a localized sub graph of the D-wave machine.

The advantage of this over classical algorithm is that there is no need to even build a system of equations, so there is memory savings and avoids cost of assembly of a linear system. The solution complexity however is same as the classical conjugate gradient method: $$\mathcal{O}(n)$$. The HHL algorithm on the other hand can give exponential speed up, but like you said does not directly give the solution, plus we do need to assemble the linear system in the first place.

• Hi Jeremy! According to this thread and to other research papers, the conjugate gradient method is not $\mathcal{O}(n)$ but rather $\mathcal{O}(s \sqrt{\kappa})$ with $s$ the sparsity of the matrix and $\kappa$ its condition number. – Nelimee Jan 10 '19 at 8:04