Constant-Factor Improvements in Quantum Algorithms for Linear Differential Equations

TL;DR

This work delivers a non-asymptotic, constant-factor resource analysis for the Linear Combination of Hamiltonian Simulations (LCHS) quantum solver applied to time-independent linear differential equations $\frac{du}{dt} = -A u$. By deriving tight bounds on integral truncation via the Lambert $W$ function and discretization via Gaussian quadrature, and by constructing an efficient block-encoded implementation using PREP/SELECT and robust amplitude amplification, the authors show LCHS can achieve 100–200x reductions in block-encoding query counts compared to prior methods like the Randomization Linear Solver (RLS). The results include explicit formulas for truncation $K$, discretization $Q$, and the total summands $M$, as well as an end-to-end error budget and practical numerical results illustrating speedups that grow as the relative cost of initial-state preparation increases. The work provides a concrete toolkit for estimating and optimizing quantum resource costs for differential-equation solvers, and identifies avenues for further optimization and application-specific improvements.

Abstract

Finding the solution to linear ordinary differential equations of the form $\partial_t u(t) = -A(t)u(t)$ has been a promising theoretical avenue for \textit{asymptotic} quantum speedups. However, despite the improvements to existing quantum differential equation solvers over the years, little is known about \textit{constant factor} costs of such quantum algorithms. This makes it challenging to assess the prospects for using these algorithms in practice. In this work, we prove constant factor bounds for a promising new quantum differential equation solver, the linear combination of Hamiltonian simulation (LCHS) algorithm. Our bounds are formulated as the number of queries to a unitary $U_A$ that block encodes the generator $A$. In doing so, we make several algorithmic improvements such as tighter truncation and discretization bounds on the LCHS kernel integral, a more efficient quantum compilation scheme for the SELECT operator in LCHS, as well as use of a constant-factor bound for oblivious amplitude amplification, which may be of general interest. To the best of our knowledge, our new formulae improve over previous state of the art by at least two orders of magnitude, where the speedup can be far greater if state preparation has a significant cost. Accordingly, for any previous resource estimates of time-independent linear differential equations for the most general case whereby the dynamics are not \textit{fast-forwardable}, these findings provide a 100-200x reduction in runtime costs. This analysis contributes towards establishing more promising applications for quantum computing.

Figures (3)

• Figure 1: a) A comparison of $K$ upper bounds, which is the size of the truncation region of the integral which subsequently impacts the block encoding sub-normalization factor of the LCHS algorithm. b) A comparison of $M$ upper bounds which modulates the number of ancilla qubits and rotation gates required in constructing the block encoding of the LCHS formula. The improvement here is due to both improvements in $K$ and $M$ bounds.
• Figure 2: A comparison of the LCHS constant factor bounds and the randomization linear solver (RLS) method from Ref jennings2024cost. The numerics are conducted such that the total error satisfies $\epsilon\leq 10^{-10}$, and the block encoding constant is $\alpha=1$, though we expect comparable speedups to persist when varying these parameters. For the equal-error data, the errors for each subroutine are evenly distributed such that the total error is less than $\epsilon\leq 10^{-10}$, and for the other cases the optimizer finds a more efficient distribution. In the optimized cases it is found that $\beta \in [0.7403, 0.8127]$, yielding $\|c\|_1 \in [1.385,1.585]$. We note that, although the randomization linear solver cost data appears to become sub-linear, the curve actually eventually returns to being linear, though with a smaller intercept; the location of this transition is governed by the $\mu_P$ parameter in jennings2024cost. This is likely a numerical artifact given that we are limited in our accuracy regarding how closely $\mu_P$ can approach zero. Most of the data on this plot can be explored interactively at the following https://www.desmos.com/calculator/wwryu4lhnx.
• Figure 3: The Speedup of LCHS over the randomization linear solver (RLS) approach jennings2024cost as a function of (a) the ratio of the state preparation oracle $U_0$ time cost to the block encoding $U_A$ time cost, and (b) the simulation time in units of the block encoding constant ($\alpha =1$) given that state preparation is free ($\chi$=0). We perform this fit only over the region where the $\mu_P \to 0^-$ limit is valid, namely where $t\mu_P << 1$. The most performant LCHS simulation data is used to compute $\ell$. As an examples of time cost, one could consider them to be measured in terms of number of $T$ gates beverland2022assessing or the active volume litinski2022active of the compiled quantum circuits.

Theorems & Definitions (40)

• Theorem 2: Time-Independent LCHS Identity
• proof
• Lemma 3
• proof
• Lemma 4: Quadrature points
• proof
• Lemma 5: LCHS Cost and Error
• proof
• Definition 6: State-Preparation-Pair
• Lemma 7: $\mathcal{S}$ block encoding