Sublinear Classical-to-Quantum Data Encoding using $n$-Toffoli Gates

TL;DR

This work tackles the challenge of loading classical data into quantum states by proposing a sublinear-depth amplitude-encoding scheme based on multi-controlled X gates. It builds an isomorphism between target states and hypercube graphs and formulates the encoding as a sequence of controlled operations whose cost is optimized via a Kronecker-decomposition approach. The method yields poly-logarithmic-depth behavior for structured data (e.g., Gaussian, Ricker) and an efficient core variant suitable for NISQ devices, with explicit probabilistic success $\rho$ and potential amplification. Overall, the paper delivers a gate-efficient, hardware-friendly framework for quantum state preparation that extends the utility window for near-term quantum architectures. Mathematical constructs such as $N=2^n$, $\boldsymbol{B}\in\mathbb{F}_2^{N\times L}$, and $\rho=\frac{1}{N}\sum_i (v_i/v_\infty)^2$ are central to the method and its performance characterization.

Abstract

Quantum state preparation, also known as encoding or embedding, is a crucial initial step in many quantum algorithms and often constrains theoretical quantum speedup in fields such as quantum machine learning and linear equation solvers. One common strategy is amplitude encoding, which embeds a classical input vector of size N=2\textsuperscript{n} in the amplitudes of an n-qubit register. For arbitrary vectors, the circuit depth typically scales linearly with the input size N, rapidly becoming unfeasible on near-term hardware. We propose a general-purpose procedure with sublinear average depth in N, increasing the window of utility. Our amplitude encoding method encodes arbitrary complex vectors of size N=2\textsuperscript{n} at any desired binary precision using a register with n qubits plus 2 ancillas and a sublinear number of multi-controlled NOT (MCX) gates, at the cost of a probabilistic success rate proportional to the sparsity of the encoded data. The core idea of our procedure is to construct an isomorphism between target states and hypercube graphs, in which specific reflections correspond to MCX gates. This reformulates the state preparation problem in terms of permutations and \emph{binary addition}. The use of MCX gates as fundamental operations makes this approach particularly suitable for quantum platforms such as \emph{ion traps} and \emph{neutral atom devices}. This geometrical perspective paves the way for more gate-efficient algorithms suitable for near-term hardware applications.

Figures (10)

• Figure 1: Overview of the full algorithm. Input data (step 1) is transformed to renormalized angles whose binary expansion is stored in a matrix (step 2, see Sec. \ref{['s:preprocessing']}). Each column can be encoded with a circuit containing fully controlled MCX (see Sec. \ref{['s:general_procedure']} and Sec. \ref{['s:W_Implementation']}), which we optimize using the tree algorithm (step 3, see Sec. \ref{['s:tree_algorithm']}). We further reduce the cost by constructing an optimized order of applying the encoding layers (step 4, see Fig. \ref{['fig:solution_to_TSP']}) to produce the encoded state (step 5, see \ref{['eq:final_superposition']}).
• Figure 2: An encoding layer for the first row of the matrix $\mathbf{B}$.
• Figure 3: Linear-path circuit that encodes the state $|\zeta\rangle$ starting from the initial state $|0\rangle$. It alternates the $L$ shift gates $\mathcal{W}_{\bm{\Delta}_{j,j+1}}$ and the controlled rotations $CR_y(\phi_j, T\to F)$ for $j\in \{0,1,\dots,L-1\}$ followed by the final disentangling shift $\mathcal{W}_{\bm{\Delta}_{L,L+1}}$. The costs $|\mathcal{W}_{\bm{\Delta}}|$ can be reduced by using a better permutation $\sigma$ of the $(\mathcal{W}, R_y)$ layers. The full protocol is described in Algorithm \ref{['al:Mcx_ampl_enc']}.
• Figure 4: Example of the path P for the input binary matrix in \ref{['eq:example_B']}. The nodes are represented by the binary vectors, which are the columns of $\mathbf{B}$ plus the initial and final all-zero state, that need to be encoded into the superposition state of the three registers.
• Figure 5: Complete graph representing all the possible encoding paths along the columns of $\mathbf P$, starting and ending with the all-zero columns $\bm P _{:,0}=\bm P _{:,L+1}$, Each edge $(\bm P_{:,i},\bm P_{:,j})$ is associated with a vector $\bm \Delta_{i,j}=\bm P_{:,i} \oplus \bm P_{:,j}$ and with a cost $|\mathcal{W}_{\bm \Delta_{i,j}}|$, which is the amount of MCX in the decomposition of the shift operator. We see in red the optimal path given by the permutation $\sigma = (4)(2)(3)(1)(5)$ for the example in \ref{['eq:example_B']}.