Fourier LCU for Non-Unitary Decompositions

• The paper introduces Fourier-LCU, a framework that decomposes non-unitary operators into linear combinations of unitaries using periodic extension and Fourier sine series for exponential error convergence.
• It converts sine series into complex-exponential forms to map operator terms onto pairs of unitaries, facilitating a block-encoding method with double-logarithmic subnormalization scaling.
• The approach employs convex optimization for coefficient regularization, achieving a Pareto-optimal trade-off between error tolerance and resource efficiency in quantum algorithm implementations.

A Fourier Linear Combination of Unitaries (Fourier-LCU) is a general analytic method for decomposing arbitrary non-unitary operators into accurate, exponentially convergent linear combinations of unitary operators. This is accomplished via smooth periodic extension and Fourier sine series techniques, yielding a block-encoding whose subnormalization parameter exhibits double-logarithmic scaling in the target error. The framework leverages convex optimization to regularize the coefficients for specific error budgets, tracing out a Pareto front for subnormalization-versus-error. These advances constitute a versatile approach for non-unitary quantum algorithms and circuits (Brearley et al., 25 Jan 2026).

1. Periodic Extension and Fourier Sine Series Construction

To represent an operator via a LCU, the core technical step is constructing a periodic extension of the identity function $f(\tau) = \tau$ within a given interval. Fixing $\tau \in [-\pi/\eta, \pi/\eta]$ for some $\eta > 1$, one extends $f(\tau)$ to $[−\pi, \pi]$ as a $2\pi$-periodic, odd, and infinitely differentiable function. This ensures analyticity and supports exponential Fourier coefficient decay.

The extension yields a truncated $m$-term sine series approximation:

$f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$

The optimal coefficients $a_k$ are determined by a continuous least-squares problem over $[−\pi/\eta, \pi/\eta]$:

$\tau \in [-\pi/\eta, \pi/\eta]$0

The resulting normal equations are $\tau \in [-\pi/\eta, \pi/\eta]$1, with

$\tau \in [-\pi/\eta, \pi/\eta]$2

Since $\tau \in [-\pi/\eta, \pi/\eta]$3, the coefficients $\tau \in [-\pi/\eta, \pi/\eta]$4 exhibit exponential decay in $\tau \in [-\pi/\eta, \pi/\eta]$5, ensuring rapid convergence.

2. Complex-Exponential Formulation and Unitary Mapping

The sine series can be rewritten using the Euler identity:

$\tau \in [-\pi/\eta, \pi/\eta]$6

Substitution yields:

$\tau \in [-\pi/\eta, \pi/\eta]$7

where $\tau \in [-\pi/\eta, \pi/\eta]$8, $\tau \in [-\pi/\eta, \pi/\eta]$9, and $\eta > 1$0. Consequently, each sine term maps to a pair of unitaries with complex weights, forming the desired LCU structure.

3. Application to Arbitrary Non-Unitary Operators

Let $\eta > 1$1 be a general (potentially non-unitary) operator. Decompose $\eta > 1$2 into Hermitian and anti-Hermitian components:

$\eta > 1$3

The LCU approximation proceeds by:

• Choosing $\eta > 1$4 so that $\eta > 1$5;
• Approximating $\eta > 1$6 and $\eta > 1$7 by sine series as above;
• Rewriting each sine term in unitary difference form.

The full LCU for $\eta > 1$8 to exponential error $\eta > 1$9 is:

$f(\tau)$0

Each $f(\tau)$1 is one of $f(\tau)$2 with real weights $f(\tau)$3 given by $f(\tau)$4 (up to phase).

4. Block-Encoding and Subnormalization Scaling

Employing standard LCU block-encoding [Childs–Wiebe 2012], one introduces $f(\tau)$5 ancillas, prepares amplitude state $f(\tau)$6, applies controlled-$f(\tau)$7 gates, and uncomputes via $f(\tau)$8:

$f(\tau)$9

with subnormalization parameter

$[−\pi, \pi]$0

Since $[−\pi, \pi]$1 decay exponentially and empirically $[−\pi, \pi]$2, the total normalization satisfies:

$[−\pi, \pi]$3

This double-logarithmic scaling in $[−\pi, \pi]$4 is a substantial improvement over previous polynomial relationships between subnormalization and error.

5. Coefficient Regularization and Pareto Front Optimization

Because the sine dictionary is overcomplete for $[−\pi, \pi]$5, there exist infinitely many coefficient sets yielding nearly identical $[−\pi, \pi]$6 error yet different $[−\pi, \pi]$7 summations (impacting $[−\pi, \pi]$8). Regularization is performed via convex optimization, exploiting the trade-off:

$[−\pi, \pi]$9

for $2\pi$0 and dictionary matrix $2\pi$1. At fixed error budget $2\pi$2, the $2\pi$3-minimization is:

$2\pi$4

Standard convex solvers or homotopy/LASSO-type path tracking yield the unique Pareto front $2\pi$5. It is proven that $2\pi$6 and $2\pi$7 are nonincreasing in $2\pi$8 and converge to a finite limit as $2\pi$9. Numerically, “sweeping” $m$0 down to zero identifies the lowest possible $m$1 at target $m$2.

6. Implementation Procedures and Practical Implications

The Fourier-LCU methodology is summarized by the following stepwise procedure:

1. Construct an analytic $m$3-periodic, odd extension of $m$4 on $m$5.
2. Compute the exponentially convergent truncated sine series via least squares.
3. Convert each $m$6 into LCUs of $m$7 for $m$8; assemble $m$9 via weighted sums.
4. Realize the decomposition as an $f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$0 block-encoding, utilizing $f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$1 scaling as $f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$2.
5. Optionally, re-optimize coefficients with $f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$3-regularized least squares to minimize $f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$4 at fixed $f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$5, thereby mapping out the Pareto front. All essential equations for $f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$6, $f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$7, $f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$8, and $f(\tau) \approx \sum_{k=1}^{m} a_k \sin(k \tau)$9 as well as the regularization trade-offs are explicitly stated; optimized $a_k$0 values for various $a_k$1 are tabulated in Table B of the corresponding source (Brearley et al., 25 Jan 2026).

A plausible implication is that non-unitary quantum algorithms leveraging Fourier-LCU can reach error targets at far lower resource cost than via polynomial-scaling frameworks, and coefficient regularization can yield trainable sparsity for practical block-encodings.