Universal Quantum Random Access Memory: A Data-Independent Unitary Construction

TL;DR

QRAM faces obstacles from data-dependent unitaries and routing-based noise. This work introduces Universal QRAM, a data-independent unitary $U_{\text{QRAM}}$ in which memory qubits act as quantum control lines, yielding a fixed, block-diagonal permutation that retrieves $K$-bit words across $N$ addresses. The construction decomposes into exactly $N K$ multi-controlled $X$ gates and is characterized by closed-form formulas for qubit counts, block structure, and dimension; it is proven to be a permutation, unique under a complete data specification, and verified for $N \in \{2,4,8,16\}$ and $K \in \{1,2,3,4\}$. The approach contrasts with bucket-brigade and QROM by eliminating data-dependent unitaries and routing, enabling seamless integration into Grover search or other coherent data-driven protocols. Overall, Universal QRAM offers a conceptually clean, implementable path toward coherent data access with straightforward uncomputation and composability in quantum algorithms, especially where classical data are stored in memory qubits or where quantum data storage is required in the memory cells.

Abstract

We present a construction for Quantum Random Access Memory (QRAM) that achieves a single, data-independent unitary operator. Unlike routing-based approaches or circuit methods that yield data-dependent unitaries, our Universal QRAM encodes data in memory qubits that act as quantum control signals within a block-diagonal permutation structure. The key insight is that memory qubits serve as control signals, enabling coherent lookup when addresses are in superposition. For N addresses with K-bit data words, the construction requires $\log_2 N + K + NK$ qubits and decomposes into exactly $NK$ multi-controlled gates. We verify the construction for $N \in \{2, 4, 8, 16\}$ and $K \in \{1, 2, 3, 4\}$, confirming that the resulting unitary is a pure permutation matrix with zero error across all data configurations. This approach simplifies QRAM implementation by separating fixed circuit structure from variable data encoding.

Figures (9)

• Figure 1: Abstract QRAM operation. The address register selects which data value is XORed into the output register. The memory register is preserved.
• Figure 2: Register layout for $N=4$, $K=1$: 2 address qubits, 1 output qubit, 4 memory qubits (7 total).
• Figure 3: Universal QRAM circuit for $N=2$, $K=1$: 1 address qubit, 1 output qubit, 2 memory qubits.
• Figure 4: Universal QRAM circuit for $N=4$, $K=1$. Each gate activates for exactly one address value (determined by the address controls) and flips the output if the corresponding memory bit is 1. Open circles ($\circ$) denote control on $\ket{0}$, filled circles ($\bullet$) denote control on $\ket{1}$.
• Figure 5: Universal QRAM circuit for $N=2$, $K=2$. Memory qubits $m_i^j$ store bit $j$ of address $i$.