Binary Tree Block Encoding of Classical Matrix

TL;DR

This work addresses efficient block-encoding of classical matrices into quantum circuits under qubit-limited regimes. It introduces Binary Tree Block-encoding (BITBLE), built on two multiplexor decoupling schemes—permutative and recursive demultiplexors—to achieve $O(n2^{2n})$ parameter-time with $\Theta(2^{2n})$ memory for $2^n\times2^n$ matrices, using several ancilla qubits only. The paper details state-preparation via multiplexors, two normalization options ($\|A\|_F$ and $\mu_p(A^T)$), and the unitary constructions that realize $(\|A\|_F,n)$-block-encodings or $(\mu_p(A^T),n+2)$-block-encodings, supported by theoretical complexity results and extensive numerical benchmarks. Across random matrices, images, and discretized Laplacians, BITBLE consistently improves the time of classical synthesis and the size metric compared with FABLE and Qiskit, highlighting practical gains for high-dimensional, structured data in quantum algorithms. The approach paves the way for scalable, low-depth block-encodings suitable for near-term fault-tolerant settings and invites further parallelization and optimization studies.

Abstract

Block-encoding is a critical subroutine in quantum computing, enabling the transformation of classical data into a matrix representation within a quantum circuit. The resource trade-offs in simulating a block-encoding can be quantified by the circuit size, the normalization factor, and the time and space complexity of parameter computation. Previous studies have primarily focused either on the time and memory complexity of computing the parameters, or on the circuit size and normalization factor in isolation, often neglecting the balance between these trade-offs. In early fault-tolerant quantum computers, the number of qubits is limited. For a classical matrix of size $2^{n}\times 2^{n}$, our approach not only improves the time of decoupling unitary for block-encoding with time complexity $\mathcal{O}(n2^{2n})$ and memory complexity $Θ(2^{2n})$ using only a few ancilla qubits, but also demonstrates superior resource trade-offs. Our proposed block-encoding protocol is named Binary Tree Block-encoding (\texttt{BITBLE}). Under the benchmark, \textit{size metric}, defined by the product of the number of gates and the normalization factor, numerical experiments demonstrate the improvement of both resource trade-off and classical computing time efficiency of the \texttt{BITBLE} protocol. The algorithms are all open-source.

Figures (15)

• Figure 1: Uniformly controlled rotation decomposition PhysRevLett.93.130502 for multiplexor operations of Eq. \ref{['eq Multiplexed Rotation']} with size $n=2$. The control nodes of the CNOT gates in this decomposition are determined by performing an Exclusive Or (XOR) operation on $n$-bit Gray codes. Specifically, these nodes correspond to the positions of the '$1$' bits (highlighted in gray) in the XOR result between two-bit Gray codes.
• Figure 2: Permutative demultiplexor implement of controlled-$R_{\bm\alpha}^{[\beta_j]_{j=0}^{2^n-1}}$ with $k=1$ and $n=3$.
• Figure 3: The first recursion of recursive demultiplexor implements controlled--$R_{\alpha}^{[\beta_j]_{j=0}^7}$ operations ($k=1,n=3$) with the controlled qubit on the second qubit.
• Figure 4: Recursive demultiplexor implement of controlled-$R_{\bm\alpha}^{[\beta_j]_{j=0}^{2^n-1}}$ with $k=1$ and $n=3$.
• Figure 5: Quantum circuit for state preparation using multiplexor operations in the case of $n=3$.

Theorems & Definitions (9)

• Definition 1: 10.1145/3313276.3316366
• Lemma 1: Amankwah2022
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2: Time of single-qubit gates' parameters computation in BITBLE protocol
• proof