Tensor-Programmable Quantum Circuits for Solving Differential Equations

Abstract

We present a quantum solver for partial differential equations based on a flexible matrix product operator representation. Utilizing mid-circuit measurements and a state-dependent norm correction, this scheme overcomes the restriction of unitary operators. Hence, it allows for the direct implementation of a broad class of differential equations governing the dynamics of classical and quantum systems. The capabilities of the framework are demonstrated for linear and non-linear partial differential equations using the example of the linearized Euler equations with absorbing boundaries and the nonlinear Burgers' equation. For a turbulence data set, we demonstrate potential advantages of the quantum tensor scheme over its classical counterparts.

Figures (8)

• Figure 1: (a) Hybrid quantum-classical routine to solve linear PDEs iteratively in a variational manner. The computation of the unitary gates $\hat{U}_{\text{MPO}}$ representing the operator $\hat{O}$ as well as the optimization routine take place on a classical computer (CC, upper part). The overlap $\bra{0}\hat{U}^{\dagger}(\boldsymbol{\theta}_{j+1})\hat{O}\hat{U}(\boldsymbol{\theta}_j)\ket{0}$ that determines the cost function, necessary to compute the solution at the next iteration step, is computed on the quantum computer (QC, lower part) using an adapted Hadamard test. The angles $\boldsymbol{\theta}_j$ describe a previous iteration step, while $\boldsymbol{\theta}_{j+1}$ are to be determined during the classical optimization process. This procedure allows to map classical differential solvers to the quantum computer. (b) Solution of the Euler equations with our quantum differential solver (QDS) for 45 time steps $dt$ encoded into 6 ansatz qubits. Depicted is the discretized velocity $u(x,t)$ evolution over time which arises is induced by a periodic pressure point source at $x=0$. (c) Solution of the non-linear Burgers' equations. Depicted is the discretized velocity field $u(x,t)$ over time, where initial Gauss peak evolves into a shock wave. A detailed description of all system and training parameters is given in \ref{['subsec:Parameter']}.
• Figure 2: Quantum circuit to encode one time-step of a non-linear PDE. Linear and non-linear operators are separated into two MPOs and an additional auxillary qubit is introduced allowing to create a weighted superposition of the linear and non-linear contributions. Double control of the global ancilla qubit and the extra auxiliary qubit can be avoided by applying $\mathrm{CNOT}$ gates before and after the non-linear operators, where the global ancilla acts as control and the extra auxiliary qubit as target. Then, the control on the global auxillary qubit can be omitted. The concrete decomposition of the $\mathrm{CNOT}$ gate is shown for the example of $n=3$.
• Figure 3: (a) Brickwall ansatz, with variational parameters. Each 2-qubit block is composed of two $\hat{R}_Y(\theta_i)$-gates and one $\mathrm{CNOT}$-gate. The ansatz can be used for a variable number of layers $L$, with each layer consisting of one column of 2-qubit blocks. (b) Evolution of the relative error $\bar{\epsilon}_{\phi}(t)$ (cf. \ref{['eq:rel_error']}) for the Euler's equation. Here, $\phi(x,t)$ corresponds to the discretized solutions $u(x,t)$ and $p(x,t)$ of the Euler equation. To represent the solutions, a brickwall ansatz with $6$ qubits and $14$ layers is used. (c) Evolution of the relative error $\bar{\epsilon}_{u}(t)$ for the Burgers' equation. To represent the solutions, a brickwall ansatz with $6$ qubits and $10$ layers is used.
• Figure 4: Representation capabilities of the brickwall and the MPS ansatz for a three-dimensional turbulent flow defined on $N$ grid points. (a) Number of parameters to represent an increasing section of an isotropic flow field of total size $N_{\textrm{tot}}=1024^3$ approximated up to an accuracy of $\bar{\epsilon}_v=0.01$. (b) Estimate of the minimal required cost $T_{\textrm{min}}$ computed for the given number of parameters from panel (a) and for an allowed maximal the sampling error $\epsilon=0.01$ The two minimal costs for the quantum scheme give the best (purple, circle) and the worst (green, diamont) case scenario for the cost contribution of the sampling error. The dashed lines corresponds to a liner interpolation in the log-log-scale.
• Figure 5: Sketch of the execution of a single Riemannian gradient step on the tensors of a unitary MPO $\mathcal{Q}$ that approximates a target MPO $\mathcal{M}$. It can be divided into three sub-steps: I) for each core $Q_j$ the gradient is computed by deriving the cost function $C$ with respect to its complex conjugate $Q_j^*$, we denote the result by $g$, II) the gradient $g$ is projected onto the tangent space of $Q_j$ via $g - \frac{1}{2} Q_j (Q_j^T g + g^T Q_j) \vcentcolon= G$Luchnikov2021, where we have defined the Riemannian gradient $G$, III) the new $Q_j$ is found by a retraction antiparallel to the Riemannian gradient, where the magnitude of the update step is controlled by the learning rate $\mu$. Using the QR decomposition as a retraction map, this last step has the form $\text{Retr}_{Q_j}^{QR}(- \mu G) = QR \left(Q_j - \mu G \right)$. We note that for the core $Q_1$, which is not in the Stiefel manifold, special measures must be taken. After step (I), $Q_j$ and the gradient $g$ must be transposed, then (II) and (III) are carried out, and the transposed result then gives the new $Q_j$.