Are Randomized Quantum Linear Systems Solvers Practical?

TL;DR

The paper quantitatively assesses the end-to-end resources required for a randomized quantum linear system solver that estimates scalar properties of the matrix inverse via a Fourier-series approximation. It develops a two-stage approach: classical construction of a Fourier-series surrogate for $A^{-1}$ and quantum sampling of time evolutions $e^{-iAt}$ using either a product-formula or a random Taylor expansion, with Hadamard-test-based estimations of overlaps. By deriving explicit non-asymptotic bounds for truncation, discretization, sampling, and gate complexities, the work both connects theory to hardware and reveals prohibitive sampling costs in practice, especially for the RTE approach, thereby challenging the practicality of Fourier-series-based randomized QLS solvers for moderate problem sizes. The findings suggest that while such methods provide a rigorous resource-analytic bridge, block-encoding-based methods with favorable asymptotic behavior may be more practical for near-term and medium-scale problems, motivating future work on preconditioning and structure-aware implementations.

Abstract

Randomized quantum algorithms have been proposed in the context of quantum simulation and quantum linear algebra with the goal of constructing shallower circuits than methods based on block encodings. While the algorithmic complexities of such randomized schemes have been shown to be suboptimal, it has been speculated that they may offer benefits in the early fault-tolerant era where circuit depth comes at a premium. In this work, we investigate the end-to-end resource requirements for a randomized quantum linear systems solver to estimate scalar properties of the matrix inverse by combining sampling from a Fourier series with Hamiltonian simulation. We derive explicit bounds on all relevant algorithmic parameters that control the total error. We analyze two algorithmic kernels for Hamiltonian simulation: a second order product formula approximation and a method called random Taylor expansion (RTE). Finally, we provide numerical demonstrations that confirm the validity of our analytical results and question the actual practicality of the randomized Fourier series-based approach for linear systems problems as we show that the sampling complexity can grow exponentially. By providing explicit bounds, our work serves as a bridge between theoretical algorithmic proposals and efficient hardware implementations while also enabling fair comparisons to alternative algorithms that exhibit optimal asymptotic complexities at the cost of large resource overheads.

Figures (4)

• Figure 1: Randomized framework for sampling from solutions to the quantum linear systems (QLS) problem $\bra{\phi} A^{-1} \ket{\psi}$ for a Hermitian matrix $A$, consisting of a classical processing step (left) which constructs a provably accurate Fourier series approximation to the inverse function, followed by repeatedly sampling a Fourier time $t$ and constructing an approximate time evolution operator $U \approx e^{-\mathrm{i} At}$ using either a Product Formula (PF) or Random Taylor Expansion (RTE). Next the quantum processing step (right) uses $\log(N) + 1$ qubits to measure $\bra{\phi} U \ket{\psi}$ for each sample. The statistical average over all samples converges to $\bra{\phi} A^{-1} \ket{\psi}$.
• Figure 2: Flowchart outlining the randomized quantum algorithm for estimating $\bra{\phi}A^{-1}\ket{\psi}$. Orange boxes are classical data processing steps, with boxes having a white background describing inputs, intermediate results, and outputs, and those filled with an orange background describing a classical algorithm. The blue box shows the Hadamard test circuit which is the only step that requires a QPU. The dashed lines indicate the iterative portion of the algorithm that uses a combination of classical and quantum processing. The dashed arrows indicate a choice in the method used to perform Hamiltonian simulation.
• Figure 3: Convergence in the root mean squared error (RMSE) as a function of the total number of simulated shots, where the exponential $e^{-iA\tau}$ corresponding to each Fourier sample is approximated by a $2^{\mathrm{nd}}$ order product formula. Results are shown for three different choices of Trotter number $r$ and for exact numerical evaluation of the real-time evolutions. To each such approximation, we add noise sampled from a Gaussian centered at 0 with width 1 to simulate the noise coming from a single shot of the Hadamard test circuit. We note that not choosing the Trotter number $r$ to depend on the Fourier time $\tau$ leads to a plateau in the RMSE as a result of the bias dominating over the sampling error.
• Figure 4: Convergence in the root mean squared error (RMSE) as a function of the total number of RTE samples for the task of approximating $\Re \bra{0}e^{-\mathrm{i} A \tau}\ket{0}$. The RTE procedure uses $r = t_{\mathrm{max}} = 100$ and simulation time $\tau$ is varied from $1$ to $80$ and hence the ratio $\tau^2/r$ varies from $0.01$ to $64$. We observe that as the ratio $\tau^2/r$ increases past $1$, the prefactor of the RTE method is severely impacted as illustrated by the large RMSE for the red and purple curves.

Theorems & Definitions (22)

• Definition 2: $\epsilon$-close integral truncation
• Definition 3: $\epsilon$-close integral discretization
• Theorem 4
• proof
• Theorem 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof