Logarithmic-depth quantum state preparation of polynomials

Abstract

Quantum state preparation is a central primitive in many quantum algorithms, yet it is generally resource intensive, with efficient constructions known only for structured families of states. This work introduces a method for preparing quantum states whose amplitudes are given by a degree$-d$ polynomial, using circuits with logarithmic depth in the number $n$ of qubits and only $\mathcal O(n)$ ancilla qubits, improving previous approaches that required linear-depth circuits. The construction first relies on a block-encoding of an affine diagonal operator based on its Pauli-basis decomposition, which involves only $n$ terms. A modified linear-combination-of-unitaries (LCU) technique is introduced to implement this decomposition in logarithmic depth, together with a novel circuit for the EXACT-one oracle that flags basis states in which exactly one qubit is in the state $|1\rangle$. It then uses a generalized quantum eigenvalue transformation (GQET) to promote this affine operator to an arbitrary degree polynomial. Theoretical analysis and numerical simulations are reported along with a proof-of-principle implementation on a trapped-ion quantum processor using $14$ qubits and more than $500$ primitive quantum gates. Because polynomial approximations are ubiquitous in scientific computing, this construction provides a scalable and resource-efficient approach to quantum state preparation, further improving the potential of quantum algorithms in fields such as chemistry, physics, engineering, and finance.

Figures (8)

• Figure 1: High-level structure of the affine block-encoding $U_{\text{aff}}$ construction of $1-2p_nL_n$ where $\theta_k = 2 \arcsin{\left(\sqrt{\frac{1}{1+2^k}}\right)}$. The circuit consists of PREPARE ($\Psi_n$), a unary SELECT ($S_n$) controlled on the one-hot subspace, and $\texttt{PREPARE}^{\dagger}$, followed by projection (postselection) onto the designated ancilla outcome. Notice that the projection is not necessary when performing the amplitude amplification with $\mathcal{O} (1)$ steps and the GQET protocol.
• Figure 2: Quantum circuit for the generalized quantum eigenvalue transform (GQET) of a matrix $A$ block-encoded in a unitary $U=A/\alpha***=(A/\alpha)\otimes \ket{0}\bra{0}+ \hdots$ by a polynomial $p(x)=\sum_{k=0}^da_kx^k=\sum_{k=0}^db_kT_k(x)$, where $T_k$ is the $k$-th Chebyshev polynomial of the first kind. The phase $(\{\theta_i\},\{\phi_i=\phi'_i-\pi\},\lambda)$ are those computed for the generalized quantum signal processing (GQSP) associated with the polynomial $P(z)=\sum_{k=0}^db_kz^k$.
• Figure 3: Scaling behavior of the SPUE construction codegit. (\ref{['fig:spue_depth']}) Circuit depth grows logarithmically with the number of qubits. (\ref{['fig:pn_conv']}) The success probability $P_n = \frac{1}{2^n} \sum\limits_{x \in B_n} \left(1-\frac{2p_nx}{1-\frac{1}{2^n}}\right)^2$ rapidly converges to a constant value $P \approx 0.3957$ .
• Figure 4: Hardware demonstration on Quantinuum H2 of a degree-one polynomial-amplitude state preparation. The plotted values correspond to the square roots of the postselected measurement frequencies of each computational-basis outcome.
• Figure 5:

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1: (Symmetric) Projected Unitary Encoding
• Theorem 4
• Definition 2: Signal processing polynomial
• Definition 3: Scaling factor
• Theorem 5
• Definition 4: Linear diagonal operator
• Proposition 1