Resource-efficient quantum algorithm for linear systems of equations

TL;DR

This work presents an original algorithmic procedure to solve the Quantum Linear System Problem (QLSP), which combines ideas from Variational Quantum Algorithms (VQA) and the framework of classical shadows, and develops the Shadow Quantum Linear Solver (SQLS), a quantum algorithm solving the QLSP avoiding the need for large controlled unitaries.

Abstract

Finding the solution to linear systems is at the heart of many applications in science and technology. Over the years a number of algorithms have been proposed to solve this problem on a digital quantum device, yet most of these are too demanding to be applied to the current noisy hardware. In this work, an original algorithmic procedure to solve the Quantum Linear System Problem (QLSP) is presented, which combines ideas from Variational Quantum Algorithms (VQA) and the framework of classical shadows. The result is the Shadow Quantum Linear Solver (SQLS), a quantum algorithm solving the QLSP avoiding the need for large controlled unitaries, requiring a number of qubits that is logarithmic in the system size. In particular, our heuristics show an exponential advantage of the SQLS in circuit execution per cost function evaluation when compared to other notorious variational approaches to solving linear systems of equations. We test the convergence of the SQLS on a number of linear systems, and results highlight how the theoretical bounds on the number of resources used by the SQLS are conservative. Finally, we apply this algorithm to a physical problem of practical relevance, by leveraging decomposition theorems from linear algebra to solve the discretized Laplace Equation in a 2D grid for the first time using a hybrid quantum algorithm.

Figures (6)

• Figure 1: Schematic diagram for the SQLS. The aim is to solve a linear system $A\Vec{x_0} = \Vec{b}$. The inputs are: a matrix $A$, a unitary $U$, both written as a linear combination of Pauli strings, and form a linear system of Equations, $A\ket{x} = \ket{b}$ in which $U\ket{0} = \ket{b}$. The solution is found by using a hybrid classical-quantum variational algorithm were the parameters $\Vec{\theta}$ of a parameterized quantum circuit, $V(\Vec{\theta})$, will be optimized through classical optimization techniques. Due to the assumptions on the form of $A$ and $U$, the cost function, $C_L$ (Eq. \ref{['eq:local cost']}), involves the calculation of linear sums of expectation values of Pauli strings, which can be calculated using a small number of shallow circuits through classical shadows. The optimization process terminates when the condition $C_L \leq \gamma$ (see Eq. \ref{['eq:bound']}) is reached, returning a set of parameters $\theta^*$ such that $V(\theta^*)\ket{0}=\ket{x^*} \approx \frac{1}{||x_0||} \Vec{x_0}$.
• Figure 2: The SQLS is compared to the VQLS in terms of resource usage. Panel (a) shows the schemes for the different circuit architectures employed by the two algorithms in evaluating a single term in the cost function (i.e., Eq. \ref{['eq:local_cost_2']}) where: (a)(i) is the schematic quantum circuit for the SQLS, (a)(ii) and (a)(iii) are the schematic circuit implementations for the VQLS. Panel (b) shows the scaling of the number of circuits per optimization step (or evaluation of the cost function), $N_{circuits}/step$, comparing between the VQLS and SQLS, respectively. The estimate is provided for a $2^{50} \times 2^{50}$ linear system of the form given in Eq. \ref{['eq:rqlsp']}, in which $k$ indicates the locality of the Pauli strings forming the linear system. Evidently, as the number of terms in the linear system $L$ increases (i.e., when the linear system becomes more complex), the SQLS has a major advantage in resource usage as compared to VQLS.
• Figure 3: A comparison of the time-to-solution for a number of different linear systems as obtained numerically by applying either the SQLS or the VQLS, respectively. We plot the average over 10 runs and the error bars are plotted using the standard deviation over the 10 runs.
• Figure 4: The potentials derived form the solution of a $16 \times 16$ discretized Laplace equation onto a $4 \times 4$ 2D grid. In (a) we show the solution that was reached using the SQLS, whilst in (b) we show the exact analytic solution. The solution reached with the SQLS has a 0.99 fidelity with respect to the analytic one. Both solutions have been normalized to have the same color range and be directly comparable.
• Figure 5: Results for the quantity $N_{circuits}/step$ as a function of the number of terms in the linear system, $L$, for the different algorithms: VQLS, VQLS with pre-processing ($\text{VQLS}^*$), and the SQLS (also with pre-processing). In this example, the numerical experiment created 30 random $2^{10} \times 2^{10}$ (i.e., 10 qubits dimensional space) linear problems with fixed number of Pauli strings, $L \in [4, 100]$, and fixed locality, $k=2$, calculating $N_{circuits}/step$ for each procedure. The plot shows that even with pre-processing, the SQLS displays an exponential advantage with the scaling when compared to the VQLS.