Efficient Quantum State Preparation with Bucket Brigade QRAM

TL;DR

This work addresses the data-loading bottleneck in quantum algorithms by introducing a memory-architecture-aware framework that embeds a Segment Tree within Bucket Brigade QRAM (BBQRAM). The authors design a memory layout that preserves the Segment Tree hierarchy, enabling logarithmic in superposition data retrieval, and define retrieval primitives plus the $U_{2CR}$ unitary to perform amplitude encoding. They prove a polylogarithmic state-preparation bound: encoding a matrix $A \in \mathbb{R}^{M\times N}$ requires $Θ(\log_2^2(MN))$ time on $Θ(\log_2(MN))$ qubits with constant ancilla under fixed precision, and they illustrate the method with a detailed numerical example. The framework provides a foundation for designing classical-to-quantum encoders that are aware of QRAM architecture, potentially enabling practical quantum advantages in linear algebra, finance, chemistry, and beyond.

Abstract

The preparation of data in quantum states is a critical component in the design of quantum algorithms. The cost of this step can significantly limit the realization of quantum advantage in domains such as machine learning, finance, and chemistry. One of the main approaches to achieve efficient state preparation is through the use of Quantum Random Access Memory (QRAM), a theoretical device for coherent data access with several proposed physical implementations. In this work, we present a framework that integrates the physical model of the Bucket Brigade QRAM (BBQRAM) with the classical data structure of the Segment Tree to achieve efficient state preparation. We introduce a memory layout that embeds a segment tree within BBQRAM memory cells by preserving the segment tree's hierarchy and supporting data retrieval in logarithmic time via specialized access primitives. We demonstrate that, under the proposed memory layout, our method encodes a matrix $A \in \mathbb{R}^{M \times N}$ in a quantum register of $Θ(\log_2(MN))$ qubits in $O(\log_2^2(MN))$ time using constant ancillary qubits under a fixed-precision assumption. We further illustrate the method through a numerical example. This framework provides theoretical support for quantum algorithms that assume negligible data loading overhead and establishes a foundation for designing classical-to-quantum encoding algorithms that are aware of the underlying physical QRAM architecture.

Figures (12)

• Figure 2: A circuit multiplexer implementing $\sum_{i=0}^{N-1} \lvert i \rangle\langle i \rvert U_i$ that prepares $N$ values in the form $\lvert x_i \rangle^t$.
• Figure 3: Bucket Brigade QRAM architecture (BBQRAM). Internal nodes act as switches that route address qubits, while leaf nodes represent memory cells storing the values $\lvert x_i \rangle$.
• Figure 4: QRAM routing algorithm for the address register $\lvert 01+ \rangle$. (Figure \ref{['fig:routing_1']}) The algorithm routes the first qubit $\lvert 0 \rangle$ through the tree until it reaches the first switch in the $\lvert \bullet \rangle$ state, which corresponds to the root. Then, the switch updates its state $\lvert \bullet \rangle$ to match the address incoming qubit $\lvert 0 \rangle$. As a result, any subsequent qubits that encounter this switch is routed to the left subtree. The same logic applies also for the next address qubit $\lvert 1 \rangle$ (Figure \ref{['fig:routing_2']}), but routing subsequent qubits to the right subtree. The last qubit register is in the superposition state $\lvert + \rangle$ (Figure \ref{['fig:routing_3']}), and linearity ensures that the switch routes the access coherently to both left and right memory cells, enabling equal superposition access to $\lvert x_2 \rangle$ and $\lvert x_3 \rangle$.
• Figure 5: Bus register traversal in BBQRAM. After establishing the access path for the address register $\lvert 01\gamma \rangle$, where $\lvert \gamma \rangle = \alpha\lvert 0 \rangle + \beta\lvert 1 \rangle$ with $\alpha, \beta \neq 0$, the bus register follows this path to reach the target memory cell. Because the memory data is encoded in the computational basis, we use CNOT gates to copy the memory contents into the bus register. This process does not violate the no-cloning theorem, as the CNOT gate copy only classical information encoded in the computational basis. Finally, the bus register returns to the root node along the same path.
• Figure 6: Retrieval from Bucket Brigade QRAM with Pipelined Routing

Theorems & Definitions (7)

• definition 1: Cascade of Controlled Unitary Gates
• Lemma 1: Cascade of Controlled Rotations Equivalence
• Lemma 2: Bucket Brigade QRAM Retrieval cost
• definition 2: Segment Tree of Squared Norms
• proposition 1: Memory Layout of Segment Tree in Bucket Brigade QRAM
• corollary 1: Quantum Retrieval Primitives
• Theorem 1: Efficient State preparation with BBQRAM