Stab-QRAM: An All-Clifford Quantum Random Access Memory for Special Data

TL;DR

Stab-QRAM's utility is highlighted as a resource-efficient oracle for applications in discrete dynamical systems, and as a core component in Quantum Linear Systems Algorithms, providing a practical pathway for executing data-intensive tasks on emerging quantum hardware.

Abstract

Quantum random access memories (QRAMs) are pivotal for data-intensive quantum algorithms, but existing general-purpose and domain-specific architectures are hampered by a critical bottleneck: a heavy reliance on non-Clifford gates (e.g., T-gates), which are prohibitively expensive to implement fault-tolerantly. To address this challenge, we introduce the Stabilizer-QRAM (Stab-QRAM), a domain-specific architecture tailored for data with an affine Boolean structure ($f(\mathbf{x}) = A\mathbf{x} + \mathbf{b}$ over $\mathbb{F}_2$), a class of functions vital for optimization, time-series analysis, and quantum linear systems algorithms. We demonstrate that the gate interactions required to implement the matrix $A$ form a bipartite graph. By applying König's edge-coloring theorem to this graph, we prove that Stab-QRAM achieves an optimal logical circuit depth of $O(\log N)$ for $N$ data items, matching its $O(\log N)$ space complexity. Critically, the Stab-QRAM is constructed exclusively from Clifford gates (CNOT and X), resulting in a zero $T$-count. This design completely circumvents the non-Clifford bottleneck, eliminating the need for costly magic state distillation and making it exceptionally suited for early fault-tolerant quantum computing platforms. We highlight Stab-QRAM's utility as a resource-efficient oracle for applications in discrete dynamical systems, and as a core component in Quantum Linear Systems Algorithms, providing a practical pathway for executing data-intensive tasks on emerging quantum hardware.

Figures (4)

• Figure 1: Analysis of logical circuit depth scaling. (a) For a fixed size ($n=m=50$), the average depth increases with matrix density $p$, remaining well below the theoretical maximum. (b) The average depth scales linearly against the storage capacity $N=2^n$ on a logarithmic axis, visually confirming the $O(\log N)$ complexity.
• Figure 2: Circuit depth versus matrix rank. A scatter plot for randomly generated matrices reveals no direct correlation between the circuit depth and the matrix rank over $\mathbb{F}_2$. This highlights that depth is a local property (maximum degree $\Delta$) rather than a global algebraic property.
• Figure 3: CNOT gate count analysis. (a) Average gate count scales linearly with density $p$ of matrix $A$ ($m = n =50$). (b) For fixed density, count scales quadratically with matrix dimension $n$ ($m=n$). Simulated data (black) aligns with theoretical expectation (red dashes).
• Figure 4: Analysis of maximum CNOT locality after mapping. Color indicates average maximum shortest-path distance $d$, with darker shades showing better locality (fewer SWAPs). (a) For fixed size ($n=m=25$), distance rises with density $p$ but falls with connectivity $k$. (b) For fixed density ($p=0.25$), distance grows with $n$, mitigated by higher $k$.