Hermitian Quantum Phase Estimation

• Hermitian Quantum Phase Estimation is a quantum algorithm that extracts eigenvalues from Hermitian operators with exponential precision.
• It employs controlled unitary evolutions and an inverse quantum Fourier transform to map phase information onto ancilla qubits.
• Practical implementations include constant-precision, iterative, and windowed filtering techniques to reduce circuit depth and enhance performance on NISQ devices.

Hermitian Quantum Phase Estimation (QPE) is a foundational quantum algorithm for extracting the eigenvalues (typically called "phases" in the context of unitaries, but corresponding directly to energies for Hermitian operators) of a given Hamiltonian or more general Hermitian operator. By leveraging quantum interference and projection, Hermitian QPE enables exponential precision in eigenvalue estimation—a cornerstone for quantum simulation, chemistry, optimization, and metrology.

1. Fundamental Concepts and Operator Framework

Hermitian QPE is applicable when the operator under investigation, $H$, is Hermitian: $H=H^\dagger$. The canonical use case is estimation of eigenvalues $E_i$ for eigenstates $|E_i\rangle$ such that $H|E_i\rangle=E_i|E_i\rangle$. The QPE algorithm proceeds by encoding the spectrum of $H$ into phase factors by simulating time evolution under $H$,

$U(t) = e^{-iHt}$

and leveraging the eigenvector relationship $U(t)|E_i\rangle = e^{-iE_it}|E_i\rangle$. By interfacing this evolution with an ancilla register and exploiting the quantum Fourier transform (QFT), Hermitian QPE efficiently "reads out" $E_i$ as a phase.

The core circuit uses two registers: a measurement register of $n$ ancilla qubits (for $M=2^n$ phase resolution) and a system register holding $|\psi\rangle$, which should have overlap with one or more eigenstates. Controlled-$U(2^j t_0)$ gates imprint the eigenenergy information onto the ancilla register, followed by an inverse QFT to transform the phase information into computational basis states, ready for measurement.

2. Algorithmic Innovations and Practical Enhancements

A range of implementations and refinements have been developed to make Hermitian QPE more practical under hardware and resource constraints:

• Constant-Precision Phase Shift Operators: The requirement for exponentially fine phase shift gates in textbook QPE is a significant hardware challenge. An alternative approach uses only a fixed, constant-precision set of controlled phase shift operators ($R_2$, $R_3$, etc.) at all bit positions. Successive modified Hadamard tests, with statistical boosting via repetition and majority voting, recover the desired bits with high probability—at the cost of a modest increase in circuit depth and sampling, but greatly relaxed physical requirements (1012.4727).

| Approach | Gate precision required | Circuit depth scaling | |-------------------------|-------------------------------------|-----------------------| | Standard QFT-QPE | Exponentially small ($\ll 2^{-n}$) | $O(n^2)$ | | Kitaev's Hadamard Test | None | $O(n \log n)$ | | Constant-precision QPE | Fixed (e.g., $R_2$, $R_3$ only) | $O(n \log n)$ |

• Circuit Depth and Resource Reduction: To mitigate hardware errors in NISQ devices, QPE variants reduce circuit depth by minimizing controlled-rotation gates, replacing some with single-qubit rotations, or removing unnecessary ancilla qubits entirely. This increases accuracy on noisy hardware, particularly for Hermitian Hamiltonians where accurate eigenvalue determination is crucial (Mohammadbagherpoor et al., 2019).
• Iterative and Adaptive Techniques: High precision is achievable using only a small number of ancilla qubits by combining QPE at different propagation (evolution) times. Each round resolves the phase (and thus eigenvalue) within a narrower interval, with overlap ("comb" structure) among intervals at incommensurate times used to resolve degeneracies and enhance precision (Li, 22 Feb 2024). This is particularly advantageous for NISQ-era devices limited in ancilla resources.

3. Mathematical Formulations

Hermitian QPE can be formalized via the following steps:

1. Eigenphase Encoding: For $|\psi\rangle = \sum_j c_j |E_j\rangle$, the action of the QPE circuit is to prepare:

$$\frac{1}{\sqrt{M}} \sum_{y=0}^{M-1} |y\rangle U^y |\psi\rangle.$$

1. Inverse QFT: Applying QFT$^{-1}$ to the ancilla register leads to a measurement histogram with peaks at $y/M \approx E_j t/2\pi \mod 1$, with intensities $|c_j|^2$, and each peak's width defined by $1/M$.
2. Constant-precision Correction: Using only $R_2$, $R_3$ phase shifts, after controlled-$U^{2^k}$, a small phase correction bounded by $|{\theta}|<\frac{1}{8}$ is applied, yielding a state measurable via repeated Hadamard tests and majority voting.
3. Resource-Success Tradeoff: The required number of trials per bit to ensure error $\epsilon$ is $m=4 \ln(1/\epsilon)$, as established via Chernoff bounds (1012.4727).
4. Iterative Propagator Methodology: For an iterative scheme with propagators at time spans $\Delta t$, $\alpha \Delta t$, ..., the additive error in phase shrinks exponentially with iteration:

$\varepsilon = \frac{C}{2^N (2^N - 1)^n}$

with $n$ as the number of rounds and $C$ an integer.

1. Resolvent-Based Generalization: The resolvent approach replaces phase kickback with the preparation of a quantum state modeling $(z \mathbb{I} - H)^{-1}$, producing ancilla amplitudes peaked at eigenvalues near $z$. This unified framework generalizes QPE to non-normal and non-Hermitian operators and to spectra on complex curves (Alase et al., 7 Oct 2024).

4. Performance and Scaling Analysis

Optimal Hermitian QPE achieves the Heisenberg limit ($\propto 1/T$ in phase resolution $T$ being the total controlled unitary rounds) and the standard quantum limit relative to the number of experimental shots $k$ ($\propto 1/\sqrt{k}$). For fixed circuit depth and constant hardware resources:

• Error Resolution and Scaling: In the curve-fitted QPE approach, error resolution behaves heuristically as $\mathcal{O}(1/(\sqrt{k} M))$, where $M = 2^n$ (Lim et al., 24 Sep 2024). For constant-precision phase-shift approaches, sampling cost is increased while maintaining the same scaling in required resources.
• Resource Frugality: Iterative QPE protocols using a single ancilla qubit but variable time evolution, or measurement-driven eigenstate projection, achieve exponential suppression of unwanted eigenstate components relative to the number of rounds, with overall cost analytically bounded (Meister et al., 2022, Li, 22 Feb 2024).
• Windowing and Filtering: QPE-based spectral filtering employs window functions (rectangular, sine, or Kaiser) for spectral isolation. The Kaiser window suppresses the Gibbs phenomenon, yielding error scaling $\mathcal{O}(\delta^{-1}\log(1/\epsilon))$, with $\delta$ the transition width and $\epsilon$ the tolerable error (Sakuma et al., 2 Jul 2025).

5. Experimental Implementations and Applications

Hermitian QPE and its modern variants have broad utility:

• Quantum Chemistry and Spectroscopy: QPE is used to extract ground and excited state energies by simulating $e^{-iHt}$; performance depends critically on initial state overlap and Trotterization precision. Iterative QPE and windowed spectral filtering strategies enable efficient low-energy state targeting and DOS calculations for real materials (e.g., antiferromagnetic MnO), providing accuracy at or below chemical precision (Ino et al., 2023, Sakuma et al., 2 Jul 2025).
• Definiteness Classification: QPE can probabilistically classify Hermitian matrices as positive definite, negative definite, or indefinite, by analyzing the MSB of the measurement register, with accuracy exceeding 97% when sufficient ancillas and samples are used (Gómez et al., 2020).
• GPU Emulation and Algorithmic Benchmarks: Practical QPE emulation using highly optimized GPU platforms (cuQuantum, CUDA) demonstrates that large-scale statevector simulation is feasible for benchmarking, though scaling is ultimately limited by memory and data transfer constraints (Akiba et al., 23 Jul 2025).

Implementation Strategy	Resource Impact	Practical Feature
Constant-precision QPE	Fixed gate set, more shots	NISQ-friendly, scalable
Iterative QPE with Propag.	Few ancillas, more rounds	Precision on shallow circuits
Windowed Filtering (Kaiser)	Slightly more ancillas	Sharp spectral filtering
Resolvent-based QPE	QLSA required	Extends to non-Hermitian
GPU-based QPE emulation	High VRAM, speedup	Fast classical simulation

6. Extensions and Limitations

Several directions extend the Hermitian QPE paradigm:

• Curve Fitting and Hybrid Classical-Quantum Estimation: Rather than the winner-take-all measurement strategy, classical curve-fitting to the full QPE histogram can extract phases approaching the Cramér-Rao lower bound, with error resolution comparable to state-of-the-art quantum estimation methods without additional quantum resources (Lim et al., 24 Sep 2024).
• Simultaneous Multiphase Estimation: Bayesian techniques generalize Hermitian QPE to estimate multiple eigenphases in parallel, efficiently exploiting resource correlations and achieving Heisenberg scaling, with robust operation even in noisy environments (Gebhart et al., 2020).
• Resilience to Decoherence and Noise: Noise models (bit flip, phase flip, depolarizing) degrade phase accuracy exponentially with error probability per qubit. Algorithmic strategies minimizing circuit depth and controlled operations, or using robust input states (e.g., Gaussian spin states), are critical for practical Hermitian QPE on NISQ-era devices (Pezzè et al., 2020, Faizan et al., 2023).

7. Impact, Outlook, and Open Questions

Hermitian QPE, supported by a growing set of algorithmic and practical refinements, is crucial for quantum simulation and measurement tasks across quantum chemistry, condensed matter, optimization, and more. Algorithmic advances—constant-precision approaches, iterative time-propagator methods, sophisticated windowing and filtering, and hybrid post-processing—have rendered QPE both more physically implementable and robust against real-world hardware limitations.

Open research questions include:

• The design of optimal window functions for QPE filtering with tight error bounds and minimal resource overhead (Sakuma et al., 2 Jul 2025).
• Trade-offs between circuit depth, classical post-processing, and robustness in hybrid NISQ-compatible QPE approaches (Lim et al., 24 Sep 2024).
• Extension of resolvent-based QPE to parametrized complex spectra and non-normal matrices, allowing spectral estimation in broad classes of quantum and classical problems (Alase et al., 7 Oct 2024).
• Integration of variational, iterative, and quantum imaginary time evolution protocols for state preparation and spectral projection (Bauer et al., 15 Apr 2025).

Hermitian QPE remains central to demonstrating quantum advantage in practical applications, benefiting from ongoing progress in algorithmic design, error mitigation, hardware acceleration, and integrated quantum-classical workflows.