[Paper Review] Quantum state preparation without coherent arithmetic

Sam McArdle, András Gilyén+1 more|arXiv (Cornell University)|Oct 26, 2022

Quantum Computing Algorithms and Architecture18 citations

TL;DR

This paper presents a quantum state preparation method that uses quantum eigenvalue transformation (QET) to prepare states with amplitudes defined by a known function, bypassing the need for coherent arithmetic or amplitude oracles. By block-encoding a sine function and applying polynomial or Fourier approximations via QET, the method reduces ancilla qubit requirements to just 4 (or 3 for definite-parity functions), achieving order-of-magnitude qubit savings over state-of-the-art approaches while maintaining comparable Toffoli gate counts for well-approximable functions.

ABSTRACT

We introduce a versatile method for preparing a quantum state whose amplitudes are given by some known function. Unlike existing approaches, our method does not require handcrafted reversible arithmetic circuits, or quantum table reads, to encode the function values. Instead, we use a template quantum eigenvalue transformation circuit to convert a low cost block encoding of the sine function into the desired function. Our method uses only 4 ancilla qubits (3 if the approximating polynomial has definite parity), providing order-of-magnitude qubit count reductions compared to state-of-the-art approaches, while using a similar number of gates if the function can be well represented by a polynomial or Fourier approximation. Like black-box methods, the complexity of our approach depends on the 'L2-norm filling-fraction' of the function. We demonstrate the algorithmic utility of our method, including preparing Gaussian and Kaiser window states.

Motivation & Objective

• To develop a scalable, low-footprint method for preparing quantum states with amplitudes defined by continuous functions.
• To eliminate the need for handcrafted coherent arithmetic circuits or quantum memory loads in amplitude oracle construction.
• To reduce qubit overhead in fault-tolerant quantum algorithms by minimizing ancilla qubit usage.
• To enable a unified circuit template applicable across diverse functions such as Gaussians, Kaiser windows, and cycloids.
• To provide a framework for efficient state preparation using polynomial or Fourier series approximations via quantum singular value transformation (QSVT).

Proposed method

• The method uses a low-cost block encoding of the sine function, $ A = \sum_x \sin(x/N)\ket{x}\!\bra{x} $, as a base resource.
• It applies quantum eigenvalue transformation (QET) to map the eigenvalues of $ A $ to the desired function values via a polynomial or Fourier approximation of $ f((b-a)\arcsin(y)+a) $.
• The transformation is implemented using a QET circuit template with $ 4 $ ancilla qubits (3 if the approximating polynomial has definite parity).
• For Fourier-based approximation, the method uses $ d $-degree Fourier series and applies $ d $ applications of $ U(A) $ and $ U(A)^\dagger $, along with $ 8d+4 $ single-qubit rotations.
• The normalization is corrected via amplitude amplification, which adds 1 qubit and is applied in $ R=1 $ round when the filling-fraction $ \mathcal{F}_{\tilde{f}}^{[\infty]} \lesssim 1 $.
• The approach avoids discretizing function values, enabling continuous approximation instead of fixed-point arithmetic.

Experimental results

Research questions

• RQ1 Can quantum state preparation be achieved without constructing custom coherent arithmetic circuits for each function?
• RQ2 To what extent can ancilla qubit count be reduced in state preparation while maintaining gate count efficiency?
• RQ3 Can a single circuit template be reused across diverse functions like Gaussians, Kaiser windows, and cycloids?
• RQ4 How does the method perform for non-smooth or non-periodic functions such as $ \sqrt{\bar{x}} $?
• RQ5 What is the resource cost of preparing multivariate functions using this QET-based framework?

Key findings

• The method reduces ancilla qubit count to 4 (or 3 for definite-parity functions), achieving order-of-magnitude savings compared to state-of-the-art amplitude-oracle-based approaches.
• For a 32-qubit Gaussian state, the method requires approximately $ 1.2 \times 10^5 $ Toffoli gates and 4 ancilla qubits, with comparable gate count to prior methods but significantly reduced qubit footprint.
• For a 32-qubit Kaiser window state, the method uses $ 1.5 \times 10^5 $ Toffoli gates and 4 ancilla qubits, again with reduced qubit cost.
• For the cycloid function over $ \bar{x} \in [0, 2\pi] $, a $ d=120 $ Fourier series achieves $ L_2 $ error $ < 10^{-3} $, requiring $ \approx 7.35 \times 10^5 $ Toffoli gates and 2 ancilla qubits.
• The method enables preparation of functions like $ \tanh $ and $ \mathrm{cycloid} $ that are difficult to implement with black-box or GR-based amplitude oracles due to lack of closed-form integrals.
• The approach is not efficient for $ \sqrt{\bar{x}} $ on $ [0,1] $ due to non-differentiability at 0 and Gibbs phenomenon in Fourier series, suggesting need for alternative methods in such cases.