Quantum variational PDE solver with machine learning

Abstract

To solve nonlinear partial differential equations (PDEs) is one of the most common but important tasks in not only basic sciences but also many practical industries. We here propose a quantum variational (QuVa) PDE solver with the aid of machine learning (ML) schemes to synergise two emerging technologies in mathematically hard problems. The core quantum processing in this solver is to calculate efficiently the expectation value of specially designed quantum operators. For a large quantum system, we only obtain data from measurements of few control qubits to avoid the exponential cost in the measurements of the whole quantum system and optimise a pathway to find possible solution sets of the desired PDEs using ML techniques. As an example, a few different types of the second-order DEs are examined with randomly chosen samples and a regression method is implemented to chase the best candidates of solution functions with another trial samples. We demonstrated that a three-qubit system successfully follows the pattern of analytical solutions of three different DEs with high fidelity since the variational solutions are given by a necessary condition to obtain the exact solution of the DEs. Thus, we believe that final solution candidate sets are efficiently extracted from the QuVa PDE solver with the support of ML techniques and this algorithm could be beneficial to search for the solutions of complex mathematical problems as well as to find good ansatzs for eigenstates in large quantum systems (e.g., for quantum chemistry).

Commutation simulator for open quantum dynamics

“…4. Apply a block controlled-SWAP gate from control C between system S and reference M to produce |Ψ 3 [16].…”

“…For the next step of the simulation, to produce |Ψ 3 , we apply a block controlled-SWAP gate between the system S and the reference M , as shown in Fig. 1 [16]. Following this, a Hadamard gate Ĥ is applied on control qubit C, to produce the final (entangled) state of the total system, before measurement…”

“…Moreover, for N = 1 1 and M = Ĥ with χ = π/2, if  is not set as the identity operator but chosen e.g. as the translation operator  [16,17], such that  |n = |n + 1 , we can also investigate any off-diagonal matrix elements, such as d dt ρ n,m (t) for n = m. Specifically, if we replace  → ( Â) p , as p actions of the translation operator  for the L-qubit system (p = 1, ..., L − 1), we can compute all the off-diagonal elements of (11), such as d dt ρ n,n+p (t). In any actual implementation of the quantum commutation simulator, one challenge is the construction of the (controlled) operator Ĥ in the quantum circuit, because the Hamiltonian operator is in general not a unitary operator.…”

Variational Quantum-Based Simulation of Waveguide Modes

“…Popular VQAs include the variational quantum eigensolver (VQE), which computes the ground state energies of Hamiltonians [17], [18] and the quantum approximate optimization algorithm (QAOA) [19], which finds approximate solutions to combinatorial optimization problems. More recently, VQAs have been proposed to solve linear systems [20]- [22] and partial differential equations [23]- [27].…”

“…𝑘 (𝜃) 𝜕𝜃 𝑗 = 𝜕 𝑗 𝜓(𝜃), 𝜓(𝜃)|𝑋 ⊗ 𝐴|𝜕 𝑗 𝜓(𝜃), 𝜓(𝜃)= 𝑋 ⊗ 𝑀 𝜕 𝑗 𝜓 ( 𝜃), 𝜓 ( 𝜃) 𝑖 𝑋 ⊗ |𝜓 𝑖 𝜓 𝑖 | 𝜕 𝑗 𝜓 ( 𝜃), 𝜓 ( 𝜃) ,(27)where we used the following notation to denote quantum expectation values: 𝐻 𝜙 = 𝜙|𝐻|𝜙 . To express the gradient(27) in terms of preparable states and simple observables, we use the fact that |𝜓 𝑖 = |𝜓(𝜃 (𝑖) ) = 𝑈 (𝜃 (𝑖) )|0 . By defining the state|Φ 𝑖 𝑗 (𝜃) = 𝐼 ⊗ 𝑈 † (𝜃 (𝑖) ) |𝜕 𝑗 𝜓(𝜃), 𝜓(𝜃) ,(28)the gradient (27) can be written as the following sum of expectation values:𝜕𝐹 𝑘 (𝜃) 𝜕𝜃 𝑗 = 𝑋 ⊗ 𝑀 𝜕 𝑗 𝜓 ( 𝜃) , 𝜓 ( 𝜃) + 𝑘−1 ∑︁ 𝑖=0 𝛽 𝑖 𝑋 ⊗ |0 0| Φ 𝑖 𝑗( 𝜃) .…”