The discrete adiabatic quantum linear system solver has lower constant factors than the randomized adiabatic solver

TL;DR

The paper investigates the discrete adiabatic quantum walk (QW) solver for the quantum linear system problem (QLSP) and compares it against the randomized adiabatic method (RM). Through extensive numerical tests on random matrices, it finds a practical constant factor of $α\approx1.84$ in the walk cost, about $1{,}200×$ smaller than the theoretical upper bound $α=2305$, making QW considerably more efficient in practice. It also shows that QW is, on average, roughly 7× faster than RM, and that including a filtering stage yields at least a 3× improvement in total cost for realistic error targets $ε$. Overall, the results suggest that the discrete adiabatic approach is highly effective for solving QLSP in realistic regimes and can outperform randomized adiabatic strategies in practice, despite the presence of conservative analytical bounds.

Abstract

The solution of linear systems of equations is the basis of many other quantum algorithms, and recent results provided an algorithm with optimal scaling in both the condition number $κ$ and the allowable error $ε$ [PRX Quantum \textbf{3}, 040303 (2022)]. That work was based on the discrete adiabatic theorem, and worked out an explicit constant factor for an upper bound on the complexity. Here we show via numerical testing on random matrices that the constant factor is in practice about 1,200 times smaller than the upper bound found numerically in the previous results. That means that this approach is far more efficient than might naively be expected from the upper bound. In particular, it is about an order of magnitude more efficient than using a randomised approach from [arXiv:2305.11352] that claimed to be more efficient.

Figures (14)

• Figure 1: A comparison between two probability distribution functions used in the Randomized method. Here the legend "JLPSS" refers to the one in jennings2023efficient and "Optimal" to the one in Sanders_2020. The gap $\Delta(v_j)$ is chosen to be $0.5$ in this test.
• Figure 2: Comparison between the quantum walk method and the Randomized method. The complexity is quantified by the number of calls to the block encoding of the matrix for the quantum walk case and by the total evolution time for the Randomized approach until we get the 2-norm error, $\Delta=0.4$ in the QW method and the infidelity error $\delta=0.4$ in for the Randomized method. The complexities are computed by the geometric mean of the 100 instances of $16$-dimensional matrices (solid line), and the box plots show their distributions. Part (a): positive-definite case. Part (b): general non-Hermitian case.
• Figure 3: The values of $\Delta$ as a function of $\epsilon$ that minimise the cost. The results are for $\alpha=0.17$ (green) and $\alpha=2$ (blue) using the cost in Eq. \ref{['eq:complex']}. Results using Eq. \ref{['eq:complex2']} are shown for $\beta=0.8$ (red) and $\beta=5.3$ (orange).
• Figure 4: The values of $\Delta$ as a function of $\epsilon$ that minimise the cost. We include results assuming various values of $\alpha$, as well as with interpolation of the value of $\alpha$ as a function of $\Delta$.
• Figure 5: The two contributions to the minimised costs as a function of $\epsilon$ for the discrete adiabatic approach and for the randomised walk. In both cases the scaling constant is fitted as a function of $\Delta$.