Quantum amplitude damping for solving homogeneous linear differential equations: A noninterferometric algorithm

In contexts where relevant problems can easily attain configuration spaces of enormous sizes, solving linear differential equations (LDEs) can become a hard achievement for classical computers; on the other hand, the rise of quantum hardware can conceptually enable such high-dimensional problems to be solved with a foreseeable number of qubits, while also yielding quantum advantage in terms of time complexity. Nevertheless, to bridge towards experimental realizations with several qubits and harvest such potential in a short-term basis, one must dispose of efficient quantum algorithms that are compatible with near-term projections of state-of-the-art hardware, in terms of both techniques and limitations. As the conception of such algorithms is no trivial task, insights on new heuristics are welcomed. This work proposes an approach by using the quantum amplitude damping operation as a resource to construct an efficient quantum algorithm for solving homogeneous LDEs. As the intended implementation involves performing amplitude damping exclusively via a simple equivalent quantum circuit, our algorithm shall be given by a gate-level quantum circuit (predominantly composed of elementary two-qubit gates) and is particularly nonrestrictive in terms of connectivity within and between some of its main quantum registers. We show that such an open quantum-system-inspired circuitry allows for constructing the real exponential terms in the solution in a noninterferometric way; we also provide a guideline for guaranteeing a lower bound on the probability of success for each realization, by exploring the decay properties of the underlying quantum operation.

Figure 1

Proposed quantum circuit for solving HDLEs when $A$ is Hermitian. It includes three main quantum registers: the work register [of size $n\equiv {log}_2\left(N\right)$], the phase register and the environment register [both of size $l\equiv {log}_2\left(L\right)$, stipulated by the intended precision]; the circuit is divided into four modules plus measurements at the end.

Figure 2

Illustrative example of the eigenvalues mapping between $A$ and $\mathcal{M}$, given by ${\left(m_i\right)}_{1\le i\le N}\equiv {\left(e^{-i2\pi a_i/\left|\right|A\left|\right|}\right)}_{1\le i\le N}$. The eigenvalues of $A$ are wrapped in a clockwise manner around the unit circle, as indicated by the dashed arrow. The black filled dots around the circle represent all phases that can be exactly represented by a single computational state in the phase register, in this example for $L=8$. These can also be regarded as the allowed levels of damping that can later be performed by the sim-AD chain. The largest eigenvalue of $A$ is always set to $m_N=0$. For all other eigenvalues, a mismatch with relation to the black dots will engender a superposition of computational states in the phase register, which is a mechanism of error in the algorithm.

Figure 3

White-box representation of a sim-AD module. This simple quantum circuitry realizes the amplitude damping quantum operation on $\rho$, whereas the auxiliary upper qubit plays the role of an environment. $R_y\left({\theta }_k\right)\equiv e^{-i{\theta }_k{\sigma }_y/2}$, where ${\sigma }_y$ is the complex Pauli matrix.

Figure 4

Illustrative example of the eigenvalues mapping between $A$ and $\mathcal{M}$, for the less restrictive Hermitian $A$ formulation. The $[-|\left|A\right||,|\left|A\right|\left|\right]$ interval is mapped to the complex phase interval $[0,2\pi (L-1)/L]$. The black filled dots around the circle represent the allowed values of damping, in this case for $l=3$.