Quantum Prolate Diagonalization

• Quantum Prolate Diagonalization (QPD) is a hybrid classical–quantum method that leverages prolate spheroidal functions to optimally extract quantum spectral information.
• It reformulates the eigenvalue problem as a signal-processing task using autocorrelation functions and time–frequency subspace filtering.
• QPD achieves Heisenberg-limited precision with rigorous error bounds through short-time quantum evolutions and efficient classical postprocessing.

Quantum Prolate Diagonalization (QPD) is a hybrid classical–quantum methodology designed to optimally extract spectral information—such as ground and excited state energies—of quantum systems through signal-processing techniques based on prolate spheroidal wavefunction theory. Unlike conventional approaches, which focus on direct diagonalization of high-dimensional matrices or the standard time-independent Schrödinger equation, QPD reframes the eigenproblem in terms of the autocorrelation function of the quantum system and employs time–frequency optimal subspace filtering to achieve Heisenberg-limited accuracy with provable error bounds. QPD is particularly notable for its ability to address eigenvalue problems in large systems by operating with a manageable number of short-time quantum evolutions, coupled with rigorous classical postprocessing grounded in band- and time-limiting operator theory.

1. Foundations: Reformulating the Eigenproblem via Autocorrelation

The principal innovation in QPD is the use of the autocorrelation function

$C(t) = \langle\Psi | e^{iHt} | \Psi\rangle,$

where $|\Psi\rangle$ is an initial state and $H$ is the system’s Hamiltonian. This object encodes the complete energy spectrum and transition information of the system within its frequency content: $$C(t) = \sum_{j} |\langle E_j | \Psi\rangle|^2 e^{i E_j t},$$ with $\{E_j\}$ the eigenvalues of $H$. QPD bypasses the exponential scaling of wavefunction-based approaches by reframing the recovery of the spectrum as a signal-processing problem—estimating the frequencies $E_j$ from sampled values of $C(t)$. The quantum processor is used to efficiently generate $C(t_k)$ at a discrete set of times $\{t_k\}$ via the Hadamard test or related ancilla-based circuits.

2. Prolate Filter Diagonalization (PFD) and Subspace Construction

Central to QPD is the application of Prolate Filter Diagonalization (PFD): a subspace projection method leveraging the time- and band-limiting properties of prolate spheroidal wave functions (PSWFs).

• Basis filtering: The autocorrelation signal is projected onto a set of PSWFs $\{f_n(t)\}$ defined to be optimally concentrated in time $[-T, T]$ and frequency $[-W_f, W_f]$. These functions are each solutions to the integral equation for maximal simultaneous concentration in a time–frequency window, ensuring superior signal resolution.
• Generalized eigenvalue problem: Matrices $A$ and $B$ are constructed as

$$A_{sl} = -i \int_{-T}^T f_s(t) (\partial_t C * f_l)(t) dt,\quad B_{sl} = \int_{-T}^T f_s(t) (C * f_l)(t) dt,$$

with $*$ denoting convolution. The QPD procedure solves the generalized eigenproblem $A \mathbf{x} = E B \mathbf{x}$ to extract energy estimates.

• Sampled signal version: By invoking a prolate sampling theorem, the method rigorously establishes that $\sim 2W_f T/\pi$ equidistant samples $C(t_k)$ suffice for accurate reconstruction, providing operational guidance on quantum resource requirements.

3. Heisenberg-Limited Precision and Error Bounds

QPD attains Heisenberg-limited scaling in its accuracy: the total required runtime to achieve error $\epsilon$ in estimating energies scales as $O(\epsilon^{-1})$. This is underpinned by sharp, analytic error bounds:

• Subspace error: The error in extracted energies is controlled by the smallest eigenvalue $\gamma_M$ of the time–frequency limiting operator (determined by the PSWF basis size $M$). A “phase transition” in achievable accuracy arises as $M$ approaches the effective dimensionality $2W_f T/\pi$, beyond which the error sharply increases due to the loss of time–frequency concentration.
• Discretization error: The error due to sampling $C(t_k)$ instead of having continuous access is rigorously bounded via the sampling theorem, provided $C(t)$ is sufficiently time-concentrated.
• Statistical (shot) noise: Quantum hardware introduces sampling noise into each $C(t_k)$, controlled by the number of circuit repetitions (shots). The impact is quantified using standard statistical inequalities (e.g., Hoeffding’s bound).

The full eigenvalue error is thus expressible as a combination of these subspace, discretization, and statistical errors, and explicit formulae are given for their dependence on $T$, $W_f$, and $M$.

4. Quantum Subsystem Implementation and Resource Efficiency

QPD generates the signal $C(t_k)$ on a quantum processor using short, moderate-depth circuits:

• Hadamard test: For each $t_k$, a Hadamard-ancilla protocol is used to evaluate the real and imaginary parts of $C(t_k)$.
• Short-time evolution: All time evolutions $e^{iHt_k}$ are restricted to times $T_{\max} = \max t_k$ significantly less than those required for conventional quantum phase estimation (QPE), thereby greatly reducing the circuit depth and mitigating hardware error accumulation.
• Parallelization and robustness: Since all $C(t_k)$ are independent, experimental runs can be parallelized, and the method exhibits robust precision even in the presence of imperfect initial state preparation—signal extraction remains accurate as long as the initial state has nontrivial support on the eigenstates of interest.

5. Comparison to Other Quantum Diagonalization Methods

QPD represents a substantial conceptual and technical departure from prior quantum eigenvalue algorithms:

Method	Signal Generation	Classical Postprocessing	Precision Scaling	Circuit Depth Scaling	Error Rigorousness
QPD	Short-time evol.	Prolate filter diagonalization	Heisenberg limit	$O(T_{\max})$ (short)	Full analytic bounds
QPE	Long-time evol.	Fourier phase extraction	Heisenberg limit	$O(\epsilon^{-1})$ (deep)	Stochastic, partial
QFD, Krylov (others)	Short-time evol.	Ritz, Lanczos, subspace solvers	Sub-Heisenberg	$O(T_{\max})$	Partial, empirical

Key features distinguishing QPD:

• Optimal time–frequency filtering leads to minimal basis dimensions and rigorous error thresholds.
• Simultaneous estimation of ground and excited state energies from the same data set.
• No reliance on phase kickbacks, multi-controlled gates, or deep circuits.

6. Applications and Extensions

QPD has been validated in quantum chemistry settings—specifically for estimating spectrum and eigenstates of molecular systems such as benzene and hydrogen chains—demonstrating chemical accuracy (errors below 1 mHartree).

The method provides:

• Efficient extraction of dense spectra from strongly correlated systems.
• High resilience to imperfect initial states and experimental noise.
• Scalability to larger systems, where classical methods or conventional QPE would be infeasible.

By leveraging the prolate spheroidal subspace, QPD automatically emphasizes the frequencies within the physical band of interest, leading to efficient resolvability and resource allocation.

7. Limitations and Prospective Developments

QPD requires:

• Careful selection of the target bandwidth $W_f$ and observation time $T$, with the practical trade-off set by the sharp “phase transition” in error.
• Sufficient quantum hardware access to perform the requisite number of Hadamard tests at short to moderate times.

Potential developments include optimization of the prolate basis for systems with complex spectra, integration with error mitigation techniques, and extensions to time-dependent or open quantum systems. The rigorous analytic framework of QPD draws a clear path for future algorithmic refinements that maintain provable performance guarantees while further reducing quantum circuit complexity.

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QPD thus provides a hybrid quantum-classical procedure for Hamiltonian spectral analysis built on optimal time–frequency subspace theory, making it one of the first approaches to combine Heisenberg-limited quantum precision with mathematically rigorous error certification and practical circuit requirements for modern quantum processors (Stroschein et al., 20 Jul 2025).