Quantum Time-Space Tradeoffs for Exponential Dynamic Programming

Abstract

We investigate the quantum algorithms for dynamic programming by Ambainis et al. (SODA'19). While giving provable complexity speedups and applicable to a variety of NP-hard problems, these algorithms have a notable drawback: they require a large amount of Quantum Random Access Memory (QRAM), which potentially could be very challenging to implement in a physical quantum computer. In this work, we study how we can improve the space complexity by trading it for time, while still retaining a speedup over the classical algorithms. We show novel quantum time-space tradeoffs, which we obtain by adjusting the parameters of these algorithms and combining them with "quantized" classical strategies.

Figures (4)

• Figure 1: Time-space tradeoffs for the divide & conquer problems. A point $(\mathcal{S}, \mathcal{T})$ denotes an algorithm with time complexity $\widetilde{\mathop{\mathrm{O}}\nolimits}(\mathcal{T}^n)$ that uses at most $\widetilde{\mathop{\mathrm{O}}\nolimits}(\mathcal{S}^n)$ QRAM space. Orange line denotes the optimization tradeoff of Theorem \ref{['thm:dnq-opt']}. Teal line denotes the improved tradeoff of Theorem \ref{['thm:dnq-imp']}. It is bounded below by $\mathcal{T} = 2/\mathcal{S}^{0.268}$ and above by $\mathcal{T} = 2/\mathcal{S}^{0.201}$ (dashed gray). Dotted line shows the function $\mathcal{T} = \mathcal{S}$, giving a barrier to improving the tradeoffs after $\mathcal{S} \approx 1.728$, which corresponds to the quantum dynamic programming algorithm of ABIKPV19 (red circle).
• Figure 2: Time-space tradeoffs for the permutation problems. A point $(\mathcal{S}, \mathcal{T})$ denotes an algorithm with time complexity $\widetilde{\mathop{\mathrm{O}}\nolimits}(\mathcal{T}^n)$ that uses at most $\widetilde{\mathop{\mathrm{O}}\nolimits}(\mathcal{S}^n)$ QRAM space. The discrete teal plot denotes the optimization tradeoff of Theorem \ref{['thm:hp-opt']}. It is bounded below by $\mathcal{T} = 2/\mathcal{S}^{0.161}$ and above by $\mathcal{T} = 2/\mathcal{S}^{0.099}$ (dashed gray). The orange line denotes the quantum pairwise scheme tradeoff of Theorem \ref{['thm:hp-ps']}. Dotted line shows the function $\mathcal{T} = \mathcal{S}$, giving a barrier to improving the tradeoffs after $\mathcal{S} \approx 1.817$, which corresponds to the quantum Hypercube Path dynamic programming algorithm of ABIKPV19 (red circle).
• Figure 3: Illustration of the recursive hypercube layering strategy. \ref{['fig:hp-ill-2-l']} depicts the case for $k=2$ layers. The initial preprocessing layer $A_1$ is highlighted in blue. Its size, determined by parameter $\alpha_1$, is constrained by the available memory, represented by the blue striped region. The middle layer $A_2$ is highlighted in red. The diagram shows three levels of recursion. At the final level, the subcube is small enough that its entire solution fits within the memory budget and can be computed classically. \ref{['fig:hp-ill-3-l']} illustrates the analogous structure for $k=3$ layers. The intermediate layer $A_2$ is highlighted in green.
• Figure 4: Quantum time-space tradeoffs for the Hypercube Path problem. A point at coordinates $(\mathcal{S}, \mathcal{T})$ denotes an algorithm with time complexity $\widetilde{\mathcal{O}}(\mathcal{T}^n)$ and QRAM space complexity $\widetilde{\mathcal{O}}(\mathcal{S}^n)$. Each colored curve corresponds to a different recursive strategy parameterized by $k$, the number of layers split per recursive call. Higher values of $k$ yield better tradeoffs. Dotted line shows the function $\mathcal{T} = \mathcal{S}$, giving a barrier to improving the tradeoffs after $\mathcal{S} \approx 1.817$, which corresponds to the quantum Hypercube Path dynamic programming algorithm of ABIKPV19 (red circle).

Theorems & Definitions (14)

• Theorem 1: Entropy approximation
• Theorem 2: Quantum minimum finding with erroneous oracle, Corollary 10 in ABBLS25
• Lemma 3: Fractalization
• proof
• Theorem 4
• proof
• Theorem 5
• proof
• Theorem 6
• proof